What Does It Mean for a Statistic to Be Resistant?
In Finding the Median for Odd and Even Samples, you learned to find the middle observation or observations in ordered data. The median depends on positions in the ordered list. That makes it respond differently from the mean when a value far from the rest of the data is present.
A statistic is resistant if it is not greatly affected by changes to a relatively small number of extreme observations. An extreme observation is unusually far from the rest of the data. Resistance is about how a statistic responds to the data; it does not say whether an extreme value is a mistake, or whether it should be removed.
The median is resistant because its value depends on the middle position or positions, not on how far away the smallest or largest observations are. The IQR, defined in Describing Spread in Context and Five-Number Summary and Boxplot Construction, is \(Q_3-Q_1\). It describes the width of the middle 50% of the data, so a change to an extreme value often leaves the quartiles and IQR unchanged.
By contrast, the mean uses every observation in its arithmetic calculation. A very large or very small value can pull the mean toward itself. Standard deviation describes typical distance from the mean, and its calculation uses squared deviations from the mean. An extreme observation can therefore have a large effect on the standard deviation as well.
Compare the Statistics Before and After a Change
A direct comparison is a useful way to see resistance. Keep all but one observation fixed, change that observation, and recalculate each statistic. For the median and quartiles, first order the data, as you did in the earlier tutorial on finding the median. For the mean and standard deviation, use the formulas from Computing the Mean and Interpreting It and Describing Spread in Context.
Record the mean, median, standard deviation, and IQR for the data as given.
Use a clearly stated change, such as replacing the largest value with a much larger value.
Check which statistics changed substantially and which changed little or not at all.
Connect the results to how each statistic uses the observations: positions for the median and quartiles, and numerical distances for the mean and standard deviation.
Worked Examples: Resistance in Context
Worked Example: Seedling Heights Before an Extreme Change
A fictional gardening group measures seven seedling heights, in centimeters: 4, 5, 5, 6, 7, 8, and 9. Find the mean, median, standard deviation, and IQR. These values will provide a baseline for comparison.
Solution—Find the median and IQR. The data are already in order. The middle, fourth observation is 6 centimeters, so the median is 6 centimeters. The lower half is 4, 5, 5, whose median is \(Q_1=5\). The upper half is 7, 8, 9, whose median is \(Q_3=8\). Thus:
Find the mean. The sum is \(4+5+5+6+7+8+9=44\), and \(n=7\):
Find the sample standard deviation. The sum of the squared observations is \(4^2+5^2+5^2+6^2+7^2+8^2+9^2=296\). Using \(s=\sqrt{\frac{\sum (x-\bar{x})^2}{n-1}}\), an equivalent calculation of the sum of squared deviations is \(\sum x^2-\frac{(\sum x)^2}{n}\):
The sample mean height is about 6.2857 centimeters, and the sample standard deviation is about 1.7995 centimeters. The median is 6 centimeters, and the IQR is 3 centimeters. In the next example, changing just one height will show how these summaries respond differently.
Worked Example: Replacing One Height With an Extreme Value
Suppose the largest seedling height in the previous example is changed from 9 centimeters to 100 centimeters, while the other six heights stay the same. The new ordered data are 4, 5, 5, 6, 7, 8, and 100. Compare the four statistics with their original values.
Solution—Recalculate the median and IQR. The middle, fourth observation is still 6 centimeters. The lower half is still 4, 5, 5, so \(Q_1=5\); the upper half is now 7, 8, 100, so \(Q_3=8\). Therefore:
Both values are unchanged from the original data. The largest observation is much farther from the other heights, but it has not changed the middle observation or either quartile in this example.
Recalculate the mean. The new sum is \(4+5+5+6+7+8+100=135\):
The mean has increased from about 6.2857 to about 19.2857 centimeters. The new sum of squared observations is \(4^2+5^2+5^2+6^2+7^2+8^2+100^2=10215\). The sample standard deviation is:
The standard deviation increased from about 1.7995 to about 35.6170 centimeters. The calculation uses squared distances from the mean, so the very large height has a substantial effect. In this comparison, the median and IQR are resistant, while the mean and standard deviation are non-resistant.
