2.9: Measures of Variance and Standard Deviation on Grouped Data
Learning Objectives
- Determine range, variance, and standard deviation of grouped discrete data
- Determine range, variance, and standard deviation of grouped relative frequency data
Section \(2.9\) Excel File (contains all of the data sets for this section)
Introduction to Measures of Spread on Grouped Data
As mentioned in the previous section, there are times when we may be given data in a summarized frequency distribution format instead of a collection of "raw" data. We now examine finding descriptive statistic measures of dispersion in grouped data: range, variance, and standard deviation. If the data is grouped over intervals, we can only estimate such measures since the grouping action has caused us to lose some data information. However, if the data is grouped into single-value classes, we can usually produce the same spread measures as if we had the raw data.
Range, Variance, and Standard Deviation of Grouped (non-interval) Data in a Frequency Table
Look at the frequency table from Text Exercise \(2.8.1\) regarding quiz scores of twenty students:
| Quiz Scores | Frequency |
|---|---|
| \( 5 \) | \( 2 \) |
| \( 6 \) | \( 6 \) |
| \( 7 \) | \( 5 \) |
| \( 8 \) | \( 4 \) |
| \( 9 \) | \( 3 \) |
With such a table, we can easily find the range with no new ideas needed. The table shows the minimum data value is \(5\) and the maximum is \(9\). Because \(9 - 5 \) \(= 4,\) the range is a score difference of \(4.\)
Next, assuming the data is population data, we examine the variance and standard deviation. Recall that variance is the average of all the various squared deviations of the individual data values from the mean of the data. In the previous section, we found the mean of this distribution to be \( \mu\) \(= \frac{ \sum \left( x_j \cdot f_j \right) }{\sum f_j}\) \(= \frac{140}{20}\) \(= 7.\) In a new column, we can construct the deviations from the mean and square those deviations as a first step. It becomes imperative that we not get tied up with all the messy numbers as we move through our work but keep our focus on what we are measuring:
Table \(\PageIndex{2}\): Computation of squared deviations from the mean
| \(x_j\) | \(f_j\) | \( \left( x_j - \mu \right)^2 \) |
|---|---|---|
| \( 5 \) | \( 2 \) | \( \left( 5 - 7 \right) ^2 =4 \) |
| \( 6 \) | \( 6 \) | \( \left( 6 - 7 \right) ^2 =1 \) |
| \( 7 \) | \( 5 \) | \( \left( 7 - 7 \right) ^2 =0 \) |
| \( 8 \) | \( 4 \) | \( \left( 8 - 7 \right) ^2 =1 \) |
| \( 9 \) | \( 3 \) | \( \left( 9 - 7 \right) ^2 =4 \) |
| Totals: | \( \sum f = 20\) |
Before we can average these squared deviation measures, we must remember that some squared deviations occur more frequently (as given by the frequency column) than others. For example, the frequency column shows that many more squared deviations are tied to the data values of \(6\) than those of the squared deviations associated with the data value of \(9.\) We must "weight" these squared deviations by the frequency of their occurrence to account for the various twenty data values. In another new column, we form the products of \( \left( x_j - \mu \right)^2 \cdot f_j \) to accomplish this, and then proceed to "average" those weighted squared deviations:
Table \(\PageIndex{3}\): Computation of the variance
| \(x_j\) | \(f_j\) | \( \left( x_j - \mu \right)^2 \) | \( \left( x_j - \mu \right)^2 \cdot f_j\) |
|---|---|---|---|
| \( 5 \) | \( 2 \) | \( 4 \) | \( 4 \cdot 2 = 8 \) |
| \( 6 \) | \( 6 \) | \( 1 \) | \( 1 \cdot 6 = 6 \) |
| \( 7 \) | \( 5 \) | \( 0 \) | \( 0 \cdot 5 = 0 \) |
| \( 8 \) | \( 4 \) | \( 1 \) | \( 1 \cdot 4 = 4 \) |
| \( 9 \) | \( 3 \) | \( 4 \) | \( 4 \cdot 3 = 12 \) |
| Totals: | \( \sum f_j = 20\) | \( \sum \left( x_j - \mu \right)^2 \cdot f_j = 30\) | |
| Variance: | \( \sigma^2=\frac{\sum\left[\left( x_j - \mu \right)^2 \cdot f_j\right]}{\sum f_j }=\frac{30}{20}=1.5 \) |
As shown in the last two rows above, with our weighted squared deviations determined, we can find a meaningful average by summing our squared deviations and dividing by the number of data values involved. We have a variance measure of \( \sigma ^2\( \)=1.5\) in the bottom right table cell. We can find the standard deviation by taking the square root of our variance: \( \sigma\) \(= \sqrt{\sigma^2}\) \(= \sqrt{1.5}\) \(\approx 1.2247.\) We note that our calculation work was for population data. If the table referenced sample data, we would have divided by \( \sum (f_j) - 1 \) instead on the last computation:
Variance and Standard Deviation from a Frequency Distribution
\[ \text{Variance for sample data: }s^{2} = \frac{\sum\left[\left( x_j - \bar{x} \right)^2 \cdot f_j\right]}{\sum{(f_j)} -1 } \nonumber\]
\[ \text{Standard Deviation for sample data: }s = \sqrt{\frac{\sum\left[\left( x_j - \bar{x} \right)^2 \cdot f_j\right]}{\sum{(f_j)} -1 }} = \sqrt{s^{2}} \nonumber\]
\[ \text{Variance for population data: } \sigma^{2} = \frac{\sum\left[\left( x_j - \mu \right)^2 \cdot f_j\right]}{\sum f_j } \nonumber\]
\[ \text{Standard Deviation for population data: } \sigma = \sqrt{\frac{\sum\left[\left( x_j - \mu \right)^2 \cdot f_j\right]}{\sum f_j}} = \sqrt{\sigma^{2}} \nonumber\]
As mentioned, working in a spreadsheet will often make the computation quicker and easier, especially when working with such columns of information.
The frequency table from Section \(2.1\) shows thirty student scores (discrete 10-point scale) for an assignment.
Table \(\PageIndex{4}\): Student quiz score data
| Student Score | Frequency |
|---|---|
| \( 3 \) | \( 1 \) |
| \( 4 \) | \( 1 \) |
| \( 5 \) | \( 3 \) |
| \( 6 \) | \( 5 \) |
| \( 7 \) | \( 5 \) |
| \( 8 \) | \( 7 \) |
| \( 9 \) | \( 5 \) |
| \( 10 \) | \( 3 \) |
Find the range, variance, and standard deviation of this grouped (non-interval) data. Assume the data is a sample from a larger population.
- Answer
-
The table shows that the minimum data value is \(3\) and the maximum data value is \(10\). Because \(10 - 3\) \(= 7,\) the range is a score difference of \(7.\)
Next, we determine the variance by averaging the squared deviations from the mean, once weighted by the frequency counts. Since this is sample data, we also remember to divide by one less in the averaging step. In the previous section, we found the mean of this distribution to be \( \bar{x}\) \(= \frac{ \sum \left( x_j \cdot f_j \right) }{\sum f_j}\) \(= \frac{218}{30}\) \(\approx 7.2667.\) We can construct the deviations from the mean and square those as the first step. Again, it becomes imperative that we not get distracted by the messy decimals we compute in our work. Keep focusing on what we measure in the computational work (squared deviations from the mean to be averaged) and calculate it accurately.
