1.3: Data, Sampling, and Variation in Data and Sampling
-
- Last updated
- Save as PDF
Data may come from a population or from a sample. Small letters like \(x\) or \(y\) generally are used to represent data values. Most data can be put into the following categories:
- Qualitative
- Quantitative
Qualitative data are the result of categorizing or describing attributes of a population. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O-, or B+. Researchers often prefer to use quantitative data over qualitative data because it lends itself more easily to mathematical analysis. For example, it does not make sense to find an average hair color or blood type.
Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous .
All data that are the result of counting are called quantitative discrete data . These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get values such as zero, one, two, or three.
All data that are the result of measuring are quantitative continuous data assuming that we can measure accurately. Measuring angles in radians might result in such numbers as \(\frac{\pi}{6}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{4}\), and so on. If you and your friends carry backpacks with books in them to school, the numbers of books in the backpacks are discrete data and the weights of the backpacks are continuous data.
Sample of Quantitative Discrete Data
The data are the number of books students carry in their backpacks. You sample five students. Two students carry three books, one student carries four books, one student carries two books, and one student carries one book. The numbers of books (three, four, two, and one) are the quantitative discrete data .
Exercise \(\PageIndex{1}\)
The data are the number of machines in a gym. You sample five gyms. One gym has 12 machines, one gym has 15 machines, one gym has ten machines, one gym has 22 machines, and the other gym has 20 machines. What type of data is this?
- Answer
-
quantitative discrete data
Sample of Quantitative Continuous Data
The data are the weights of backpacks with books in them. You sample the same five students. The weights (in pounds) of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data because weights are measured.
Exercise \(\PageIndex{2}\)
The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are 144 sq. feet, 160 sq. feet, 190 sq. feet, 180 sq. feet, and 210 sq. feet. What type of data is this?
- Answer
-
quantitative continuous data
Exercise \(\PageIndex{3}\)
You go to the supermarket and purchase three cans of soup (19 ounces) tomato bisque, 14.1 ounces lentil, and 19 ounces Italian wedding), two packages of nuts (walnuts and peanuts), four different kinds of vegetable (broccoli, cauliflower, spinach, and carrots), and two desserts (16 ounces Cherry Garcia ice cream and two pounds (32 ounces chocolate chip cookies).
Name data sets that are quantitative discrete, quantitative continuous, and qualitative.
Solution
One Possible Solution:
- The three cans of soup, two packages of nuts, four kinds of vegetables and two desserts are quantitative discrete data because you count them.
- The weights of the soups (19 ounces, 14.1 ounces, 19 ounces) are quantitative continuous data because you measure weights as precisely as possible.
- Types of soups, nuts, vegetables and desserts are qualitative data because they are categorical.
Try to identify additional data sets in this example.
Sample of qualitative data
The data are the colors of backpacks. Again, you sample the same five students. One student has a red backpack, two students have black backpacks, one student has a green backpack, and one student has a gray backpack. The colors red, black, black, green, and gray are qualitative data .
Exercise \(\PageIndex{4}\)
The data are the colors of houses. You sample five houses. The colors of the houses are white, yellow, white, red, and white. What type of data is this?
- Answer
-
qualitative data
Collaborative Exercise \(\PageIndex{1}\)
Work collaboratively to determine the correct data type (quantitative or qualitative). Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words "the number of."
- the number of pairs of shoes you own
- the type of car you drive
- where you go on vacation
- the distance it is from your home to the nearest grocery store
- the number of classes you take per school year.
- the tuition for your classes
- the type of calculator you use
- movie ratings
- political party preferences
- weights of sumo wrestlers
- amount of money (in dollars) won playing poker
- number of correct answers on a quiz
- peoples’ attitudes toward the government
- IQ scores (This may cause some discussion.)
Items a, e, f, k, and l are quantitative discrete; items d, j, and n are quantitative continuous; items b, c, g, h, i, and m are qualitative.
Exercise \(\PageIndex{5}\)
Determine the correct data type (quantitative or qualitative) for the number of cars in a parking lot. Indicate whether quantitative data are continuous or discrete.
- Answer
-
quantitative discrete
Exercise \(\PageIndex{6}\)
A statistics professor collects information about the classification of her students as freshmen, sophomores, juniors, or seniors. The data she collects are summarized in the pie chart Figure \(\PageIndex{1}\). What type of data does this graph show?
- Answer
-
This pie chart shows the students in each year, which is qualitative data .
Exercise \(\PageIndex{7}\)
The registrar at State University keeps records of the number of credit hours students complete each semester. The data he collects are summarized in the histogram. The class boundaries are 10 to less than 13, 13 to less than 16, 16 to less than 19, 19 to less than 22, and 22 to less than 25.
What type of data does this graph show?
- Answer
-
A histogram is used to display quantitative data: the numbers of credit hours completed. Because students can complete only a whole number of hours (no fractions of hours allowed), this data is quantitative discrete.
Qualitative Data Discussion
Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts (frequencies) and percentages or proportions (relative frequencies). The percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.
| De Anza College | Foothill College | |||||
|---|---|---|---|---|---|---|
| Number | Percent | Number | Percent | |||
| Full-time | 9,200 | 40.9% | Full-time | 4,059 | 28.6% | |
| Part-time | 13,296 | 59.1% | Part-time | 10,124 | 71.4% | |
| Total | 22,496 | 100% | Total | 14,183 | 100% |
Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. There are no strict rules concerning which graphs to use. Two graphs that are used to display qualitative data are pie charts and bar graphs.
- In a pie chart , categories of data are represented by wedges in a circle and are proportional in size to the percent of individuals in each category.
- In a bar graph , the length of the bar for each category is proportional to the number or percent of individuals in each category. Bars may be vertical or horizontal.
- A Pareto chart consists of bars that are sorted into order by category size (largest to smallest).
Look at Figures \(\PageIndex{3}\) and \(\PageIndex{4}\) and determine which graph (pie or bar) you think displays the comparisons better.
It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the “best” graph depending on the data and the context. Our choice also depends on what we are using the data for.
Percentages That Add to More (or Less) Than 100%
Sometimes percentages add up to be more than 100% (or less than 100%). In the graph, the percentages add to more than 100% because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than 100%.
| Characteristic/Category | Percent |
|---|---|
| Full-Time Students | 40.9% |
| Students who intend to transfer to a 4-year educational institution | 48.6% |
| Students under age 25 | 61.0% |
| TOTAL | 150.5% |
Omitting Categories/Missing Data
The table displays Ethnicity of Students but is missing the "Other/Unknown" category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart.
| Frequency | Percent | |
|---|---|---|
| Asian | 8,794 | 36.1% |
| Black | 1,412 | 5.8% |
| Filipino | 1,298 | 5.3% |
| Hispanic | 4,180 | 17.1% |
| Native American | 146 | 0.6% |
| Pacific Islander | 236 | 1.0% |
| White | 5,978 | 24.5% |
| TOTAL | 22,044 out of 24,382 | 90.4% out of 100% |
The following graph is the same as the previous graph but the “Other/Unknown” percent (9.6%) has been included. The “Other/Unknown” category is large compared to some of the other categories (Native American, 0.6%, Pacific Islander 1.0%). This is important to know when we think about what the data are telling us.
This particular bar graph in Figure \(\PageIndex{4}\) can be difficult to understand visually. The graph in Figure \(\PageIndex{5}\) is a Pareto chart . The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.
Pie Charts: No Missing Data
The following pie charts have the “Other/Unknown” category included (since the percentages must add to 100%). The chart in Figure \(\PageIndex{6}\) is organized by the size of each wedge, which makes it a more visually informative graph than the unsorted, alphabetical graph in Figure \(\PageIndex{6}\).