Interpret in context. “After the 9-centimeter height was changed to 100 centimeters, the median height and IQR remained 6 centimeters and 3 centimeters, while the mean and standard deviation increased substantially.” This reports what happened without assuming the changed value is an error or should be removed.
Worked Example: Resistant Does Not Mean Unchanged
A fictional repair shop records the time, in hours, needed to complete seven routine device repairs: 10, 11, 12, 12, 13, 14, and 15. To see whether resistant statistics must stay exactly the same after a change, compare these data with a version in which the largest time is 16 hours rather than 15.
Solution—Find the original median and IQR. The middle observation is 12 hours. The lower half is 10, 11, 12, so \(Q_1=11\). The upper half is 13, 14, 15, so \(Q_3=14\). Thus the original IQR is \(14-11=3\) hours.
Find the original mean and standard deviation. The sum is 87 hours and the sum of squared observations is 1099:
Recalculate after changing 15 to 16. The new ordered data are 10, 11, 12, 12, 13, 14, and 16. The median is still 12 hours, and the lower and upper halves still have medians 11 and 14 hours. The IQR is still \(14-11=3\) hours. The new sum is 88 hours and the new sum of squares is 1130:
The median and IQR did not change, while the mean increased by about 0.1428 hour and the standard deviation increased by about 0.2698 hour. A resistant statistic may change when data change; the point is that it is generally less affected by a few extreme observations than a non-resistant statistic. Here, even the mean and standard deviation changed by much less than they did when 9 was replaced by 100 in the seedling example.
Choosing and Describing Resistant Statistics
Resistance matters when deciding how to summarize a distribution. As discussed in Choosing Mean or Median to Describe Center and Choosing IQR or Standard Deviation to Describe Spread, the median and IQR are often suitable for a skewed distribution or one with outliers. The mean and standard deviation are often useful for an approximately symmetric distribution without strong outliers.
These are guidelines, not rules that make an unusual observation disappear. First describe the distribution and check the context of any extreme value. A value might be a measurement error, a valid but unusual case, or an observation that reveals meaningful variation. The choice of summary should fit the data and the question being answered.
| Statistic | What it summarizes | Resistance to extreme observations |
|---|---|---|
| Mean, \(\bar{x}\) | Center, using the arithmetic average | Non-resistant |
| Median | Center, using the middle position or positions | Resistant |
| Standard deviation, \(s\) | Spread around the mean | Non-resistant |
| IQR | Spread of the middle 50%, from \(Q_1\) to \(Q_3\) | Resistant |
Common Mistakes and AP Exam Tips
- Calling the median or IQR completely unaffected. “Resistant” does not mean “never changes.” Say that these statistics are generally not greatly affected by a few extreme observations, and check the actual values when comparing data sets.
- Calling the mean resistant because it uses all the data. Using every value does not protect the mean from an extreme value. In fact, a very large or small observation can pull the average noticeably.
- Forgetting that standard deviation is non-resistant. Standard deviation is calculated using distances from the mean, including squared deviations. An extreme value can greatly increase the spread it reports.
- Confusing the IQR with the range. The IQR measures the width from \(Q_1\) to \(Q_3\), not the distance from the minimum to the maximum. It does not directly include the extremes.
- Assuming an extreme value should automatically be deleted. A statistic’s sensitivity does not determine whether an observation is valid. Investigate the context and report the data appropriately.
For a full-credit comparison, name the statistics, report what changed, and connect the pattern to resistance. For example: “Replacing the largest height with 100 centimeters left the median at 6 centimeters and the IQR at 3 centimeters, but increased the mean from about 6.2857 to 19.2857 centimeters and the standard deviation from about 1.7995 to 35.6170 centimeters. This illustrates that the median and IQR are resistant, whereas the mean and standard deviation are non-resistant.”
Check Your Understanding
For each question, explain your reasoning in terms of how the statistic uses the observations.
- Which two of the mean, median, IQR, and standard deviation are resistant to a few extreme observations?
- A very large value is added to a data set. Which is more likely to change substantially: the mean or the median? Explain why.
- Why can a single extreme observation have a strong effect on standard deviation?
- Does “resistant” mean a statistic must remain exactly the same after any data change? Explain.
- A distribution of household repair costs is right-skewed and has an unusually large cost. Which measures of center and spread would usually be more appropriate to report: mean and standard deviation, or median and IQR? Give a reason.