Table \(\PageIndex{5}\): Computation of square deviations from the mean
\(x_j\) \(f_j\) \( \left( x_j - \mu \right)^2 \) \( 3 \) \( 1 \) \( \left( 3 - 7.2667 \right) ^2 \approx 18.2044 \) \( 4 \) \( 1 \) \( \left( 4 - 7.2667 \right) ^2 \approx 10.6711\) \( 5 \) \( 3 \) \( \left( 5 - 7.2667 \right) ^2 \approx 5.1378\) \( 6 \) \( 5 \) \( \left( 6 - 7.2667 \right) ^2 \approx 1.6044 \) \( 7 \) \( 5 \) \( \left( 7 - 7.2667 \right) ^2 \approx 0.0711\) \( 8 \) \( 7 \) \( \left( 8 - 7.2667 \right) ^2 \approx 0.5378 \) \( 9 \) \( 5 \) \( \left( 9 - 7.2667 \right) ^2 \approx 3.0044\) \( 10 \) \( 3 \) \( \left( 10 - 7.2667 \right) ^2 \approx 7.4711\) Totals: \( \sum f_j = 30\) Next, we weight our various squared deviations by their frequency of occurrence, forming the products of \( \left( x_j - \mu \right)^2 \cdot f_j \) to accomplish this:
Table \(\PageIndex{6}\): Computation of variance
\(x_j\) \(f_j\) \( \left( x_j - \mu \right)^2 \) \( \left( x_j - \mu \right)^2 \cdot f_j\) \( 3 \) \( 1 \) \( \left( 3 - 7.2667 \right) ^2 \approx 18.2044 \) \( 18.2044 \cdot 1 \approx 18.2044 \) \( 4 \) \( 1 \) \( \left( 4 - 7.2667 \right) ^2 \approx 10.6711\) \( 10.6711 \cdot 1 \approx 10.6711 \) \( 5 \) \( 3 \) \( \left( 5 - 7.2667 \right) ^2 \approx 5.1378\) \( 5.1378 \cdot 3 \approx 15.4133 \) \( 6 \) \( 5 \) \( \left( 6 - 7.2667 \right) ^2 \approx 1.6044 \) \( 1.6044 \cdot 5 \approx 8.0222 \) \( 7 \) \( 5 \) \( \left( 7 - 7.2667 \right) ^2 \approx 0.0711 \) \( 0.0711 \cdot 5 \approx 0.3556 \) \( 8 \) \( 7 \) \( \left( 8 - 7.2667 \right) ^2 \approx 0.5378 \) \( 0.5378 \cdot 7 \approx 3.7645 \) \( 9 \) \( 5 \) \( \left( 9 - 7.2667 \right) ^2 \approx 3.0044 \) \( 3.0044 \cdot 5 \approx 15.0222 \) \( 10 \) \( 3 \) \( \left( 10 - 7.2667 \right) ^2 \approx 7.4711 \) \( 7.4711 \cdot 3 \approx 22.4133 \) Totals: \( \sum f_j = 30\) Variance: \( s = \frac{\sum\left[\left( x_j - \mu \right)^2 \cdot f_j\right]}{\sum{(f_j)} - 1 } \approx 3.2368 \) Again, our last row of the table shows the "averaging" of those weighted squared variations when working with sample data. If the data were given as population data, we would have divided by the sum of the frequencies instead of one less than that sum. This would have resulted in a slightly different value of \( \sigma ^2\) \(\approx 3.1289.\) When calculating, we must vigilantly remember the difference between sample and population variance. These data measures are distinct and continue to emphasize one, among other, reasons for knowing if the data is sample data or population data. As shown by the bottom right measure in the table, this grouped sample data has a variance measure of \( s^{2}\) \(\approx 3.2368.\) The sample standard deviation measure is: \( s \) \(= \sqrt{s^{2}}\) \(\approx \sqrt{3.2368}\) \(\approx 1.7991.\) We can compare this result with the calculated variance for the ungrouped data set to see that we have produced the same measure.
Range, Variance, and Standard Deviation of Grouped (non-interval) Data in a Relative Frequency Table
What must we do to measure the variation of the data if, instead of a frequency distribution table, we have a relative frequency distribution of population data? The approach presented here only produces valid measures in population data since the relative frequency measures do not always disclose the sample size. Recall the sample size is essential for the computation of the sample variance as we must use \(n-1\) in our averaging step. We start with the same example of twenty quiz scores.
| Quiz Scores | Relative Frequency \(P(x_j)\) |
|---|---|
| \( 5 \) | \(\frac{2}{20}=10\%\) |
| \( 6 \) | \(\frac{2}{20}=30\%\) |
| \( 7 \) | \(\frac{5}{20}=25\%\) |
| \( 8 \) | \(\frac{4}{20}=20\%\) |
| \( 9 \) | \(\frac{3}{20}=15\%\) |
| Totals: | \(\sum P(x_j) = 100\%\) |
Again, we can easily find the range--the table shows the minimum data value is \(5\) and the maximum data value is \( 9, \) leading us to a range of \(9 - 5\) \(= 4.\)
Next, we use the same ideas as above to determine the variance and standard deviation of the data from this table. We need to find the average squared deviations of the data values from the mean. In Section \(2.8,\) we found the mean of this relative frequency distribution to be \( \mu\) \(= \sum \left( x_j \cdot P(x_j) \right)\) \(= 7.\) We will add a column to our table to create the squared deviations from the mean.
Table \(\PageIndex{7}\): Computation of the squared deviations from the mean
| \(x_j\) | \( P(x_j) \) | \((x_j - \mu )^2\) |
|---|---|---|
| \( 5 \) | \(\frac{2}{20}=10\%\) | \(( 5 - 7 )^2 =4\) |
| \( 6 \) | \(\frac{2}{20}=30\%\) | \((6 - 7)^2 =1\) |
| \( 7 \) | \(\frac{5}{20}=25\%\) | \((7 - 7)^2 =0\) |
| \( 8 \) | \(\frac{4}{20}=20\%\) | \((8 - 7)^2 =1\) |
| \( 9 \) | \(\frac{3}{20}=15\%\) | \((9 - 7)^2 =4\) |
| Totals: | \(\sum P(x_j) = 100\%\) |
Before we can average these squared deviation measures, we must weight these squared deviations by the relative frequency of occurrence to account for the fact that some data values, such as the \(6,\) occur with different relative frequency than others, such as the \(9.\) To do so, we form a new column for the products of \( \left( x_j - \mu \right)^2 \cdot P(x_j) \) and then proceed to "average" those weighted squared deviations.
Table \(\PageIndex{8}\): Computation of the variance
| \(x_j\) | \(P(x_j)\) | \(\left(x_j-\mu\right)^2\) | \(\left( x_j - \mu \right)^2 \cdot P(x_j) \) |
|---|---|---|---|
| \( 5 \) | \(\frac{2}{20}=10\%\) | \( 4 \) | \(4 \cdot 0.10 = 0.40 \) |
| \( 6 \) | \(\frac{2}{20}=30\%\) | \( 1 \) | \(1 \cdot 0.30 = 0.30 \) |
| \( 7 \) | \(\frac{5}{20}=25\%\) | \( 0 \) | \(0 \cdot 0.25= 0.00 \) |
| \( 8 \) | \(\frac{4}{20}=20\%\) | \( 1 \) | \(1 \cdot 0.20 = 0.20 \) |
| \( 9 \) | \(\frac{3}{20}=15\%\) | \( 4 \) | \(4 \cdot 0.15 = 0.60 \) |
| Totals: | \(\sum P(x_j) = 100\%\) | \(\sum{\left((x_j - \mu )^2\cdot P(x_j) \right)}= 1.50\) | |
| Variance: | \(\sigma^2=\frac{\sum\left[\left( x_j - \mu \right)^2 \cdot P(x_j)\right]}{\sum P(x_j) }=\frac{1.50}{1}=1.5\) |
As shown in the last two rows, with our weighted squared deviations determined, we can find the average by summing our squared deviations and then dividing by the total weighting of the relative frequency measures, which will always be the value \(1.00\) \(=100\%.\) We again get the same measure of variance, \(1.5,\) as we did earlier when the data was in frequency table form.
Variance and Standard Deviation from a Relative Frequency Distribution
\[ \text{Variance for population data: } \sigma^{2} = \sum{\left[ (x_j - \mu )^2 \cdot P(x_j) \right]}\nonumber\]
\[ \text{Standard Deviation for population data: } \sigma = \sqrt{\sum{\left[ (x_j - \mu)^2 \cdot P(x_j) \right]}}= \sqrt{\sigma^{2}} \nonumber\]
We should note that the computation work here is simpler when the grouped data is given in relative frequency rather than just frequency format. This is one reason we often look at data in relative frequency form.
We take the frequency table shown below from Section \(2.1\) regarding thirty student scores (discrete \(10\)-point scale) for an assignment but with the distribution given in relative frequency format instead. This time, we assume the data is population data—our focus is only on these thirty students and not some larger group.