Sampling
Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. A sample should have the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling . In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample . Any group of n individuals is equally likely to be chosen by any other group of n individuals if the simple random sampling technique is used. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has 31 members not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all 31 names in a hat, shake the hat, close her eyes, and pick out three names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number, as in Table \(\PageIndex{2}\):
| ID | Name | ID | Name | ID | Name |
|---|---|---|---|---|---|
| 00 | Anselmo | 11 | King | 21 | Roquero |
| 01 | Bautista | 12 | Legeny | 22 | Roth |
| 02 | Bayani | 13 | Lundquist | 23 | Rowell |
| 03 | Cheng | 14 | Macierz | 24 | Salangsang |
| 04 | Cuarismo | 15 | Motogawa | 25 | Slade |
| 05 | Cuningham | 16 | Okimoto | 26 | Stratcher |
| 06 | Fontecha | 17 | Patel | 27 | Tallai |
| 07 | Hong | 18 | Price | 28 | Tran |
| 08 | Hoobler | 19 | Quizon | 29 | Wai |
| 09 | Jiao | 20 | Reyes | 30 | Wood |
| 10 | Khan |
Lisa can use a table of random numbers (found in many statistics books and mathematical handbooks), a calculator, or a computer to generate random numbers. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are as follows:
0.94360; 0.99832; 0.14669; 0.51470; 0.40581; 0.73381; 0.04399
Lisa reads two-digit groups until she has chosen three class members (that is, she reads 0.94360 as the groups 94, 43, 36, 60). Each random number may only contribute one class member. If she needed to, Lisa could have generated more random numbers.
The random numbers 0.94360 and 0.99832 do not contain appropriate two digit numbers. However the third random number, 0.14669, contains 14 (the fourth random number also contains 14), the fifth random number contains 05, and the seventh random number contains 04. The two-digit number 14 corresponds to Macierz, 05 corresponds to Cuningham, and 04 corresponds to Cuarismo. Besides herself, Lisa’s group will consist of Marcierz, Cuningham, and Cuarismo.
To generate random numbers:
- Press MATH.
- Arrow over to PRB.
- Press 5:randInt(. Enter 0, 30).
- Press ENTER for the first random number.
- Press ENTER two more times for the other 2 random numbers. If there is a repeat press ENTER again.
Note: randInt(0, 30, 3) will generate 3 random numbers.
Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample.
To choose a stratified sample , divide the population into groups called strata and then take a proportionate number from each stratum. For example, you could stratify (group) your college population by department and then choose a proportionate simple random sample from each stratum (each department) to get a stratified random sample. To choose a simple random sample from each department, number each member of the first department, number each member of the second department, and do the same for the remaining departments. Then use simple random sampling to choose proportionate numbers from the first department and do the same for each of the remaining departments. Those numbers picked from the first department, picked from the second department, and so on represent the members who make up the stratified sample.
To choose a cluster sample , divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four departments from your college population, the four departments make up the cluster sample. Divide your college faculty by department. The departments are the clusters. Number each department, and then choose four different numbers using simple random sampling. All members of the four departments with those numbers are the cluster sample.
To choose a systematic sample , randomly select a starting point and take every n th piece of data from a listing of the population. For example, suppose you have to do a phone survey. Your phone book contains 20,000 residence listings. You must choose 400 names for the sample. Number the population 1–20,000 and then use a simple random sample to pick a number that represents the first name in the sample. Then choose every fiftieth name thereafter until you have a total of 400 names (you might have to go back to the beginning of your phone list). Systematic sampling is frequently chosen because it is a simple method.
A type of sampling that is non-random is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favor certain outcomes) in others.
Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased (they may favor a certain group). It is better for the person conducting the survey to select the sample respondents.
True random sampling is done with replacement . That is, once a member is picked, that member goes back into the population and thus may be chosen more than once. However for practical reasons, in most populations, simple random sampling is done without replacement . Surveys are typically done without replacement. That is, a member of the population may be chosen only once. Most samples are taken from large populations and the sample tends to be small in comparison to the population. Since this is the case, sampling without replacement is approximately the same as sampling with replacement because the chance of picking the same individual more than once with replacement is very low.
In a college population of 10,000 people, suppose you want to pick a sample of 1,000 randomly for a survey. For any particular sample of 1,000 , if you are sampling with replacement ,
- the chance of picking the first person is 1,000 out of 10,000 (0.1000);
- the chance of picking a different second person for this sample is 999 out of 10,000 (0.0999);
- the chance of picking the same person again is 1 out of 10,000 (very low).
If you are sampling without replacement ,
- the chance of picking the first person for any particular sample is 1000 out of 10,000 (0.1000);
- the chance of picking a different second person is 999 out of 9,999 (0.0999);
- you do not replace the first person before picking the next person.
Compare the fractions 999/10,000 and 999/9,999. For accuracy, carry the decimal answers to four decimal places. To four decimal places, these numbers are equivalent (0.0999).
Sampling without replacement instead of sampling with replacement becomes a mathematical issue only when the population is small. For example, if the population is 25 people, the sample is ten, and you are sampling with replacement for any particular sample , then the chance of picking the first person is ten out of 25, and the chance of picking a different second person is nine out of 25 (you replace the first person).
If you sample without replacement , then the chance of picking the first person is ten out of 25, and then the chance of picking the second person (who is different) is nine out of 24 (you do not replace the first person).
Compare the fractions 9/25 and 9/24. To four decimal places, 9/25 = 0.3600 and 9/24 = 0.3750. To four decimal places, these numbers are not equivalent.
When you analyze data, it is important to be aware of sampling errors and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough. Factors not related to the sampling process cause nonsampling errors . A defective counting device can cause a nonsampling error.
In reality, a sample will never be exactly representative of the population so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error.
In statistics, a sampling bias is created when a sample is collected from a population and some members of the population are not as likely to be chosen as others (remember, each member of the population should have an equally likely chance of being chosen). When a sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied.
Exercise \(\PageIndex{8}\)
A study is done to determine the average tuition that San Jose State undergraduate students pay per semester. Each student in the following samples is asked how much tuition he or she paid for the Fall semester. What is the type of sampling in each case?
- A sample of 100 undergraduate San Jose State students is taken by organizing the students’ names by classification (freshman, sophomore, junior, or senior), and then selecting 25 students from each.
- A random number generator is used to select a student from the alphabetical listing of all undergraduate students in the Fall semester. Starting with that student, every 50th student is chosen until 75 students are included in the sample.
- A completely random method is used to select 75 students. Each undergraduate student in the fall semester has the same probability of being chosen at any stage of the sampling process.
- The freshman, sophomore, junior, and senior years are numbered one, two, three, and four, respectively. A random number generator is used to pick two of those years. All students in those two years are in the sample.
- An administrative assistant is asked to stand in front of the library one Wednesday and to ask the first 100 undergraduate students he encounters what they paid for tuition the Fall semester. Those 100 students are the sample.
Answer
a. stratified; b. systematic; c. simple random; d. cluster; e. convenience
Example \(\PageIndex{9}\): Calculator
You are going to use the random number generator to generate different types of samples from the data. This table displays six sets of quiz scores (each quiz counts 10 points) for an elementary statistics class.
| #1 | #2 | #3 | #4 | #5 | #6 |
|---|---|---|---|---|---|
| 5 | 7 | 10 | 9 | 8 | 3 |
| 10 | 5 | 9 | 8 | 7 | 6 |
| 9 | 10 | 8 | 6 | 7 | 9 |
| 9 | 10 | 10 | 9 | 8 | 9 |
| 7 | 8 | 9 | 5 | 7 | 4 |
| 9 | 9 | 9 | 10 | 8 | 7 |
| 7 | 7 | 10 | 9 | 8 | 8 |
| 8 | 8 | 9 | 10 | 8 | 8 |
| 9 | 7 | 8 | 7 | 7 | 8 |
| 8 | 8 | 10 | 9 | 8 | 7 |
Instructions: Use the Random Number Generator to pick samples.
