Comparing How Much Two Groups Vary
In Comparing Centers Using Medians and Means, you learned to compare the same measure of center for two groups and describe the difference in context. The same principle applies to spread: compare IQR with IQR or standard deviation with standard deviation. A larger value for a measure means more spread according to that measure.
The choice of measure matters. As discussed in Choosing IQR or Standard Deviation to Describe Spread, the IQR is often more useful for skewed distributions or distributions with outliers, while standard deviation is often useful for approximately symmetric distributions without strong outliers. Once you choose a suitable measure, use that same measure for both groups.
A spread comparison describes a feature of the distributions. It does not mean that every observation in the group with the larger spread is farther from its center, or that the groups do not overlap. The comparison is about the summaries, not a claim about every individual value.
Compare Matching Measures
The IQR is the distance from the first quartile, \(Q_1\), to the third quartile, \(Q_3\). It describes the width of the middle half of the ordered observations. Standard deviation describes a typical distance of observations from their mean. These measures describe spread in different ways, so an IQR for one group should not be directly subtracted from a standard deviation for another.
For either matching measure, subtract one group’s value from the other’s. For example, if comparing Group B with Group A:
A positive difference means Group B has the larger value for that measure. A negative result means Group A’s value is larger. Either way, explain the direction in words and use the units of the original variable. The difference between two IQRs or two standard deviations has the same units as the data.
Check that both summaries refer to the same quantitative variable and use the same units.
Use the distributions’ shape and unusual values to guide whether the IQR or standard deviation is more informative.
Subtract IQRs from IQRs or standard deviations from standard deviations.
Name the measure, state which group has the larger spread by that measure, and include the variable and units.
The IQR and standard deviation are both nonnegative and use the same units as the observations, but they do not summarize spread in the same way. A larger IQR means a wider middle half. A larger standard deviation means observations are typically farther from their mean. Neither measure describes the full span of values, and neither implies that all observations lie a fixed distance from the center.
Worked Example: Comparing IQRs of Daily Sunlight
Worked Example: Comparing IQRs of Daily Sunlight
A fictional environmental team summarizes daily hours of sunlight at two monitoring sites. Site A has \(Q_1=5\) hours and \(Q_3=8\) hours. Site B has \(Q_1=4\) hours and \(Q_3=11\) hours. Compare the sites’ spreads using the IQR.
Calculate each IQR. The IQR is \(Q_3-Q_1\). For Site A:
For Site B:
Find the difference. Subtract Site A’s IQR from Site B’s:
Compare in context. The IQR of daily sunlight at Site B is 4 hours greater than the IQR at Site A. In other words, the middle half of daily sunlight measurements has a width of 7 hours at Site B, compared with 3 hours at Site A.
This comparison describes the widths of the middle halves; it does not say that every day at Site B differs more from its center than every day at Site A. Nor does it describe the full range of sunlight measurements.
Comparing Standard Deviations
When standard deviation is the suitable measure, compare the sample standard deviations, \(s\), or compare population standard deviations if the summaries describe entire populations. Do not mix the two types without a reason. As covered in Standard Deviation Using a Calculator, calculator output labels \(Sx\) as the sample standard deviation and \(\sigma x\) as the population standard deviation.
For raw data, calculate each group’s standard deviation from its own observations. Then subtract the two values and interpret the difference as a difference between typical distances from the means. The calculation does not mean that the group with the larger standard deviation has every observation farther from its mean.
Worked Example: Comparing Sample Standard Deviations of Lap Times
Worked Example: Comparing Sample Standard Deviations of Lap Times
Two fictional groups of cyclists record lap times, in seconds, during practice. The observations are:
| Group A (seconds) | Group B (seconds) |
|---|---|
| 9, 11, 13, 15, 17, 19 | 4, 8, 12, 16, 20, 24 |
Suppose the groups’ distributions are approximately symmetric without strong outliers. Compare their variability using the sample standard deviations.
State. We will compare the sample standard deviations of lap times for Groups A and B. Since both groups have the same quantitative variable and units, a difference in their standard deviations can be interpreted in seconds.
Plan. Standard deviation is a suitable measure here because the distributions are described as approximately symmetric without strong outliers. We will calculate each sample standard deviation using the sample formula, which divides the sum of squared deviations from the sample mean by \(n-1\), and then take the square root. Both groups have six observations, so each calculation uses \(n-1=5\).
Do: Group A. The mean lap time is:
The squared deviations from 14 are \(25, 9, 1, 1, 9,\) and \(25\), whose sum is 70. Therefore:
Do: Group B. The mean lap time is:
The squared deviations from 14 are \(100, 36, 4, 4, 36,\) and \(100\), whose sum is 280. Thus:
Subtract Group A’s standard deviation from Group B’s:
Using the exact expressions gives \(\sqrt{56}-\sqrt{14}=\sqrt{14}\approx3.7417\) seconds; the small difference in the subtraction above is due to rounding the displayed values first. We can report the difference as approximately 3.74 seconds.