Table \(\PageIndex{9}\): Relative frequency distribution of student scores
| \(x_j\) | \(P(x_j)\) |
|---|---|
| \(3\) | \( \frac{1}{30} \approx 0.0333 = 3.33\% \) |
| \(4\) | \( \frac{1}{30} \approx 0.0333 = 3.33\% \) |
| \(5\) | \( \frac{3}{30} = 0.1000 = 10.00\% \) |
| \(6\) | \( 0.1667 = 16.67\% \) |
| \(7\) | \( 0.1667 = 16.67\% \) |
| \(8\) | \( 0.2333 = 23.33\% \) |
| \(9\) | \( 0.1667 = 16.67\% \) |
| \(10\) | \( 0.1000 = 10.00\% \) |
| Totals: | \( \sum P(x_j) = 1.0000 = 100\%\) |
Find this grouped population data's range, variance, and standard deviation.
- Answer
-
The given table clearly shows that the minimum data value is \(3\) and the maximum is \(10,\) thus a range measure of \(7.\)
For variance, we must first find the squares on the deviations from the mean, as shown in the added column below. Recall that, in Section \(2.8,\) we found the mean of the relative frequency distribution as \(\mu\) \(=\sum{x_j\cdot P(x_j)}\) \(\approx 7.2667.\) We will use this value in our computation work.
Table \(\PageIndex{10}\): Computation of squared deviations from the mean
\(x_j\) \(P(x_j)\) \( \left(x_j-\mu\right)^2\) \(3\) \( 0.0333 = 3.33\% \) \(\left( 3 - 7.2667\right)^2\approx 18.2044 \) \(4\) \( 0.0333 = 3.33\% \) \(\left( 4 - 7.2667\right)^2 \approx 10.6711\) \(5\) \( 0.1000 = 10.00\% \) \(\left( 5 - 7.2667\right)^2 \approx 5.1378\) \(6\) \( 0.1667 = 16.67\% \) \(\left( 6 - 7.2667\right)^2 \approx 1.6044\) \(7\) \( 0.1667 = 16.67\% \) \(\left( 7 - 7.2667\right)^2 \approx 0.0711\) \(8\) \( 0.2333 = 23.33\% \) \(\left( 8 - 7.2667\right)^2 \approx 0.5378 \) \(9\) \( 0.1667 = 16.67\% \) \(\left( 9 - 7.2667\right)^2 \approx 3.0044\) \(10\) \( 0.1000 = 10.00\% \) \(\left( 10 - 7.2667\right)^2 \approx 7.4711\) Totals: \( \sum P(x_j) = 1.0000 = 100\%\) Now we weight, through multiplication, our various squared deviations by their relative frequency of occurrence. This forms the products \(\left( x_j-\mu\right)^2 \cdot P(x_j)\) in our next added column.
Table \(\PageIndex{11}\): Computation of the variance
\(x_j\) \(P(x_j)\) \(\left(x_j-\mu\right)^2\) \(\left(x_j-\mu\right)^2\cdot P(x_j)\) \(3\) \( 0.0333 = 3.33\% \) \(18.2044 \) \(18.2044 \cdot 0.0333 \approx 0.6068\) \(4\) \( 0.0333 = 3.33\% \) \(10.6711\) \(10.6711 \cdot 0.0333 \approx 0.3557 \) \(5\) \( 0.1000 = 10.00\% \) \(5.1378\) \(5.1378 \cdot 0.1000 \approx 0.5138\) \(6\) \( 0.1667 = 16.67\% \) \(1.6044\) \(1.6044 \cdot 0.1667 \approx 0.2674 \) \(7\) \( 0.1667 = 16.67\% \) \( 0.0711 \) \( 0.0711 \cdot 0.1667 \approx 0.0119 \) \(8\) \( 0.2333 = 23.33\% \) \( 0.5378 \) \( 0.5378 \cdot 0.2333 \approx 0.1255 \) \(9\) \( 0.1667 = 16.67\% \) \( 3.0044 \) \( 3.0044 \cdot 0.1667 \approx 0.5007 \) \(10\) \( 0.1000 = 10.00\% \) \( 7.4711 \) \( 7.4711 \cdot 0.1000 \approx 0.7471 \) Totals: \( \sum P(x_j) = 1.0000 = 100\%\) Variance: \(\sigma^2=\sum{\left[ \left( x_j - \mu \right)^2 \cdot P(x_j)\right]}\approx 3.1289\) Our last row illustrates the production of the variance as the average of the weighted squared deviations. The original table of data has a population variance of \(\sigma^{2}\) \(\approx 3.1289\) and population standard deviation of \(\sigma\) \(=\sqrt{\sigma^{2}}\) \(\approx \sqrt{3.1289}\) \(\approx 1.7689.\)
We have computed the three desired measures of variation in the data and grouped them within a relative frequency table.
We can compute population variance and standard deviation even when given data in a relative frequency table. However, we cannot do so for sample data given only in a relative frequency table.
Section Summary
This section has demonstrated that we can often compute range, variance, and standard deviation even after the data has been grouped into a frequency table or a relative frequency table. We also remind ourselves that the formulas developed in this section came from the meaning of each measure. We do not memorize the formulas; we recall what each measure means and how and why we performed the computations. This section demonstrated how we can adjust our process to produce the same measures, but it also showed that, due to the "column" computation work, the use of a spreadsheet makes the process much easier.
The following optional section explores how we might estimate the range, variance, and standard deviation if our data has been grouped into interval classes. The ideas are similar, but we can only roughly estimate such measures because we have lost individual data representation (a loss of information about the data).


15: Population and Community Ecology
-
- 15.1: Introduction
- This fish—actually a group of species including the silver, black, grass, and big head carp—has been farmed and eaten in China for over 1000 years. It is one of the most important aquaculture food resources worldwide. In the United States, however, Asian carp is considered a dangerous invasive species that disrupts community structure and composition to the point of threatening native species.
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- 15.2: Population Demographics and Dynamics
- Populations are individuals of a species that live in a particular habitat. Ecologists measure characteristics of populations: size, density, and distribution pattern. Life tables are useful to calculate life expectancies of individual population members. Survivorship curves show the number of individuals surviving at each age interval plotted versus time.
-
- 15.4: The Human Population
- Earth’s human population is growing exponentially. Humans have increased their carrying capacity through technology, urbanization, and harnessing the energy of fossil fuels. The age structure of a population allows us to predict population growth. Unchecked human population growth could have dire long-term effects on human welfare and Earth’s ecosystems.
-
- 15.5: Community Ecology
- Communities include all the different species living in a given area. The variety of these species is referred to as biodiversity. Many organisms have developed defenses against predation and herbivory, including mechanical defenses, warning coloration, and mimicry. Two species cannot exist indefinitely in the same habitat competing directly for the same resources. Species may form symbiotic relationships such as commensalism, mutualism, or parasitism. Community structure is described by its fou
Thumbnail: A group of multicolored horses standing on a hill (CC0 License; Martin Bennie , via Unsplash )


1.6: Stage Properties
Categories of Props
Stage properties, or “props,” are those items that are not permanently attached to the scenery, or costumes, that add to the visual picture of the shows. If that definition seems a bit vague, you may have discovered why props are such a varied and interesting section of technical theatre. Props can be divided into the following categories by their use in the final show:
- Set Props / Furnishings: These are large items that are rarely, if ever, moved by the cast in the course of the show. These include tables, chairs, settees etc.
- Set Dressing: These are items that are not touched by the actors but complete the look of the set. These include books on a book case, pictures on a wall, fancy plates in a china cabinet, etc.
- Hand Props: These are items carried by and used by the actors in the course of stage directions or stage business. These include such things as a cell phone an actor uses, a coffee mug they drink from, a journal in which they take a note, etc..
- Costume Props: These are items that are not generally considered clothing, but would be designed by the costume designer and worn by the actors. These include hats, eye glasses, wallets, pocket watches, etc.
Some props have additional requirements which they must meet. These additional categories of props are a description in addition to any item above.
- Practical Props: Any item that is a practical prop is also one of the other categories listed. A practical prop is one that must perform its real-world function. For example, a cell phone (hand prop) that an actor needs to hold and imagine that they are using to make a call is not practical. However, if that cell phone needs to ring on cue, it is now practical hand prop. Similarly, a lamp that sits on the stage is set dressing, until it is determined that it must also turn on. Generally, if the prop must light up or make a noise, especially if cued somehow from off stage, it is considered a practical prop.