-
Create a stratified sample by column. Pick three quiz scores randomly from each column.
- Number each row one through ten.
- On your calculator, press Math and arrow over to PRB.
- For column 1, Press 5:randInt( and enter 1,10). Press ENTER. Record the number. Press ENTER 2 more times (even the repeats). Record these numbers. Record the three quiz scores in column one that correspond to these three numbers.
- Repeat for columns two through six.
- These 18 quiz scores are a stratified sample.
-
Create a cluster sample by picking two of the columns. Use the column numbers: one through six.
- Press MATH and arrow over to PRB.
- Press 5:randInt( and enter 1,6). Press ENTER. Record the number. Press ENTER and record that number.
- The two numbers are for two of the columns.
- The quiz scores (20 of them) in these 2 columns are the cluster sample.
-
Create a simple random sample of 15 quiz scores.
- Use the numbering one through 60.
- Press MATH. Arrow over to PRB. Press 5:randInt( and enter 1, 60).
- Press ENTER 15 times and record the numbers.
- Record the quiz scores that correspond to these numbers.
- These 15 quiz scores are the systematic sample.
-
Create a systematic sample of 12 quiz scores.
- Use the numbering one through 60.
- Press MATH. Arrow over to PRB. Press 5:randInt( and enter 1, 60).
- Press ENTER. Record the number and the first quiz score. From that number, count ten quiz scores and record that quiz score. Keep counting ten quiz scores and recording the quiz score until you have a sample of 12 quiz scores. You may wrap around (go back to the beginning).
Example \(\PageIndex{10}\)
Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).
- A soccer coach selects six players from a group of boys aged eight to ten, seven players from a group of boys aged 11 to 12, and three players from a group of boys aged 13 to 14 to form a recreational soccer team.
- A pollster interviews all human resource personnel in five different high tech companies.
- A high school educational researcher interviews 50 high school female teachers and 50 high school male teachers.
- A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital.
- A high school counselor uses a computer to generate 50 random numbers and then picks students whose names correspond to the numbers.
- A student interviews classmates in his algebra class to determine how many pairs of jeans a student owns, on the average.
Answer
a. stratified; b. cluster; c. stratified; d. systematic; e. simple random; f.convenience
Exercise \(\PageIndex{11}\)
Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).
A high school principal polls 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors regarding policy changes for after school activities.
Answer
stratified
If we were to examine two samples representing the same population, even if we used random sampling methods for the samples, they would not be exactly the same. Just as there is variation in data, there is variation in samples. As you become accustomed to sampling, the variability will begin to seem natural.
Example \(\PageIndex{12}\): Sampling
Suppose ABC College has 10,000 part-time students (the population). We are interested in the average amount of money a part-time student spends on books in the fall term. Asking all 10,000 students is an almost impossible task. Suppose we take two different samples.
First, we use convenience sampling and survey ten students from a first term organic chemistry class. Many of these students are taking first term calculus in addition to the organic chemistry class. The amount of money they spend on books is as follows:
$128; $87; $173; $116; $130; $204; $147; $189; $93; $153
The second sample is taken using a list of senior citizens who take P.E. classes and taking every fifth senior citizen on the list, for a total of ten senior citizens. They spend:
$50; $40; $36; $15; $50; $100; $40; $53; $22; $22
a. Do you think that either of these samples is representative of (or is characteristic of) the entire 10,000 part-time student population?
Answer
a. No. The first sample probably consists of science-oriented students. Besides the chemistry course, some of them are also taking first-term calculus. Books for these classes tend to be expensive. Most of these students are, more than likely, paying more than the average part-time student for their books. The second sample is a group of senior citizens who are, more than likely, taking courses for health and interest. The amount of money they spend on books is probably much less than the average parttime student. Both samples are biased. Also, in both cases, not all students have a chance to be in either sample.
b. Since these samples are not representative of the entire population, is it wise to use the results to describe the entire population?
Answer
b. No. For these samples, each member of the population did not have an equally likely chance of being chosen.
Now, suppose we take a third sample. We choose ten different part-time students from the disciplines of chemistry, math, English, psychology, sociology, history, nursing, physical education, art, and early childhood development. (We assume that these are the only disciplines in which part-time students at ABC College are enrolled and that an equal number of part-time students are enrolled in each of the disciplines.) Each student is chosen using simple random sampling. Using a calculator, random numbers are generated and a student from a particular discipline is selected if he or she has a corresponding number. The students spend the following amounts:
$180; $50; $150; $85; $260; $75; $180; $200; $200; $150
c. Is the sample biased?
Answer
Students often ask if it is "good enough" to take a sample, instead of surveying the entire population. If the survey is done well, the answer is yes.
Exercise \(\PageIndex{12}\)
A local radio station has a fan base of 20,000 listeners. The station wants to know if its audience would prefer more music or more talk shows. Asking all 20,000 listeners is an almost impossible task.
The station uses convenience sampling and surveys the first 200 people they meet at one of the station’s music concert events. 24 people said they’d prefer more talk shows, and 176 people said they’d prefer more music.
Do you think that this sample is representative of (or is characteristic of) the entire 20,000 listener population?
Answer
The sample probably consists more of people who prefer music because it is a concert event. Also, the sample represents only those who showed up to the event earlier than the majority. The sample probably doesn’t represent the entire fan base and is probably biased towards people who would prefer music.
Collaborative Exercise \(\PageIndex{8}\)
As a class, determine whether or not the following samples are representative. If they are not, discuss the reasons.
- To find the average GPA of all students in a university, use all honor students at the university as the sample.
- To find out the most popular cereal among young people under the age of ten, stand outside a large supermarket for three hours and speak to every twentieth child under age ten who enters the supermarket.
- To find the average annual income of all adults in the United States, sample U.S. congressmen. Create a cluster sample by considering each state as a stratum (group). By using simple random sampling, select states to be part of the cluster. Then survey every U.S. congressman in the cluster.
- To determine the proportion of people taking public transportation to work, survey 20 people in New York City. Conduct the survey by sitting in Central Park on a bench and interviewing every person who sits next to you.
- To determine the average cost of a two-day stay in a hospital in Massachusetts, survey 100 hospitals across the state using simple random sampling.
Variation in Data
Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage:
15.8; 16.1; 15.2; 14.8; 15.8; 15.9; 16.0; 15.5
Measurements of the amount of beverage in a 16-ounce can may vary because different people make the measurements or because the exact amount, 16 ounces of liquid, was not put into the cans. Manufacturers regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range.
Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the same purpose. This is completely natural. However, if two or more of you are taking the same data and get very different results, it is time for you and the others to reevaluate your data-taking methods and your accuracy.
Variation in Samples
It was mentioned previously that two or more samples from the same population , taken randomly, and having close to the same characteristics of the population will likely be different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their college sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen's sample will be different from Jung's sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be different. Neither would be wrong, however.
Think about what contributes to making Doreen’s and Jung’s samples different.
If Doreen and Jung took larger samples (i.e. the number of data values is increased), their sample results (the average amount of time a student sleeps) might be closer to the actual population average. But still, their samples would be, in all likelihood, different from each other. This variability in samples cannot be stressed enough.
Size of a Sample
The size of a sample (often called the number of observations) is important. The examples you have seen in this book so far have been small. Samples of only a few hundred observations, or even smaller, are sufficient for many purposes. In polling, samples that are from 1,200 to 1,500 observations are considered large enough and good enough if the survey is random and is well done. You will learn why when you study confidence intervals.
Be aware that many large samples are biased. For example, call-in surveys are invariably biased, because people choose to respond or not.