Conclude. Group B’s sample standard deviation for lap times is about 3.74 seconds greater than Group A’s. This indicates that Group B’s lap times are typically farther from their mean lap time than Group A’s, according to the sample standard deviation. It does not mean each cyclist’s time is farther from the mean.
Choosing a Measure Before Comparing
Sometimes you are given more than one spread summary for each group. In that case, use the distribution’s shape and unusual features to choose a useful measure, as in Choosing Appropriate Summary Statistics. For example, if both groups are skewed or have outliers, the IQR may provide a more representative comparison of their middle halves. If both are approximately symmetric without strong outliers, standard deviations may be useful.
When the two groups have different shapes, be especially careful not to select a measure just because it produces a larger or smaller difference. Explain the measure you use and why it is informative. If you report both IQRs and standard deviations, label each comparison separately. Do not subtract an IQR from a standard deviation: although both are measured in the original units, they summarize spread differently.
Worked Example: Choosing IQRs for Skewed Water Use
Worked Example: Choosing IQRs for Skewed Water Use
A fictional town compares household water use in two neighborhoods, measured in hundreds of liters per day. Both distributions are right-skewed, and each has a few unusually high-use households. Neighborhood A has an IQR of 5 units and a standard deviation of 11 units. Neighborhood B has an IQR of 8 units and a standard deviation of 16 units. Compare the spreads using a suitable measure.
Choose the measure. The distributions are right-skewed and have unusually high values. As discussed in Choosing IQR or Standard Deviation to Describe Spread, the IQR is generally more resistant to such values, so it is a suitable measure for comparing the middle halves.
Calculate the difference. Neighborhood B’s IQR minus Neighborhood A’s IQR is:
Compare in context. The IQR of household water use in Neighborhood B is 3 hundreds of liters per day greater than in Neighborhood A. Thus, the middle half of the observed household water-use values is wider in Neighborhood B by 300 liters per day.
The standard deviations could also be compared with each other, giving \(16-11=5\) hundreds of liters per day, but they are more affected by the high values in these right-skewed distributions. The IQR comparison is the more useful one for describing the spread of the middle half. We do not compare Neighborhood B’s IQR of 8 directly with Neighborhood A’s standard deviation of 11.
Common Mistakes and AP Exam Tips
- Comparing unlike measures. Subtracting one group’s IQR from another group’s standard deviation does not give a useful spread difference. Compare IQR with IQR or standard deviation with standard deviation.
- Reporting the numbers without the direction. “The IQRs are 3 and 7 hours” does not say which group is more variable. State that the group with the IQR of 7 hours has an IQR 4 hours greater.
- Leaving out the variable or units. “Group B’s spread is 4 greater” is incomplete. Identify the measure, variable, groups, and units.
- Calling an IQR difference a difference in ranges. The IQR describes the width of the middle half, not the full span from the minimum to the maximum.
- Interpreting standard deviation as a guaranteed distance. Standard deviation describes a typical distance from the mean; it does not say every observation is within one standard deviation of the mean.
- Claiming that one group’s values are all more spread out. A larger IQR or standard deviation is a comparison of summaries, not proof that every observation in one group is farther from its center.
- Ignoring shape and unusual values. A calculation alone does not establish that the measure is the most useful one. Briefly connect the choice of IQR or standard deviation to the distributions.
For a full-credit AP response, name the groups and variable, identify the matching spread measure, report the difference with units, and state which group has the greater spread according to that measure. When the choice of measure matters, briefly justify it using the distributions’ shape or unusual values. Keep the conclusion descriptive and tied to the groups summarized.
Check Your Understanding
For each item, identify the matching measure, calculate or describe the difference, and interpret it in context.
- Group A’s IQR for weekly reading time is 3 hours, and Group B’s is 7 hours. Which group has the greater IQR, and by how many hours?
- Two garden plots have sample standard deviations of plant height of 4 centimeters and 6 centimeters. State the difference and identify which plot’s heights typically lie farther from their mean, according to standard deviation.
- One group’s IQR is 5 minutes and another group’s standard deviation is 9 minutes. Can you subtract these values to compare their spreads directly? Explain.
- Two distributions are right-skewed and contain unusually high observations. Which measure is often more useful for comparing their spread, IQR or standard deviation? Give a brief reason.
- A group has a larger standard deviation than another group. Does that establish that every observation in the first group is farther from its mean? Explain what the comparison supports.