- Consumable Props: Like practical props, consumable props fit into an additional category. A consumable is one that is “used up” over the course of one or more performances. Food that is eaten by the cast on stage is an example of a consumable prop, as are letters that are torn to pieces by the cast. If a show has a practical flash light, its batteries and lamp would be considered consumable.
- Rehearsal Props: Often the “real” props are not available for the rehearsal period. In those cases where the actors need certain props to rehearse, but do not have access to the real prop, a substitute, or rehearsal prop, is acquired. Rehearsal props need to be approximately the same size and weight as the real prop, but are generally a cheap substitute that does not have the correct look. For example, even though a play may be set in the 1800s, the cast could rehearse with a modern composition book instead of a period diary, assuming the composition book was approximately the same size as the final prop will be.
One of the most confusing things about props is that an item, such as a hat, may be in different categories depending on the show. For example, if the hat is attached to a mannequin in a window and never touched by the cast, it might be considered furnishings. In another show, the hat may be hanging on a hat stand, but not used by the cast, making it set dressing. Yet, another show, an actor, playing a maid, might hand the hat to another actor who carries it off stage with them, making it a hand prop. In a final example an actor might wear the hat, making it a costume prop.
Another confusing layer is what department will supply that hat. As the costume department often has more hats in their storage, they may supply that hat regardless of its use in the show. Props may be supplied by the props department, the scenery department or the costume department depending on the prop and the production team. Some props may take collaboration between departments. For example, a table lamp may be supplied by the scenery department but may be wired and made practical by the electrics department. In all cases, clear communication between departments, facilitated by the stage manager, is essential. Often all the designers and department heads will meet to examine the prop lists developed by each department and the stage manager. At this point they will collaborate on who will supply each prop.
Creating a Prop List
A prop list, can be created much like we were creating our breakdowns. If so, it might look like this:
Initial Prop List
|
Page |
Prop |
Type |
Note |
|
2 |
Bed |
Furnishings |
With curtains |
|
2 |
Chest of Drawers |
Furnishings |
|
|
2 |
Table with 2 chairs |
Furnishings |
|
|
2 |
Assorted decorations |
Dressing |
|
|
2 |
Small Looking Glass |
Hand |
|
|
3 |
Hat |
Costume |
|
|
3 |
2 other hats |
hand |
|
|
3 |
Bolster |
Hand |
"not very protuberant" |
|
5 |
Rasher of Bacon |
Hand |
|
|
5 |
Box of matches |
hand |
almost empty |
|
5 |
candlestick |
hand |
with small candle |
|
6 |
Pork Chop |
Hand |
|
|
6 |
Gridiron |
Hand |
|
|
6 |
Fork |
hand |
|
|
7 |
Tea things |
hand |
|
|
7 |
2 receipts |
hand |
|
|
8 |
Roll |
Hand |
Eaten |
|
8 |
Pipe |
Hand |
|
|
14 |
2 trick dice |
hand |
always roll 6 |
|
15 |
2 trick coins |
hand |
always heads |
|
15 |
additional coins |
hand |
|
|
16 |
Letter in envelope from Margate |
hand |
envelope will need to be replaced nightly |
|
18 |
Letter in envelope from Margate |
hand |
Envelope will need to be replaced nightly |
|
19 |
Letter in envelope from Margate |
hand |
Envelope will need to be replaced nightly |
Like many shows, this initial prop list is mostly hand props. Hand props are the most obvious props in the script. Set dressing is almost non-existent on the list. Set dressing props are determined by the scenic designer based on his or her design, and will often be presented as a list to the prop department.
You might notice that the gridiron is used several times in the show, yet only appears on the prop list once. That is because if it appeared several times, the prop department would supply a new gridiron for each use. Similarly, since there are three letters form Margate, the prop department will supply three, which is what the show requires.
After this list is created, the production team will decide who will be responsible for acquiring each prop. For example, there are three hats, one a costume prop and two hand props. The costume department might decide to provide all three hats so that they work together to create a pleasing image on stage. Similarly, the scene shop may decide that such a large piece as the bed is too large to be built in the prop shop, and decide that it will be created in the scene shop.
Acquiring Props
There are basically three methods of acquiring props for a show: build the prop, buy the prop, or borrow the prop. Each method has its own advantages and disadvantages. After determining which department will provide the prop, the department will examine the script to determine which method should be used to acquire the prop.
Building is perhaps the easiest method to understand. The script is examined, the prop is sketched, and then the department builds it. This method allows the prop to be created to meet the exact needs of the show. In some cases, this is the only method available, especially when doing a fantasy or science fiction based show. The disadvantage of this is that it can take a great deal of time and expertise to build certain props, which may be a limiting factor. One other consideration is that it is sometimes most costly to build a prop rather than buying it. For example, it is possible to melt wax and dip wicks to create the candle for the prop list. This task however would likely a props artisan several long days to create the one candle needed for the show. It is probably much more cost effective to spend a few dollars buying a candle.
Buying the prop is often the fastest method of acquiring a prop, especially if you are doing a play set in contemporary times. With the advent of internet searches, even historical reproductions can often be quickly purchased and shipped to the theatre.  There is a stereotype in musicals of people dancing on tables. The average table that you buy is not strong enough to have several people tap dancing on it for 8 performances a week for many weeks on end. In that case, even if the table was purchased, the scenery or property department would have to spend time reinforcing and strengthening the table. It is not uncommon however to purchase a prop and modify it for use. For example the hats needed may be purchased as a simple hat, and the costume crafts person may add flowers, and hat bands and other embellishments to make it suit the play.
The last category is borrowing a prop. If you are located in a major theatre, film or television market there are often prop rental companies that have an extensive supply of all sorts of varied props. Often these companies have a website that you can peruse to find what you need. If you are not in a major market, it is common for props people to build a relationship with all the other theatre’s nearby and contact them to see if they will rent out their props to your production. When renting props, you will have to provide the rental company with proof of insurance so that if the prop is damaged, they will be able to replace it. Additionally, rental agreements usually prohibit the prop from being changed in any way. This means that if you don’t like the color of the prop, or some other decoration you are stuck with it. The other challenge with renting props is the cost. Props are generally rented by the week. To save money, producers usually do not like to rent the prop for more than a week before opening night, which means the cast will have limited abilities to rehearse with it. The producer will need to be involved in any rental as there are required legal agreements that must be signed. In general, it is the producer’s responsibility to sign and pay for these items, not the prop technician’s.
Another type of borrowing is to “borrow” the prop from the theatre company’s own storage. After a show closes, many companies take the props from the production and store them in a prop storage room. These props can then be used as rehearsal props or as show props for future productions. Different theatres have different rules on modifying props from storage, but they are often far more flexible than a rental company would be.
One last note on borrowing props: it is not a good idea for the prop technician (or any production team member) to loan a prop to a production. Too often, this is the prop that is accidentally broken. Since there is not legal agreement, or proof of insurance issued, there is no recourse or ability to have the prop repaired or replaced. Certainly, in no circumstance should a theatre ever borrow (or rent) a truly irreplaceable and unrepairable prop. There are too many things that could go wrong.
At some point, the stage manager will create a list of those props that are needed for rehearsal. In some theatre companies it is the responsibility of the stage manager to acquire rehearsal props, and in some theatres it is the responsibility of the prop department. Regardless, when that list is generated, rehearsal props need to be provided to the rehearsal room as soon as possible.