Collaborative Exercise \(\PageIndex{8}\)
Divide into groups of two, three, or four. Your instructor will give each group one six-sided die. Try this experiment twice. Roll one fair die (six-sided) 20 times. Record the number of ones, twos, threes, fours, fives, and sixes you get in the following table (“frequency” is the number of times a particular face of the die occurs):
| First Experiment (20 rolls) | Second Experiment (20 rolls) | |||
|---|---|---|---|---|
| Face on Die | Frequency | Face on Die | Frequency | |
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 | ||||
| 6 |
Did the two experiments have the same results? Probably not. If you did the experiment a third time, do you expect the results to be identical to the first or second experiment? Why or why not?
Which experiment had the correct results? They both did. The job of the statistician is to see through the variability and draw appropriate conclusions.
Critical Evaluation
We need to evaluate the statistical studies we read about critically and analyze them before accepting the results of the studies. Common problems to be aware of include
- Problems with samples: A sample must be representative of the population. A sample that is not representative of the population is biased. Biased samples that are not representative of the population give results that are inaccurate and not valid.
- Self-selected samples: Responses only by people who choose to respond, such as call-in surveys, are often unreliable.
- Sample size issues: Samples that are too small may be unreliable. Larger samples are better, if possible. In some situations, having small samples is unavoidable and can still be used to draw conclusions. Examples: crash testing cars or medical testing for rare conditions
- Undue influence: collecting data or asking questions in a way that influences the response
- Non-response or refusal of subject to participate: The collected responses may no longer be representative of the population. Often, people with strong positive or negative opinions may answer surveys, which can affect the results.
- Causality: A relationship between two variables does not mean that one causes the other to occur. They may be related (correlated) because of their relationship through a different variable.
- Self-funded or self-interest studies: A study performed by a person or organization in order to support their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not automatically assume that the study is good, but do not automatically assume the study is bad either. Evaluate it on its merits and the work done.
- Misleading use of data: improperly displayed graphs, incomplete data, or lack of context
- Confounding: When the effects of multiple factors on a response cannot be separated. Confounding makes it difficult or impossible to draw valid conclusions about the effect of each factor.
References
- Gallup-Healthways Well-Being Index. http://www.well-beingindex.com/default.asp (accessed May 1, 2013).
- Gallup-Healthways Well-Being Index. http://www.well-beingindex.com/methodology.asp (accessed May 1, 2013).
- Gallup-Healthways Well-Being Index. http://www.gallup.com/poll/146822/ga...questions.aspx (accessed May 1, 2013).
- Data from www.bookofodds.com/Relationsh...-the-President
- Dominic Lusinchi, “’President’ Landon and the 1936 Literary Digest Poll: Were Automobile and Telephone Owners to Blame?” Social Science History 36, no. 1: 23-54 (2012), ssh.dukejournals.org/content/36/1/23.abstract (accessed May 1, 2013).
- “The Literary Digest Poll,” Virtual Laboratories in Probability and Statistics http://www.math.uah.edu/stat/data/LiteraryDigest.html (accessed May 1, 2013).
- “Gallup Presidential Election Trial-Heat Trends, 1936–2008,” Gallup Politics http://www.gallup.com/poll/110548/ga...9362004.aspx#4 (accessed May 1, 2013).
- The Data and Story Library, lib.stat.cmu.edu/DASL/Datafiles/USCrime.html (accessed May 1, 2013).
- LBCC Distance Learning (DL) program data in 2010-2011, http://de.lbcc.edu/reports/2010-11/f...hts.html#focus (accessed May 1, 2013).
- Data from San Jose Mercury News
Review
Data are individual items of information that come from a population or sample. Data may be classified as qualitative, quantitative continuous, or quantitative discrete.
Because it is not practical to measure the entire population in a study, researchers use samples to represent the population. A random sample is a representative group from the population chosen by using a method that gives each individual in the population an equal chance of being included in the sample. Random sampling methods include simple random sampling, stratified sampling, cluster sampling, and systematic sampling. Convenience sampling is a nonrandom method of choosing a sample that often produces biased data.
Samples that contain different individuals result in different data. This is true even when the samples are well-chosen and representative of the population. When properly selected, larger samples model the population more closely than smaller samples. There are many different potential problems that can affect the reliability of a sample. Statistical data needs to be critically analyzed, not simply accepted.
Footnotes
- lastbaldeagle. 2013. On Tax Day, House to Call for Firing Federal Workers Who Owe Back Taxes. Opinion poll posted online at: www.youpolls.com/details.aspx?id=12328 (accessed May 1, 2013).
- Scott Keeter et al., “Gauging the Impact of Growing Nonresponse on Estimates from a National RDD Telephone Survey,” Public Opinion Quarterly 70 no. 5 (2006), http://poq.oxfordjournals.org/content/70/5/759.full (accessed May 1, 2013).
- Frequently Asked Questions, Pew Research Center for the People & the Press, www.people-press.org/methodol...wer-your-polls (accessed May 1, 2013).
Glossary
- Cluster Sampling
- a method for selecting a random sample and dividing the population into groups (clusters); use simple random sampling to select a set of clusters. Every individual in the chosen clusters is included in the sample.
- Continuous Random Variable
- a random variable (RV) whose outcomes are measured; the height of trees in the forest is a continuous RV.
- Convenience Sampling
- a nonrandom method of selecting a sample; this method selects individuals that are easily accessible and may result in biased data.
- Discrete Random Variable
- a random variable (RV) whose outcomes are counted
- Nonsampling Error
- an issue that affects the reliability of sampling data other than natural variation; it includes a variety of human errors including poor study design, biased sampling methods, inaccurate information provided by study participants, data entry errors, and poor analysis.
- Qualitative Data
- See Data .
- Quantitative Data
- See Data .
- Random Sampling
- a method of selecting a sample that gives every member of the population an equal chance of being selected.
- Sampling Bias
- not all members of the population are equally likely to be selected
- Sampling Error
- the natural variation that results from selecting a sample to represent a larger population; this variation decreases as the sample size increases, so selecting larger samples reduces sampling error.
- Sampling with Replacement
- Once a member of the population is selected for inclusion in a sample, that member is returned to the population for the selection of the next individual.
- Sampling without Replacement
- A member of the population may be chosen for inclusion in a sample only once. If chosen, the member is not returned to the population before the next selection.
- Simple Random Sampling
- a straightforward method for selecting a random sample; give each member of the population a number. Use a random number generator to select a set of labels. These randomly selected labels identify the members of your sample.
- Stratified Sampling
- a method for selecting a random sample used to ensure that subgroups of the population are represented adequately; divide the population into groups (strata). Use simple random sampling to identify a proportionate number of individuals from each stratum.
- Systematic Sampling
- a method for selecting a random sample; list the members of the population. Use simple random sampling to select a starting point in the population. Let k = (number of individuals in the population)/(number of individuals needed in the sample). Choose every kth individual in the list starting with the one that was randomly selected. If necessary, return to the beginning of the population list to complete your sample.


15.14C: Pyelonephritis
- Identify the main symptoms of pyelonephritis
Pyelonephritis is an inflammation of the kidney tissue, calyces, and renal pelvis. It is commonly caused by bacterial infection that has spread up the urinary tract or travelled through the bloodstream to the kidneys. A similar term is “pyelitis” which means inflammation of the pelvis and calyces. In other words, pyelitis together with nephritis is collectively known as pyelonephritis. Severe cases of pyelonephritis can lead to pyonephrosis (pus accumulation around the kidney), sepsis (a systemic inflammatory response of the body to infection), kidney failure and even death.
Pyelonephritis presents with fever, accelerated heart rate, painful urination, abdominal pain radiating to the back, nausea, and tenderness at the costovertebral angle on the affected side. Pyelonephritis that has progressed to urosepsis may be accompanied by signs of septic shock, including rapid breathing, decreased blood pressure, violent shivering, and occasionally delirium. Pyelonephritis requires antibiotic therapy, and sometimes surgical intervention, as well as treatment of any underlying causes to prevent its recurrence.
Key Points
- Pyelonephritis is an inflammation of the kidney tissue and renal area of the pelvis.