Revised Prop List
This prop list is updated to show how the prop will be acquired, if it is practical or consumable.
|
Page |
Prop |
Type |
Note |
Pract. |
Cons. |
Rehers |
Buy |
Build |
Rent |
|
2 |
Bed |
Furnishings |
With curtains |
X |
|||||
|
2 |
Chest of Drawers |
Furnishings |
X |
||||||
|
2 |
Table with 2 chairs |
Furnishings |
X |
||||||
|
2 |
Assorted decorations |
Dressing |
|||||||
|
2 |
Small Looking Glass |
Hand |
X |
X |
|||||
|
3 |
Hat |
Costume |
X |
X |
|||||
|
3 |
2 other hats |
hand |
X |
||||||
|
3 |
Bolster |
Hand |
"not very protuberant" |
X |
|||||
|
5 |
Rasher of Bacon |
Hand |
X |
X |
|||||
|
5 |
Box of matches |
hand |
almost empty |
X |
X |
X |
X |
||
|
5 |
candlestick |
hand |
with small candle |
X |
X |
||||
|
6 |
Pork Chop |
Hand |
X |
X |
|||||
|
6 |
Gridiron |
Hand |
X |
X |
|||||
|
6 |
Fork |
hand |
X |
X |
|||||
|
7 |
Tea things |
hand |
X |
||||||
|
7 |
2 receipts |
hand |
X |
X |
|||||
|
8 |
Roll |
Hand |
X |
X |
X |
||||
|
8 |
Pipe |
Hand |
X |
||||||
|
14 |
2 trick dice |
hand |
always roll 6 |
X |
X |
||||
|
15 |
2 trick coins |
hand |
always heads |
X |
X |
||||
|
15 |
additional coins |
hand |
X |
||||||
|
16 |
Letter in envelope from Margate |
hand |
envelope will need to be replaced nightly |
X |
X |
X |
|||
|
18 |
Letter in envelope from Margate |
hand |
Envelope will need to be replaced nightly |
X |
X |
X |
|||
|
19 |
Letter in envelope from Margate |
hand |
Envelope will need to be replaced nightly |
X |
X |
X |
Prop lists are in flux throughout the rehearsal process up to opening night. The stage manager is often sending out updates, and there will be many meetings and discussions about props.
Props in the Theatre
Eventually all the props (along with the set, and the costumes, and the actors) will move into the theatre. Generally props are stored in a room of their own, or in a specially constructed cabinet. It is important that the props crew organized the prop storage area so that each prop in the show (and any back-up props or additional supplies (like batteries)) have a specific place that they live. A list of all the items in the prop cabinet needs to be created. When a prop leaves the storage area it must be signed out by a prop technician. When the prop is returned, it should be signed back in. It is very easy to not bother checking props at the end of the night, but if the prop crew does not, something will eventually be lost. A helpful trick is to take a photo of the properly loaded prop storage area and keep the photo(s) with the list. People are often much faster to noctice if something is out of place compared with the photo than notice an error on the list.
For the performance, props will often be placed on prop tables so that they are easily accessible and in the correct location for whomever is grabbing or replacing a prop. The prop table should be laid out exactly the same way every performance so that each prop as its own, clearly defined space. One way to do this is to cover the prop table with butcher paper or craft paper. Once the prop table has been laid out, trace each prop with a thick black marker, and label what it is. This makes it easy to spot missing props, or misplaced props because they traceout brings the organizational structure to everyone’s attention. Ideally there will be a prop table on each side of the stage so that someone does not have a long distance to go to grab the prop they need.
Additional Materials
- History For Hire prop shop
- Omega Cinema Props (one of the largest prop companies on Los Angeles)
- Thurston James has written one of the most comprehensive book series on theatre props .


8.1: Introduction
Contract and workflow management are often major time commitments for infrastructure managers. Contract management is the process of ordering and monitoring outside organizations to do some work required for infrastructure management. Workflow management is the process of organizing and scheduling tasks related to infrastructure management. The extent to which tasks are contracted out will depend upon the internal resources available for infrastructure management.
As with many processes, information technology aids are available and useful for both contract and workflow management. As tasks are defined, they can be logged into a database, monitored and progress recorded. Contract documents and records are quite likely to be kept digitally by both the principal and the contractor. Either specialized or general purpose software can be used for these purposes.
Of course, contract and workflow management is not limited to the domain of infrastructure management. Large firms such as Amazon have very sophisticated financial, inventory and order tracking software. The transportation provider firm Uber has large numbers of contractor drivers as well as suppliers. Similarly, hospitals have systems in place for contracting and workflow management. We will focus on contracting and workflow management for infrastructure, although the processes are similar for other applications (Monczka et. al 2015; Van Der Aalst and Kees, 2004).


6.4: Solve Simple Interest Applications
- Use the simple interest formula
- Solve simple interest applications
Before you get started, take this readiness quiz.
- Solve 0.6y = 45. If you missed this problem, review Example 5.7.4 .
- Solve \(\dfrac{n}{1.45}\) = 4.6. If you missed this problem, review Example 5.7.5 .
Use the Simple Interest Formula
Do you know that banks pay you to let them keep your money? The money you put in the bank is called the principal , \(P\), and the bank pays you interest , \(I\). The interest is computed as a certain percent of the principal; called the rate of interest , \(r\). The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable for time, t, represents the number of years the money is left in the account.
If an amount of money, \(P\), the principal, is invested for a period of t years at an annual interest rate r, the amount of interest, \(I\), earned is
\[I = Prt \nonumber\]
where
- \(I\) = interest
- \(P\) = principal
- \(r\) = rate
- \(t\) = time
Interest earned according to this formula is called simple interest.
The formula we use to calculate simple interest is I = Prt. To use the simple interest formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information by listing all four variables and filling in the given information.
Find the simple interest earned after 3 years on $500 at an interest rate of 6%.
Solution
Organize the given information in a list.
I = ?, P = $500, r = 6%, t = 3 years
We will use the simple interest formula to find the interest.
| Write the formula. | I = Prt |
| Substitute the given information. Remember to write the percent in decimal form. | I = (500)(0.06)(3) |
| Simplify. | I = 90 |
| Check your answer. Is $90 a reasonable interest earned on $500 in 3 years? | In 3 years the money earned 18%. If we rounded to 20%, the interest would have been 500(0.20) or $100. Yes, $90 is reasonable. |
| Write a complete sentence that answers the question. | The simple interest is $90. |
Find the simple interest earned after 4 years on $800 at an interest rate of 5%.
- Answer
-
$160
Find the simple interest earned after 2 years on $700 at an interest rate of 4%.
- Answer
-
$56
In the next example, we will use the simple interest formula to find the principal.
Find the principal invested if $178 interest was earned in 2 years at an interest rate of 4%.
Solution
Organize the given information in a list.
I = $178, P = ?, r = 4%, t = 2 years
We will use the simple interest formula to find the principal.
| Write the formula. | I = Prt |
| Substitute the given information. | 178 = P(0.04)(2) |
| Divide. | $$\dfrac{178}{0.08} = \dfrac{0.08P}{0.08}$$ |
| Simplify. | 2,225 = P |
| Check your answer. Is it reasonable that $2,225 would earn $178 in 2 years? | $$178 \stackrel{?}{=} 2,225 (0.04)(2)$$ |
| $$178 = 178 \; \checkmark$$ | |
| Write a complete sentence that answers the question. | The principal is $2,225. |
Find the principal invested if $495 interest was earned in 3 years at an interest rate of 6%.
- Answer
-
$2,750
Find the principal invested if $1,246 interest was earned in 5 years at an interest rate of 7%.
- Answer
-
$3,560
Now we will solve for the rate of interest.
Find the rate if a principal of $8,200 earned $3,772 interest in 4 years.
Solution
Organize the given information.
I = $3,772, P = $8,200, r = ?, t = 4 years
We will use the simple interest formula to find the rate.
| Write the formula. | I = Prt |
| Substitute the given information. | 3,772 = 8,200r(4) |
| Multiply. | 3,772 = 32,800r |
| Divide. | $$\dfrac{3,772}{32,800} = \dfrac{32,800r}{32,800}$$ |
| Simplify. | 0.115 = r |
| Write as a percent. | 11.5% = r |
| Check your answer. Is 11.5% a reasonable rate if $3,772 was earned in 4 years? | $$3,772 \stackrel{?}{=} 8,200 (0.115)(4)$$ |
| $$3,772 = 3,772 \; \checkmark$$ | |
| Write a complete sentence that answers the question. | The rate was 11.5%. |
Find the rate if a principal of $5,000 earned $1,350 interest in 6 years.
- Answer
-
4.5%
Find the rate if a principal of $9,000 earned $1,755 interest in 3 years.
- Answer
-
6.5%
Solve Simple Interest Applications
Applications with simple interest usually involve either investing money or borrowing money. To solve these applications, we continue to use the same strategy for applications that we have used earlier in this chapter. The only difference is that in place of translating to get an equation, we can use the simple interest formula.
We will start by solving a simple interest application to find the interest.
Nathaly deposited $12,500 in her bank account where it will earn 4% interest. How much interest will Nathaly earn in 5 years?
Solution
We are asked to find the Interest, I. Organize the given information in a list.
I = ?, P = $12,500, r = 4%, t = 5 years
| Write the formula. | I = Prt |
| Substitute the given information. | I = (12,500)(0.04)(5) |
| Simplify. | I = 2,500 |
| Check your answer. Is $2,500 a reasonable interest on $12,500 over 5 years? | At 4% interest per year, in 5 years the interest would be 20% of the principal. Is 20% of $12,500 equal to $2,500? Yes. |
| Write a complete sentence that answers the question. | The interest is $2,500. |
Areli invested a principal of $950 in her bank account with interest rate 3%. How much interest did she earn in 5 years?
- Answer
-
$142.50
Susana invested a principal of $36,000 in her bank account with interest rate 6.5%. How much interest did she earn in 3 years?
- Answer
-
$7,020
There may be times when you know the amount of interest earned on a given principal over a certain length of time, but you don't know the rate. For instance, this might happen when family members lend or borrow money among themselves instead of dealing with a bank. In the next example, we'll show how to solve for the rate.
Loren lent his brother $3,000 to help him buy a car. In 4 years his brother paid him back the $3,000 plus $660 in interest. What was the rate of interest?
Solution
We are asked to find the rate of interest, r. Organize the given information.
I = 660, P = $3,000, r = ?, t = 4 years
| Write the formula. | I = Prt |
| Substitute the given information. | 660 = (3,000)r(4) |
| Multiply. | 660 = (12,000)r |
| Divide. | $$\dfrac{660}{12,000} = \dfrac{(12,000)r}{12,000}$$ |
| Simplify. | 0.055 = r |
| Change to percent form. | 5.5% = r |
| Check your answer. Is 5.5% a reasonable interest rate to pay your brother? | $$660 \stackrel{?}{=} (3,000)(0.055)(4)$$ |
| $$660 = 660 \; \checkmark$$ | |
| Write a complete sentence that answers the question. | The rate of interest was 5.5%. |
Jim lent his sister $5,000 to help her buy a house. In 3 years, she paid him the $5,000, plus $900 interest. What was the rate of interest?
- Answer
-
6%
Hang borrowed $7,500 from her parents to pay her tuition. In 5 years, she paid them $1,500 interest in addition to the $7,500 she borrowed. What was the rate of interest?
- Answer
-
4%
There may be times when you take a loan for a large purchase and the amount of the principal is not clear. This might happen, for instance, in making a car purchase when the dealer adds the cost of a warranty to the price of the car. In the next example, we will solve a simple interest application for the principal.
Eduardo noticed that his new car loan papers stated that with an interest rate of 7.5%, he would pay $6,596.25 in interest over 5 years. How much did he borrow to pay for his car?
Solution
We are asked to find the principal, P. Organize the given information.
I = 6,596.25, P = ?, r = 7.5%, t = 5 years
| Write the formula. | I = Prt |
| Substitute the given information. | 6,596.25 = P(0.075)(5) |
| Multiply. | 6,596.25 = 0.375P |
| Divide. | $$\dfrac{6,596.25}{0.375} = \dfrac{0.375P}{0.375}$$ |
| Simplify. | 17,590 = P |
| Check your answer. Is $17,590 a reasonable amount to borrow to buy a car? | $$6,596.25 \stackrel{?}{=} (17,590)(0.075)(5)$$ |
| $$6,596.25 = 6,596.25 \; \checkmark$$ | |
| Write a complete sentence that answers the question. | The amount borrowed was $17,590. |
Sean's new car loan statement said he would pay $4,866.25 in interest from an interest rate of 8.5% over 5 years. How much did he borrow to buy his new car?
- Answer
-
$11,450
In 5 years, Gloria's bank account earned $2,400 interest at 5%. How much had she deposited in the account?
- Answer
-
$9,600
In the simple interest formula, the rate of interest is given as an annual rate, the rate for one year. So the units of time must be in years. If the time is given in months, we convert it to years.
Caroline got $900 as graduation gifts and invested it in a 10-month certificate of deposit that earned 2.1% interest. How much interest did this investment earn?
Solution
We are asked to find the interest, I. Organize the given information.
I = ?, P = $900, r = 2.1%, t = 10 months
| Write the formula. | I = Prt |
| Substitute the given information, converting 10 months to \(\dfrac{10}{12}\) of a year. | $$I = \$900 (0.021) \left(\dfrac{10}{12}\right)$$ |
| Multiply. | I = 15.75 |
| Check your answer. Is $15.75 a reasonable amount of interest? | If Caroline had invested the $900 for a full year at 2% interest, the amount of interest would have been $18. Yes, $15.75 is reasonable. |
| Write a complete sentence that answers the question. | The interest earned was $15.75. |
Adriana invested $4,500 for 8 months in an account that paid 1.9% interest. How much interest did she earn?
- Answer
-
$57.00
Milton invested $2,460 for 20 months in an account that paid 3.5% interest How much interest did he earn?
- Answer
-
$143.50
Practice Makes Perfect
Use the Simple Interest Formula
In the following exercises, use the simple interest formula to fill in the missing information.
| Interest | Principal | Rate | Time (years) |
|---|---|---|---|
| $1200 | 3% | 5 |
| Interest | Principal | Rate | Time (years) |
|---|---|---|---|
| $1500 | 2% | 4 |
| Interest | Principal | Rate | Time (years) |
|---|---|---|---|
| $4410 | 4.5% | 7 |
| Interest | Principal | Rate | Time (years) |
|---|---|---|---|
| $2212 | 3.2% | 6 |
| Interest | Principal | Rate | Time (years) |
|---|---|---|---|
| $577.08 | $4580 | 2 |
| Interest | Principal | Rate | Time (years) |
|---|---|---|---|
| $528.12 | $3260 | 3 |
In the following exercises, solve the problem using the simple interest formula.
- Find the simple interest earned after 5 years on $600 at an interest rate of 3%.
- Find the simple interest earned after 4 years on $900 at an interest rate of 6%.
- Find the simple interest earned after 2 years on $8,950 at an interest rate of 3.24%.
- Find the simple interest earned after 3 years on $6,510 at an interest rate of 2.85%.
- Find the simple interest earned after 8 years on $15,500 at an interest rate of 11.425%.
- Find the simple interest earned after 6 years on $23,900 at an interest rate of 12.175%.
- Find the principal invested if $656 interest was earned in 5 years at an interest rate of 4%.
- Find the principal invested if $177 interest was earned in 2 years at an interest rate of 3%.
- Find the principal invested if $70.95 interest was earned in 3 years at an interest rate of 2.75%.
- Find the principal invested if $636.84 interest was earned in 6 years at an interest rate of 4.35%.
- Find the principal invested if $15,222.57 interest was earned in 6 years at an interest rate of 10.28%.
- Find the principal invested if $10,953.70 interest was earned in 5 years at an interest rate of 11.04%.
- Find the rate if a principal of $5,400 earned $432 interest in 2 years.
- Find the rate if a principal of $2,600 earned $468 interest in 6 years.
- Find the rate if a principal of $11,000 earned $1,815 interest in 3 years.
- Find the rate if a principal of $8,500 earned $3,230 interest in 4 years.
Solve Simple Interest Applications
In the following exercises, solve the problem using the simple interest formula.
- Casey deposited $1,450 in a bank account with interest rate 4%. How much interest was earned in 2 years?
- Terrence deposited $5,720 in a bank account with interest rate 6%. How much interest was earned in 4 years?
- Robin deposited $31,000 in a bank account with interest rate 5.2%. How much interest was earned in 3 years?
- Carleen deposited $16,400 in a bank account with interest rate 3.9%. How much interest was earned in 8 years?
- Hilaria borrowed $8,000 from her grandfather to pay for college. Five years later, she paid him back the $8,000, plus $1,200 interest. What was the rate of interest?
- Kenneth lent his niece $1,200 to buy a computer. Two years later, she paid him back the $1,200, plus $96 interest. What was the rate of interest?
- Lebron lent his daughter $20,000 to help her buy a condominium. When she sold the condominium four years later, she paid him the $20,000, plus $3,000 interest. What was the rate of interest?