- Pyelonephritis is most commonly caused by a bacterial infection ascending up the upper urinary tract.
- Severe cases of pyelonephritis can lead to sepsis, kidney failure, and death.
- Pyelonephritis presents with fever, accelerated heart rate, painful urination, abdominal pain, and nausea.
- Pyelonephritis requires antibiotic therapy, and sometimes surgical intervention, as well as treatment of any underlying causes to prevent recurrence.
Key Terms
- pyelonephritis : An ascending urinary tract infection that has reached the pelvis or the kidney.
- Urinary tract : The organ system that produces, stores, and eliminates urine. In humans it includes two kidneys, two ureters, the bladder, and the urethra.


2.3: Isaac and Family
In this chapter you will
- explore how composers and songwriters have wrestled with one of the most difficult and unsettling stories in the Bible, that in which Abraham attempts to sacrifice his son
- consider how music has given voice to those whose perspectives we are not given the chance to hear as fully as we might wish in the story in Genesis, including Sarah (the mother of Isaac and wife of Abraham), Hagar, and one other character who is not actually mentioned in the story at all
The Binding of Isaac
There are several disturbing, troubling, and distressing parts of the Bible. Many of them have rarely been explored through music. Some have been explored surprisingly often, at times with interesting adaptations to cope with the most difficult parts of the text in question (see, for instance, the discussion of Psalm 137 in chapter 8 on the Psalms). Few passages, however, connect with readers at as quite a visceral level as the story of the binding of Isaac (often referred to in Jewish tradition simply as the Akedah, the Hebrew word for “binding”). Child sacrifice is difficult for most modern English speakers to fathom. Life in a world in which children who survived into adulthood were the exception rather than the rule is unimaginable to us. To people in that kind of context, offering one’s firstborn child in the hope that capricious divine forces would show favor to the rest of one’s offspring might have made a tragic sort of sense. Many Bible readers also take comfort in the fact that the Jewish Scriptures reflect a trajectory away from the practice of child sacrifice. Its influence in that regard is one that can presumably be appreciated and viewed positively by just about anyone today.
Yet the logic of the story of Abraham’s near sacrifice of Isaac in Genesis 22 presupposes that a divine voice might ask one to sacrifice one’s child and that the appropriate response would be to do so. Musical treatments of the story, however few they might be, provide real insights into the challenges for modern readers (as well as for many others all throughout history) of knowing what to do with the story and how to best understand it. Composer Samuel Adler inserts the figure of Satan into the story (as do some midrashic retellings of the story) and imagines the conversations that might have transpired not only between father and son but also between them and this additional malevolent character. Ofer Ben-Amots , on the other hand, sets aside words entirely in his musical treatment of the story, which echoes music used in Jewish liturgy for mourning and is in some traditions specifically associated with the Shoah (also known as the Holocaust—i.e., the murder of some six million Jews by the Nazis during the Second World War). Here is a performance of Ben-Amots’s “Akëda” by Laura Farré Rozada, shared on her YouTube channel.
Roxanna Panufnik has also offered an instrumental interpretation of the story. Aharon Harlap’s treatment , on the other hand, sets the text for choir without instruments.
More than one interpreter has wondered whether the solution might not be to place the blame for things on another figure, whether the devil or Abraham’s own mind. The latter may be assumed in the lyrics of Joan Baez’s song “Isaac and Abraham.” Interestingly, Baez also attributes the sparing of Isaac to Abraham as well. There may be voices in his mind, but we are never told that any come from a source outside. [1]
Unless one reads the lyrics, it would be easy to miss that Bob Dylan’s song “Highway 61 Revisited” is about this story. The well-known modern songwriter and singer Leonard Cohen has also offered his treatment of it. [2]
English composer Benjamin Britten set two very different versions of the story to music. His “Canticle II” (with text taken from the Chester Miracle Plays ) ends with Abraham sparing Isaac. This performance features countertenor Leandro Marziotte and tenor Raphael Höhn accompanied by Carolien Drewes on piano. [3]
When he revisits the story in the context of his War Requiem , however, he sets Wilfred Owen’s poem “The Parable of the Old Man and the Young,” which ends the story with the following lines:
But the old man would not so, but slew his son,
And half the seed of Europe, one by one.
Closely connected to this is a recent composition by Delvyn Case that sets the Elohist version of the story to music—that is, one of the sources that scholars have detected behind the version we now have in Genesis in its present form. Have a listen to the world premiere performance, which has been shared by the composer on his YouTube channel, and read more about the work and the sources of inspiration behind it on the composer’s website .
Others have explored the story in music, and the music (or the cessation thereof) is an important part of the interpretation offered. Igor Stravinsky depicts a steady flow of words and music until the command to sacrifice Isaac is given, after which there is an abrupt silence that conveys a reaction of astonishment and shock more effectively than any combination of notes or words could. Also needing a mention here are the works (including several based on a libretto by Pietro Metastasio) that understand Isaac and his near sacrifice as a type of the sacrifice of Jesus, reflecting a long history of Christian interpretation of the story. [4]
Sarah
The words of Benjamin Britten’s “Canticle II” at one point depict Isaac wishing his mother were there. Readers of the Bible may or may not have noticed how Abraham’s wife Sarah is impacted by Abraham attempting to sacrifice Isaac. Immediately following this story, we are told that Sarah dies. Did Abraham’s action cause her death? Some have considered that as a possible interpretation, including Alicia Jo Rabins in her song “River So Wide.” [5]
Sarah’s story is also included in Andrea Clearfield’s suite Women of Valor , which tells a number of biblical women’s stories from their own point of view. [6] We are not told whether Abraham and Isaac ever spoke again after the attempt to sacrifice him. Noticing such details opens many possible avenues of interpretation, which composers and songwriters have explored in a variety of ways. What they have produced can assist those wrestling with a text that continues to haunt its readers.
Hagar
The story of Hagar and Ishmael is often given less attention than that of Sarah and Isaac, although Hagar’s story is part of the readings for Rosh Hashanah , the Jewish New Year. Hagar’s traumatic experience, courage, and powerful emotion have unsurprisingly led more than one composer to gravitate to the text. Sometimes what the text emphasizes sparks a creative response, while at others the voices that are not heard or are downplayed may invite exploration, expanding on and extending what is said in the text. Andrea Clearfield has explored Hagar’s perspective (just as she has done with Sarah’s perspective and that of other female characters in the Bible in her Women of Valor suite mentioned above), and Sally Beamish’s opera Hagar in the Wilderness does so as well. If you have an opportunity to hear the latter work, additional information is online in an interview of Beamish by Kate Molleson as well as a video interview with both Beamish and librettist Clara Glynn . When he was just fourteen years old, Franz Schubert set a poem about Hagar to music in his song “Hagars Klage” (“Hagar’s Lament”). Kathie Lee Gifford and Nicole C. Mullen have explored the perspective of Hagar together with that of Ruth and others in their oratorio The God Who Sees . In his score to the 1966 movie The Bible: In the Beginning , Toshiro Mayuzumi provides an instrumental exploration of “Hagar the Egyptian.”
Abraham’s Daughter
Finally, even some of the biggest fans of the Hunger Games movies may have missed that Arcade Fire’s song “Abraham’s Daughter” on the soundtrack to the movie depicts this character not mentioned in the Bible playing a heroic role in the story not unlike that which Katniss plays in the movies (and the novels on which they were based).
For Further Reading
Albright, Daniel. Music’s Monisms: Disarticulating Modernism . Chicago: University of Chicago Press, 2021.
Dowling Long, Siobhán. The Sacrifice of Isaac: The Reception of a Biblical Story in Music . Sheffield, UK: Sheffield Phoenix Press, 2013.
Payne, Anthony. “Stravinsky’s ‘Abraham and Isaac’ and ‘Elegy for J. F. K.’” Tempo 73 (1965): 12–15.