- Pablo borrowed $50,000 to start a business. Three years later, he repaid the $50,000, plus $9,375 interest. What was the rate of interest?
- In 10 years, a bank account that paid 5.25% earned $18,375 interest. What was the principal of the account?
- In 25 years, a bond that paid 4.75% earned $2,375 interest. What was the principal of the bond?
- Joshua's computer loan statement said he would pay $1,244.34 in interest for a 3 year loan at 12.4%. How much did Joshua borrow to buy the computer?
- Margaret's car loan statement said she would pay $7,683.20 in interest for a 5 year loan at 9.8%. How much did Margaret borrow to buy the car?
- Caitlin invested $8,200 in an 18-month certificate of deposit paying 2.7% interest. How much interest did she earn form this investment?
- Diego invested $6,100 in a 9-month certificate of deposit paying 1.8% interest. How much interest did he earn form this investment?
- Airin borrowed $3,900 from her parents for the down payment on a car and promised to pay them back in 15 months at a 4% rate of interest. How much interest did she owe her parents?
- Yuta borrowed $840 from his brother to pay for his textbooks and promised to pay him back in 5 months at a 6% rate of interest. How much interest did Yuta owe his brother?
Everyday Math
-
Interest on savings
Find the interest rate your local bank pays on savings accounts.
- What is the interest rate?
- Calculate the amount of interest you would earn on a principal of $8,000 for 5 years.
-
Interest on a loan
Find the interest rate your local bank charges for a car loan.
- What is the interest rate?
- Calculate the amount of interest you would pay on a loan of $8,000 for 5 years.
Writing Exercises
- Why do banks pay interest on money deposited in savings accounts?
- Why do banks charge interest for lending money?
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at<EMAIL_ADDRESS>."


2.2: Seasonal Plants
Learning Objectives
- Describe seasonal plants common to the horticulture industry.
Planning combinations of woody and herbaceous plants with different life cycles and high visual impact generates year round interest in exterior and interior plantings. When visual interest is planned for one period such as early summer, borders and containers can have a poor appearance the rest of the year. Optimizing the use of grasses, bulbs, perennials, annuals, biennials, shrubs, climbers, and trees can provide a succession of plant forms, colours, textures, and habits throughout the seasons. In temperate regions, year round interest is maximized by selecting plants with at least two, and even three or four seasons of interest.
Conifers and broadleaf evergreens shrubs are often used for year round colour and spatial structure. For example, Taxus cuspidata ‘Capitata’ (upright yew) provides reliable winter colour and a framework that can be enhanced with other shapes, textures, and colours. On the other hand, a planting of broadleaf evergreens such as Skimmia japonica (Japanese skimmia) offers winter colour and structure as well as showy spring flowers and colourful fruit in the autumn. Distinctive plant shapes and the bark of trees such as Cryptomeria japonica (Japanese cedar) and Morus alba ‘Pendula’ or species with persistent fruit like Sorbus aucuparia (European mountain ash) also contribute structure and winter interest.
Practice: Recognize woody plants for winter interest.Some deciduous shrubs and trees like Caryopteris x clandonensis (bluebeard), Cercidiphyllum japonicum (katsura), and Rhus typhina (staghorn sumac) have interesting branching patterns throughout all seasons. The bark and buds of Ribes sanguineum (flowering currant, winter currant), Magnolia x soulangeana (saucer magnolia), Liriodendron tulipifera, and Styrax japonicus (Japanese snowbell, Japanese snowcone) provide winter interest and interesting buds forecast the appearance of foliage and flowers. Climbers with variegated or textured foliage and colourful flowers like Actinidia kolomikta (actinitdia) and Campsis radicans (trumpet vine) also contribute vertical structure. View the images seasonal plant characteristics available at this link to the KPU Plant Database [New Tab] [1]
The appearance of plants before, during, and after flowering is an important consideration for planning seasonal interest. For example, the herbaceous specimen plant Gunnera manicata (gunnera, giant rhubarb) provides a bold shape and texture for at least half to perhaps three quarters of the year. With planning, the eye catching winter stems and seed heads of grasses and perennial species such as Pennisetum alopecuroides (fountain grass), Pennisetum setaceum ‘Rubrum’ (red fountain grass), and Perovskia atriplicifolia (Russian sage) can serve as distractions from seasonal voids. Layering various heights of ground covers, bulbs, annuals and perennials under and around woody shrubs and trees allows a succession of foliage shapes, sizes, textures, and colours to become prominent as the year progresses. In this way, emphasis is placed on year round interest and not only the seasonal show of flowers. A planting calendar is a useful tool for working out the succession of flowers and colour palettes as well as other planting design features. Figure 8.1 shows an example of a basic planting calendar that allows the planner to visualize the times of the year that are most colourful and interesting and those that could use additional development.
As the succession of spring bulbs like Anemone blanda (Greek windflower, blue wood) and Hyacinthus cvs. (hyacinth) finish flowering and foliage fades, deciduous shrubs such as Spiraea x vanhouttei (bridal wreath spirea) and an array of herbaceous annuals, biennials and perennials come into flower in early and mid spring. Examples of spring blooming perennials include Aubrieta x cultorum (common rock cress), Brunnera macrophylla (Siberian bugloss), Papaver orientale (oriental poppy), Pulmonaria saccharata (lungwort), and Dicentra spectabilis (bleeding heart). From late spring and early to mid summer, the flowers and foliage of broadleaf evergreen shrubs such as Daphne cneorum (garland daphne) and herbaceous species like Thymus pseudolanuginosus (woolly thyme), Heuchera cvs. (coralbells, alumroot), and Phlox paniculata (common phlox) take prominence. The progression of seasonal foliage and bloom continues in mid to late summer and through autumn with perennials such as Actaea simplex Atropurpurea Group (cimicifuga), Aster spp. (common aster), Astrantia major (masterwort, astrantia), Coreopsis spp. & cvs. ( coreopsis), Geranium spp. & cvs. (geranium), and Gaillardia cvs. (blanket flower). The texture and seed heads of perennials like Hylotelephium spectabile (autumn joy sedum, stonecrop), and grasses such as Andropogon gerardii (big bluestem), Calamagrostis x acutiflora (feather reed grass), and Molinia arundinacea ‘Skyracer’ (tall moor grass) extend the visual interest from late autumn into winter. Year round interest is fulfilled by evergreens and the flowers of winter blooming shrubs and perennials. View images of the seasonal plant characteristics available at this link to the KPU Plant Database [New Tab] [2] . Read more about seasonal plant combinations at this link to Gardenia Seasonal Garden Ideas [New Tab] [3] .
Practice: Recognize plants for seasonal interest.


41.E: Osmotic Regulation and Excretion (Exercises)
41.1: Osmoregulation and Osmotic Balance
Review Questions
When a dehydrated human patient needs to be given fluids intravenously, he or she is given:
- water, which is hypotonic with respect to body fluids
- saline at a concentration that is isotonic with respect to body fluids
- glucose because it is a non-electrolyte
- blood
- Answer
-
B
The sodium ion is at the highest concentration in:
- intracellular fluid
- extracellular fluid
- blood plasma
- none of the above
- Answer
-
B
Cells in a hypertonic solution tend to:
- shrink due to water loss
- swell due to water gain
- stay the same size due to water moving into and out of the cell at the same rate
- none of the above
- Answer
-
A
Free Response
Why is excretion important in order to achieve osmotic balance?
- Answer
-
Excretion allows an organism to rid itself of waste molecules that could be toxic if allowed to accumulate. It also allows the organism to keep the amount of water and dissolved solutes in balance.
Why do electrolyte ions move across membranes by active transport?
- Answer
-
Electrolyte ions often require special mechanisms to cross the semi-permeable membranes in the body. Active transport is the movement against a concentration gradient.
41.2: The Kidneys and Osmoregulatory Organs
Review Questions
The macula densa is/are:
- present in the renal medulla.
- dense tissue present in the outer layer of the kidney.
- cells present in the DCT and collecting tubules.
- present in blood capillaries.
- Answer
-
C
The osmolarity of body fluids is maintained at ________.
- 100 mOsm
- 300 mOsm
- 1000 mOsm
- it is not constantly maintained
- Answer
-
B
The gland located at the top of the kidney is the ________ gland.
- adrenal
- pituitary
- thyroid
- thymus
- Answer
-
A
Free Response
Why are the loop of Henle and vasa recta important for the formation of concentrated urine?
- Answer
-
The loop of Henle is part of the renal tubule that loops into the renal medulla. In the loop of Henle, the filtrate exchanges solutes and water with the renal medulla and the vasa recta (the peritubular capillary network). The vasa recta acts as the countercurrent exchanger. The kidneys maintain the osmolality of the rest of the body at a constant 300 mOsm by concentrating the filtrate as it passes through the loop of Henle.
Describe the structure of the kidney.
- Answer
-
Externally, the kidneys are surrounded by three layers. The outermost layer is a tough connective tissue layer called the renal fascia. The second layer is called the perirenal fat capsule, which helps anchor the kidneys in place. The third and innermost layer is the renal capsule. Internally, the kidney has three regions—an outer cortex, a medulla in the middle, and the renal pelvis in the region called the hilum of the kidney, which is the concave part of the “bean” shape.
41.3: Excretion Systems
Review Questions
Active transport of K + in Malpighian tubules ensures that:
- water follows K + to make urine
- osmotic balance is maintained between waste matter and bodily fluids
- both a and b
- neither a nor b
- Answer
-
C
Contractile vacuoles in microorganisms:
- exclusively perform an excretory function
- can perform many functions, one of which is excretion of metabolic wastes
- originate from the cell membrane
- both b and c
- Answer
-
D
Flame cells are primitive excretory organs found in ________.
- arthropods
- annelids
- mammals
- flatworms
- Answer
-
D
Free Response
Why might specialized organs have evolved for excretion of wastes?
- Answer
-
The removal of wastes, which could otherwise be toxic to an organism, is extremely important for survival. Having organs that specialize in this process and that operate separately from other organs provides a measure of safety for the organism.
Explain two different excretory systems other than the kidneys.
- Answer
-
(1) Microorganisms engulf food by endocytosis—the formation of vacuoles by involution of the cell membrane within the cells. The same vacuoles interact and exchange metabolites with the intracellular environment. Cellular wastes are excreted by exocytosis when the vacuoles merge with the cell membrane and excrete wastes into the environment. (2) Flatworms have an excretory system that consists of two tubules. The cells in the tubules are called flame cells; they have a cluster of cilia that propel waste matter down the tubules and out of the body. (3) Annelids have nephridia which have a tubule with cilia. Excretion occurs through a pore called the nephridiopore. Annelids have a system for tubular reabsorption by a capillary network before excretion. (4) Malpighian tubules are found in some species of arthropods. They are usually found in pairs, and the number of tubules varies with the species of insect. Malpighian tubules are convoluted, which increases their surface area, and they are lined with microvilli for reabsorption and maintenance of osmotic balance. Metabolic wastes like uric acid freely diffuse into the tubules. Potassium ion pumps line the tubules, which actively transport out K + ions, and water follows to form urine. Water and electrolytes are reabsorbed when these organisms are faced with low-water environments, and uric acid is excreted as a thick paste or powder. By not dissolving wastes in water, these organisms conserve water.
41.4: Nitrogenous Wastes
Review Questions
BUN is ________.
- blood urea nitrogen
- blood uric acid nitrogen
- an indicator of blood volume
- an indicator of blood pressure
- Answer
-
A
Human beings accumulate ________ before excreting nitrogenous waste.
- nitrogen
- ammonia
- urea
- uric acid
- Answer
-
C
Free Response
In terms of evolution, why might the urea cycle have evolved in organisms?
- Answer
-
It is believed that the urea cycle evolved to adapt to a changing environment when terrestrial life forms evolved. Arid conditions probably led to the evolution of the uric acid pathway as a means of conserving water.
Compare and contrast the formation of urea and uric acid.
- Answer
-
The urea cycle is the primary mechanism by which mammals convert ammonia to urea. Urea is made in the liver and excreted in urine. The urea cycle utilizes five intermediate steps, catalyzed by five different enzymes, to convert ammonia to urea. Birds, reptiles, and insects, on the other hand, convert toxic ammonia to uric acid instead of urea. Conversion of ammonia to uric acid requires more energy and is much more complex than conversion of ammonia to urea.
41.5: Hormonal Control of Osmoregulatory Functions
Review Questions
Renin is made by ________.
- granular cells of the juxtaglomerular apparatus
- the kidneys
- the nephrons
- All of the above.
- Answer
-
A
Patients with Addison's disease ________.
- retain water
- retain salts
- lose salts and water
- have too much aldosterone
- Answer
-
C
Which hormone elicits the “fight or flight” response?
- epinephrine
- mineralcorticoids
- anti-diuretic hormone
- thyroxine
- Answer
-
A
Free Response
Describe how hormones regulate blood pressure, blood volume, and kidney function.
- Answer
-
Hormones are small molecules that act as messengers within the body. Different regions of the nephron bear specialized cells, which have receptors to respond to chemical messengers and hormones. The hormones carry messages to the kidney. These hormonal cues help the kidneys synchronize the osmotic needs of the body. Hormones like epinephrine, norepinephrine, renin-angiotensin, aldosterone, anti-diuretic hormone, and atrial natriuretic peptide help regulate the needs of the body as well as the communication between the different organ systems.
How does the renin-angiotensin-aldosterone mechanism function? Why is it controlled by the kidneys?
- Answer
-
The renin-angiotensin-aldosterone system acts through several steps to produce angiotensin II, which acts to stabilize blood pressure and volume. Thus, the kidneys control blood pressure and volume directly. Renin acts on angiotensinogen, which is made in the liver and converts it to angiotensin I. ACE (angiotensin converting enzyme) converts angiotensin I to angiotensin II. Angiotensin II raises blood pressure by constricting blood vessels. It triggers the release of aldosterone from the adrenal cortex, which in turn stimulates the renal tubules to reabsorb more sodium. Angiotensin II also triggers the release of anti-diuretic hormone from the hypothalamus, which leads to water retention. It acts directly on the nephrons and decreases GFR.


1.3: Terms- Question at Issue
The Question at Issue
Good research begins by finding a Question at Issue . A good Question at Issue is one that:
- Is current
- Reasonable/concerned people can disagree upon
- Can be answered within a narrow frame of time
Finding a current question means looking for something that is still in debate, not a question that was settled years ago. Plenty of issues that were once at the forefront of academic discourse have now faded into the background. No one is really arguing about whether blackboards are a good idea, for instance, and the argument about whether the motorcar will ever replace the horse-drawn carriage is over. While it may be interesting to go back and study settled questions, or to speculate on what would have happened if they had been decided differently, Questions at Issue engage with current topics.
Finding issues about which there is disagreement may seem easy enough, but it can be more challenging than you would think. Remember, we have to look at what the concerned and reasonable people in the argumentative community would consider an actual at-issue question, not whether reports or other parties would like the debate to be renewed. Take, for instance, the issue of Climate Change. Broad scientific consensus points toward humans being the cause. This means that the concerned and knowledgeable parties — our reasonable people — have spoken on this issue. Yet debate still remains in broadcast and online media over whether climate change is real, which might inspire someone to consider “Is climate change real?” or “Do humans cause climate change?” as Questions at Issue. Because the answer to that question — a resounding “yes!” — has been settled by the expert community, someone wanting to research questions about climate change would have to look for other, unsettled, current questions surrounding the topic. They might study, instead, the degree to which the climate is changing, or how news and opinion media have had an impact on public knowledge about the topic.
Finally, a Question at Issue should be narrow enough that the person asking the question can find an answer within their defined time frame. Should specific drugs be decriminalized/legalized? That’s certainly a question that’s at issue in many communities. However, trying to answer that gigantic question would take years of study. Unless you’re ready to sign up for law and medical school right now, consider narrower questions instead: Should my town decriminalize possession of ___ drug? might be a better question to start with.
Figuring out a question that’s at issue isn’t always easy. It’s rarely going to be the first question that comes up about a topic. You may need to conduct some preliminary research just to figure out a good question at issue — and that’s good! Early research and reading can help narrow your topic before you even start writing, and a focused question at issue will guide the rest of your process and project.