Roncace, Mark, and Dan W. Clanton Jr. “Popular Music.” In Teaching the Bible through Popular Culture and the Arts , edited by Mark Roncace and Patrick Gray, 15–51. Atlanta: SBL, 2007.
Stratton, Jon. “Jews, Judaism and Popular Music.” In The Bloomsbury Handbook of Religion and Popular Music , edited by Christopher Partridge and Marcus Moberg, 121–30. London: Bloomsbury Academic, 2017.
———. Jews, Race and Popular Music . London: Routledge, 2017.
Straus, Joseph N. Stravinsky’s Late Music . New York: Cambridge University Press, 2004.
- The recording of a live performance embedded here has been shared on YouTube by George Minister, who also has a recording of Baez singing a calypso version of the Lord’s Prayer on his channel. ↵
- Official audio recording from Leonard Cohen’s YouTube channel. Licensed by SME on behalf of Columbia. ↵
- Shared by Leandro Marziotte on his own YouTube channel. ↵
- A connection between Jesus and Isaac must have been made before some of our earliest Christian writings were composed, since already in Paul’s letters the language of God as one who “did not spare his only Son” ( Romans 8:32 ) seems to already be traditional. ↵
- Shared by singer-songwriter Alicia Jo Rabins on her own YouTube channel. ↵
- Additional information may be found in chapter 34 about Clearfield as well as on the composer’s website . Judith Lang Zaimont has composed “ Parable: A Tale of Abram and Isaac .” She has also composed a work with the title “A Woman of Valor,” the singular focusing on the phrase from Proverbs 31. More details about both works may be found in the liner notes provided on the Milken Archive website. ↵


20.5A: Antigens and Antigen Receptors
Antigens are molecules that initiate the immune response and can be bound by antibodies.
- Distinguish between antigens and antigen receptors
Key Points
- An antigen is a molecule that initiates the production of an antibody and causes an immune response.
- Antigens are typically proteins, peptides, or polysaccharides. Lipids and nucleic acids can combine with those molecules to form more complex antigens, like lipopolysaccharide, a potent bacterial toxin.
- An epitope is a molecular surface feature of an antigen that can be bound by an antibody. A paratope is the molecular surface feature of an antibody that binds to an epitope.
- Antigens are classified as exogenous (entering from outside) endogenous (generated within cells ), an autoantigen, a tumor antigen, or a native antigen.
- Antigenic specificity is the ability of host cells to recognize an antigen by its unique molecular structure, such as the relationship between antigen epitopes and antibody paratopes.
Key Terms
- antigen : A substance that induces an immune response, usually foreign, but self antigens and internally produced antigens exist as well.
- autoantigen : Any antigen that stimulates auto antibodies in the organism that produced it. These are “self” antigens that are involved in autoimmune disease pathogenesis.
EXAMPLES
Fluorescein, along with other haptens such as biotin, is used in various cell and molecular biological techniques. Fluorescein is often conjugated to a protein to allow scientists to examine its location using a fluorescent microscope.
In immunology, an antigen is a substance that evokes an immune response. Formally they are defined as a substance that causes the production of antibodies specific to that antigen, however they also cause T cell mediated immune responses, and may lead to an inflammatory response. The substance may be from the external environment or formed within the body. The immune system will try to destroy or neutralize any antigen that is recognized as a foreign and potentially harmful invader.”Self” antigens are usually tolerated by the immune system; whereas “non-self” antigens can be identified as invaders and can be attacked by the immune system.
Molecular Structure of Antigens
At the molecular level, an antigen is characterized by its ability to be “bound” at the antigen-binding site of an antibody. Antibodies tend to discriminate between the specific molecular structures presented on the surface of the antigen. Antigens are usually either proteins, peptides, or polysaccharides. This includes parts (coats, capsules, cell walls, flagella, fimbrae, and toxins) of bacteria, viruses, and other microorganisms. Lipids and nucleic acids are antigenic only when combined with proteins and polysaccharides. For example, the combination of lipids and polysaccharides are lipopolysaccharides (LPS), which are the primary component of gram negative bacterial endotoxin. LPS forms the cell wall of gram negative bacteria and causes a powerful immune response when bound. Cells present their immunogenic-antigens to the immune system via a major histocompatibility (MHC) molecule. Depending on the antigen presented and the type of the histocompatibility molecule, several types of immune cells can become activated due to an antigen.
Antigens have several structural components of interaction that may be bound by different classes of antibodies. Each of these distinct structural components is considered to be an epitope, also called an antigenic determinent. Therefore, most antigens have the potential to be bound by several distinct antibodies, each of which is specific to a particular epitope. The antigen binding receptor on an antibody is called a paratope, and is specific to the epitope of the antigen. Using the “lock and key” metaphor, the antigen itself can be seen as a string of keys – any epitope being a “key” – each of which can match a different lock.
Types of Antigens
Antigens are categorized into broad classes of antigens based on their origin. So many different molecules can function as an antigen in the body, and there is considerable diversity even within these categories.
These are the main classes of antigens that are involved in immune system activation. Their diversity is analogous to the immense diversity of the diseases that the immune system works to overcome.
Exogenous Antigens
Exogenous antigens are antigens that have entered the body from the outside, for example by inhalation, ingestion, or injection. Exogenous antigens are the most common kinds of antigens, and includes pollen or foods that may cause allergies, as well as the molecular components of bacteria and other pathogens that could cause an infection.
Endogenous Antigens
Endogenous antigens are that have been generated within previously-normal cells as a result of normal cell metabolism or because of viral or intracellular bacterial infection (which both change cells from the inside in order to reproduce). The fragments are then presented on the surface of the infected cells in the complex with MHC class I molecules.
Autoantigens
Autoantigens are normal “self” protein or complex of proteins or nucleic acid that is attacked by the host’s immune system, causing an autoimmune disease. These antigens should, under normal conditions, not be the target of the immune system, but due to mainly genetic and environmental factors, the normal immunological tolerance for such an antigen has been lost.
Tumor Antigens (Neoantigens)
These antigens are presented by MHC I or MHC II molecules on the surface of tumor cells.These antigens result from a tumor-specific mutation during malignant transformation of normal cells into cancer cells. Despite expressing this antigen, many tumors have developed ways to evade antigen recognition and immune system killing.
Native Antigens
A native antigen is an antigen that is not yet processed by an APC to smaller parts. T cells cannot bind native antigens, but require that they be digested and processed by APCs, whereas B cells can be activated by native ones without prior processing.


0.1.2: Goal Setting
Adobe Stock Image
Now that we've looked at proficiency, the three modes of communication, and the proficiency path here at Butler, it's important for you to set a goal for the semester and your Spanish career.
Learning a language is a journey and every step of the way can be embraced. Awareness of the journey can help us focus and achieve our goals. We can set achievable goals and improve our language proficiency.
Stop and think:
What are your goals for Spanish class this semester? What are your proficiency goals
S - Specific- What do you want to achieve? (Ex: Proficiency level, class grade, ability to _____)
M - Measurable- How will you know when you have achieved it?
A - Attainable- How will you achieve it?
R - Relevant- Why is this important to you right now? Why is it worthwhile?
T - Time-bound- By when?
Works Cited:
Wikipedia contributors. "SMART criteria." Wikipedia, The Free Encyclopedia . Wikipedia, The Free Encyclopedia, 12 Nov. 2023. Web. 23 Dec. 2023.


4.7: Post-test
- True or False: Meeting a baby’s physical and emotional needs, such as feeding them when hungry, changing them when soiled, and consoling them when they cry helps babies to develop a sense of trust in their caregivers and environment.
-
Which of the following is the
correct
way
to pick up an infant?
- Pick the infant up by one arm.
- Pick the infant up by her waist and allow the head to roll backwards.
- Pick the infant up using one hand to support the head and neck and the other to hold her bottom.
- Pick the infant up by grabbing both arms and allowing the head to freely tilt back so the baby can see you.
-
Which of the following are the best ways to provide a safe environment for a child?
Select all that apply.
- Allow children to use the stairs unsupervised so they can explore their environment.
- Allow children to ride bikes, skateboard and roller skate without helmets or supervision.
- Lock up medications, cleaning supplies, and hazardous chemicals.
- Keep sharp objects such as knives and scissors out of reach of children.
- Teach children to hold your hand and look both ways while crossing the street.
- Allow children to leave shoelaces untied.
- True or False: In order to promote a sense of independence and encourage exploration, children should be allowed to play with anything they want, including cleaning supplies, medications, sharp objects, and electrical outlets.
-
Which of the following are ways a HHA/PCA can support the development of self-esteem in a child?
Select
all
that
apply
.
- Find out what the child’s strengths and talents are and encourage growth in these areas.
- Allow children to explore their environment while providing supervision and ensuring safety.
- Encourage children to play make believe and to use their imagination during pretend play activities.
- Allow children to choose toys and activities during play time.
- Give children small responsibilities such as picking up toys and helping with household chores.
-
Which of the following are possible issues teens may face during the adolescent period of development?
Select
all
that
apply
.
- Concerns about their appearance
- Peer pressure
- Experimentation with drugs, alcohol, and cigarette smoking.
- Eating disorders such as anorexia and bulimia
- Thoughts of depression or suicidality
-
Which of the following are signs of abuse or neglect that a HHA/PCA should immediately report to a supervisor?
Select
all
that
apply
.
- Unexplained bruises
- Burn or teeth marks on the child
- Inadequate clothing for the weather (e.g. child has no coat in the winter)
- Child is left home alone for two days
- A caretaker is observed shaking an infant to stop him from crying
- The child reports their uncle touched them “down there”
- A father strikes the mother across the face and pushes her against a wall
- An inadequate supply of food is in the home and the children are often hungry
- A parent or caregiver is intoxicated
- True or False: Homes where substance abuse occurs also have higher rates of child abuse and neglect.
- True or False: You should immediately report signs of substance abuse or mental health issues from children or their parents to your supervisor.
-
Which is the best way to help children cope with life and family stresses?
- Change routines and rules frequently.
- Point out the child’s flaws and how they contribute to their family problems.
- Teach children positive coping skills and provide reassurance when they are upset.
- Don’t allow children to express their thoughts and feelings. Children should be seen and not heard.
- Answer
-
1. True
2. C
3. C, D, and E
4. False
5. All are ways to support self–esteem
6. All are possible issues a teen could face
7. All are signs that should be reported
8. True
9. True
10. C


6.5: Inferential statistics for cardinal data
-
- Last updated
- Save as PDF
Where one variable is nominal (more precisely, nominal with two values) and one is cardinal, the a widely-used test is the t -test, of which there are two well known versions, Welch’s t -test and Student’s t -test, that differ in terms of the requirements that the data must meet in order for them to be applicable. In the following, I will introduce Welch’s t -test, which can be applied more broadly, although it still has some requirements that I will return to below.
6.5.1 Welch’s t -test
Let us return to the case study of the length of modifiers in the two English possessive constructions. Here is the research hypothesis again, paraphrased slightly from (15) and (16) in Chapter 5:
(12) H1 : The S -POSSESSIVE will be used with short modifiers, the OF -POSSESSIVE will be used with long modifiers.
Prediction: The mean LENGTH (in “number of words”) of modifiers of the S -POSSESSIVE should be smaller than that of the modifiers of the OF -POSSESSIVE.
The corresponding null hypothesis is stated in (13):
(13) H0 : There is no relationship between LENGTH and TYPE OF POSSESSIVE.
Prediction: There will be no difference between the mean length of the modifiers of the S -POSSESSIVE and the OF -POSSESSIVE.
Table 6.12 shows the length in number of words for the modifiers of the s - and of -possessives (as already reported in Table 5.9), together with a number of additional pieces of information that we will turn to next.
First, note that one case that was still included in Table 5.9 is missing: Example (b 19) from that Table, which had a modifier of length 20. This is treated here as a so-called outlier , i.e., a value that is so far away from the mean that it can be considered an exception. There are different opinions on if and when outliers should be removed that we will not discuss here, but for expository reasons alone it is reasonable here to remove it (and for our results, it would not have made a difference if we had kept it).
In order to calculate Welch’s t -test, we determine three values on the basis of our measurements of LENGTH: the number of measurements N , the mean length for each group (x̄), and a value called “sample variance” ( s 2 ). The number of measurements is easy to determine – we just count the cases in each group: 20 s -possessives and 24 of -possessives. We already calculated the mean lengths in Chapter 5: for the s -possessive, the mean length is 1.9 words, for the of -possessive it is 3.83 words. As we already discussed in Chapter 5, this difference conforms to our hypothesis: s -possessives are, on average, shorter than of -possessives.
The question is, again, how likely it is that this difference is due to chance. When comparing group means, the crucial question we must ask in order to determine this is how large the variation is within each group of measurements: put simply, the more widely the measurements within each group vary, the more likely it is that the differences across groups have come about by chance.
Table 6.12: Length of the modifier in the sample of s - and of -possessives from Table 5.9
The first step in assessing the variation consists in determining for each measurement, how far away it is from its group mean. Thus, we simply subtract each measurement for the S -POSSESSIVE from the group mean of 1.9, and each measurement for the OF -POSSESSIVE from the group mean of 3.83. The results are shown in the third column of each sub-table in Table 6.12. However, we do not want to know how much each single measurement deviates from the mean, but how far the group S -POSSESSIVE or OF -POSSESSIVE as a whole varies around the mean. Obviously, adding up all individual values is not going to be helpful: as in the case of observed and expected frequencies of nominal data, the result would always be zero. So we use the same trick we used there, and calculate the square of each value – making them all positive and weighting larger deviations more heavily. The results of this are shown in the fourth column of each sub-table. We then calculate the mean of these values for each group, but instead of adding up all values and dividing them by the number of cases, we add them up and divide them by the total number of cases minus one. This is referred to as the sample variance:
(14)
The sample variances themselves cannot be very easily interpreted (see further below), but we can use them to calculate our test statistic, the t-value, using the following formula (x̄ stands for the group mean, s 2 stands for the sample variance, and N stands for the number of cases; the subscripts 1 and 2 indicate the two subsamples:
(15)
Note that this formula assumes that the measures with the subscript 1 are from the larger of the two samples (if we don’t pay attention to this, however, all that happens is that we get a negative t -value, whose negative sign we can simply ignore). In our case, the sample of of -possessives is the larger one, giving us:
As should be familiar by now, we compare this t-value against its distribution to determine the probability of error (i.e., we look it up in the table in Section 14.4 in the Statistical Tables at the end of this book. Before we can do so, however, we need to determine the degrees of freedom of our sample. This is done using the following formula:
(16)
Again, the subscripts indicate the sub-samples, s 2 is the sample variance, and N is the number of items the degrees of freedom for the two groups (df 1 and df 2 ) are defined as N − 1. If we apply the formula to our data, we get the following:
As we can see in the table of critical values, the t -value is smaller than 0.01. A t -test should be reported in the following format:
(17) Format for reporting the results of a t test
( t ([deg. freedom]) = [ t value], p < (or >) [sig. level]).
Thus, a straightforward way of reporting our results would be something like this: “This study has shown that for modifiers that are realized by lexical NPs, s -possessives are preferred when the modifier is short, while of -possessives are preferred when the modifier is long. The difference between the constructions is very significant ( t (25.50) = 3.5714, p < 0.01)”.
As pointed out above, the value for the sample variance does not, in itself, tell us very much. We can convert it into something called the sample standard deviation , however, by taking its square root. The standard deviation is an indicator of the amount of variation in a sample (or sub-sample) that is frequently reported; it is good practice to report standard deviations whenever we report means.
Finally, note that, again, the significance level does not tell us anything about the size of the effect, so we should calculate an effect size separately. The most widely-used effect size for data analyzed with a t -test is Cohen’s d , also referred to as the standardized mean difference. There are several ways to calculate it, the simplest one is the following, where σ is the standard deviation of the entire sample:
(18)
For our case study, this gives us
This standardized mean difference can be converted to a correlation coefficient by the formula in (19):
(19)
For our case study, this gives us
Since this is a correlation coefficient, it can be interpreted as described in Table 6.6 above. It falls into the moderate range, so a more comprehensive way of summarizing the results of this case study would be the following: “This study has shown that length has a moderate, statistically significant influence on the choice of possessive constructions with lexical NPs in modifier position: s -possessives are preferred when the modifier is short, while of -possessives are preferred when the modifier is long. ( t (25.50) = 3.5714, p < 0.01, r = 0.41)”.
6.5.2 Normal distribution requirement
In the context of corpus linguistics, there is one fundamental problem with the t -test in any of its variants: it requires data that follow what is called the normal distribution . Briefly, the normal distribution is a probability distribution where most measurements fall in the middle, decreasing on either side until they reach zero. Figure 6.1a shows some examples. As you can see, the curve may be narrower or wider (depending on the standard deviation) and it may be positioned at different points on the x axis (depending on the mean), but it is always a symmetrical bell curve.
You will often read that many natural phenomena approximate this distribution – examples mentioned in textbooks are, invariably, the size and weight of organisms, frequently other characteristics of organisms such as skin area, blood pressure or IQ, and occasionally social phenomena like test scores and salaries. Figure 6.1b show the distribution of body weight in a sample of swans collected for an environmental impact study (Fite 1979), and, indeed, it seems to follow, roughly, a normal distribution if we compare it to the bell curve superimposed on the figure.
Figure 6.1: The normal distribution and linguistic data
Unfortunately, cardinal measurements derived from language data (such as the length or of words, constituents, sentences, etc. or the distance to the last mention of a referent) are rarely (if ever) normally distributed (see, e.g., McEnery & Hardie 2012: 51). Figure 6.1c shows the distribution of the constituent length, in number of words, of of -phrases modifying nouns in the SUSANNE corpus (with of -phrases with a length of more than 20 words removed as outliers). As you can see, they do not follow a normal distribution at all – there are many more short of -phrases than long ones, shifting the distribution further to the left and making it much narrower than it should be.
There are three broad ways of dealing with this issue. First, we could ignore it and hope that the t -test is robust enough to yield meaningful results despite this violation of the normality requirement. If this seems like a bad idea, this is because it is fundamentally a bad idea – and statisticians warn against it categorically. However, many social scientists regularly adopt this approach – just like we did in the case study above. And in practice, this may be less of a problem than one might assume, since the t -test has been found to be fairly robust against violations of the normality requirement. However, we should not generally rely on this robustness, as linguistic data may depart from normality to quite an extreme degree. More generally, ignoring the prerequisites of a statistical procedure is not exactly good scientific practice – the only reason I did it above was so you would not be too shocked when you see it done in actual research (which, inevitably, you will).
Second, and more recommendably, we could try to make the data fit the normality requirement. One way in which this is sometimes achieved in the many cases where data do not follow the normal distribution is to log-transform the data (i.e., use the natural logarithm of the data instead of the data themselves). This often, but not always, causes the data to approximate a normal distribution more closely. However, this does not work in all cases (it would not, for example, bring the distribution in Figure 6.1c much closer to a normal distribution, and anyway, transforming data carries its own set of problems.
Thus, third, and most recommendably, we could try to find a way around having to use a t -test in the first place. One way of avoiding a t -test is to treat our non-normally distributed cardinal data as ordinal data, as described in Chapter 5. We can then use the Mann-Whitney U -test, which does not require a normal distribution of the data. I leave it as an exercise to the reader to apply this test to the data in Table 6.12 (you know you have succeeded if your result for U is 137, p < 0.01).
Another way of avoiding the t -test is to find an operationalization of the phenomenon under investigation that yields rank data, or, even better, nominal data in the first place. We could, for example, code the data in Table 6.12 in terms of a very simple nominal variable: LONGER CONSTITUENT (with the variables HEAD and MODIFIER). For each case, we simply determine whether the head is longer than the modifier (in which case we assign the value head) or whether the modifier is longer than the head (in which case we assign the value MODIFIER; we discard all cases where the two have the same length. This gives us Table 6.13.
Table 6.13: The influence of length on the choice between the two possessives
The χ 2 value for this table is 0.7281, which at one degree of freedom means that the p value is larger than 0.05, so we would have to conclude that there is no influence of length on the choice of possessive construction. However, the deviations of the observed from the expected frequencies go in the right direction, so this may simply be due to the fact that our sample is too small (obviously, a serious corpus-linguistic study would not be based on just 33 cases).
The normal-distribution requirement is only one of several requirements that our data set must meet in order for particular statistical methods to be applicable.
For example, many procedures for comparing group means – including the more widely-used Student t -test – can only be applied if the two groups have the same variance (roughly, if the measurements in both groups are spread out from the group means to the same extent), and there are tests to tell us this (for example, the F test). Also, it makes a difference whether the two groups that we are comparing are independent of each other (as in the case studies presented here), or if they are dependent in that there is a correspondence between measures in the two groups. For example, if we wanted to compare the length of heads and modifiers in the s-possessive, we would have two groups that are dependent in that for any data point in one of the groups there is a corresponding data point in the other group that comes from the same corpus example. In this case, we would use a paired test (for example, the matched-pairs Wilcoxon test for ordinal data and Student’s paired t -test for cardinal data).


22.2: Forming Planets from the Remnants of Exploding Stars
If we were to take an inventory of the elements that make up Earth, we would find that 95% of Earth’s mass comes from only four elements: oxygen, magnesium, silicon, and iron. Most of the remaining 5% comes from aluminum, calcium, nickel, hydrogen, and sulfur. We know that the big bang made hydrogen, helium, and lithium, but where did the rest of the elements come from?
The answer is that the other elements were made by stars. Sometimes stars are said to “burn” their fuel, but burning is not at all what is going on within stars. The burning that happens when wood in a campfire is turned to ash and smoke is a chemical reaction—heat causes the atoms that were in the wood and in the surrounding atmosphere to exchange partners. Atoms group in different ways, but the atoms themselves do not change. What stars do is change the atoms. The heat and pressure within stars cause smaller atoms to smash together and fuse into new, larger atoms. For example, when hydrogen atoms smash together and fuse, helium is formed. Large amounts of energy are released when some atoms fuse and that energy is what causes stars to shine.
It takes larger stars to make elements as heavy as iron and nickel. Our Sun is an average star; after it uses up its hydrogen fuel to make helium, and then some of that helium is fused to make small amounts of beryllium, carbon, nitrogen, oxygen, and fluorine, it will be at the end of its life. It will stop making atoms and will cool down and bloat until its middle reaches the orbit of Mars. In contrast, large stars end their lives in spectacular fashion, exploding as supernovae and casting off newly formed atoms—including the elements heavier than iron—into space. It took many generations of stars creating heavier elements and casting them into space before heavier elements were abundant enough to form planets like Earth.
Until recently, astronomers have only been able to see stars that already contain heavier elements in small amounts, but not the first-generation stars that started out before any of the heavier elements were produced. That changed in June of 2015 when it was announced that a distant galaxy called CR7 had been found that contained stars made only of hydrogen and helium. The galaxy is so far away that it shows us a view of the universe from only 800 million years after the big bang. [1]
- Read more about the discovery of CR7 here: " Astronomers spot first-generation stars, made from big bang ," ( http://bit.ly/CR7galaxy ) ↵