Why the Choice of Summary Matters
A single measure of center can make a data set sound quite different from the way its values are actually distributed. For example, a few unusually long waiting times can pull the mean upward even when most waiting times are much shorter. A center summary is most useful when it represents the pattern a reader sees in the data, rather than being dominated by an unusual observation.
The shape of a distribution and its sensitivity to extreme observations therefore guide the choice of summary. A roughly symmetric distribution without prominent outliers is commonly summarized by its mean and standard deviation. A skewed distribution or one with extreme observations is commonly summarized by its median and interquartile range. These are conventional pairings: each center is matched with a measure of spread that responds to unusual values in a similar way.
“Resistant” does not mean completely unaffected. It means that a measure usually changes less than a sensitive measure when one or a few observations are made unusually large or small. The median depends on the ordered positions of observations, and the IQR depends on the positions of the quartiles. The mean, by contrast, uses the numerical value of every observation. The standard deviation is also affected by how far observations lie from the mean, so an extreme value can influence it substantially.
Reading Shape Before Choosing
A graph, such as a dotplot, histogram, or boxplot, helps reveal the features that matter for this decision. A roughly symmetric distribution has a similar spread on either side of its center, without a long tail in just one direction. A skewed distribution has a longer tail on one side. An outlier or other unusually extreme observation may also be visible. A graph does not make the choice automatically, but it gives evidence about whether the mean or median is likely to describe a typical observation well.
For a roughly symmetric distribution without prominent outliers, the mean often describes the balance point of the observations. Standard deviation is its conventional partner because it describes spread around that same mean. When the distribution is substantially skewed or contains an extreme value, the median often better describes the middle of the data, while the IQR describes the spread without being driven by the most extreme observations.
These are guidelines, not rules that prohibit reporting other summaries. A report can include both the mean and median, for instance, when it is useful to show how strongly they differ. But when choosing a main pair to represent a distribution, match the center and spread to the shape and sensitivity of the data.
Worked Example: A Roughly Symmetric Set of Measurements
Suppose a hypothetical set of six trail sections has the following measured lengths, in kilometers: \(4, 6, 7, 9, 10, 12\). The values are approximately balanced around the middle, and none is isolated far from the others. Choose a suitable pair of summaries.
Mean. The total length is \(4+6+7+9+10+12=48\) km, so
Median. There are six ordered observations, so the median is the average of the third and fourth values:
For context, the lower half is \(4,6,7\), whose median is \(Q_1=6\) km. The upper half is \(9,10,12\), whose median is \(Q_3=10\) km. Thus \(IQR=10-6=4\) km. The mean and median agree, which is consistent with the roughly balanced values.
The data have no evident extreme observation and are approximately symmetric, so the conventional choice is the mean, \(8\) km, paired with the standard deviation \(s\) as the measure of spread. The IQR is also a valid spread summary, but the mean-and-standard-deviation pair is a natural description for this shape. No numerical value of \(s\) is needed to make the choice.
Why the Two Pairs Behave Differently
The mean’s sensitivity to a changed extreme value can be seen directly from its formula. Suppose the largest observation \(x_n\) is increased by \(c\), while the other \(n-1\) observations stay fixed and the ordering does not change. The sum increases by \(c\), so the mean increases by \(c/n\):
This shows that the changed value contributes directly to the mean. In contrast, if the extreme value remains at the end of the ordered list and the middle positions and quartile positions do not change, the median and quartiles can stay the same. That is why the median and IQR are less sensitive to an extreme value in many data sets. This is a justification for their resistance, not a guarantee that they never change.
The same reasoning helps explain why the spread measures are paired as they are. Standard deviation describes spread in relation to the mean, and large deviations from the mean can have a substantial influence on it. The IQR is based on the first and third quartiles, so an extreme value beyond those quartile positions often has little effect on it. Pairing measures with similar sensitivity helps give a coherent account of both center and spread.
Worked Example: A Right-Skewed Set with an Extreme Value
Imagine that a hypothetical clinic records these waiting times, in minutes, for six appointments: \(2, 3, 3, 4, 5, 23\). Most waits are short, while one is much longer. Compare the center and spread summaries and choose a main pair.
Mean. The total is \(2+3+3+4+5+23=40\) minutes. Therefore,
Median. The middle two observations are 3 and 4, so
Using the median-of-halves convention, the lower half is \(2,3,3\), giving \(Q_1=3\) minutes. The upper half is \(4,5,23\), giving \(Q_3=5\) minutes. Hence
The mean is higher than the median because the 23-minute wait pulls the arithmetic average upward. The distribution has a long right tail, so the median, \(3.5\) minutes, is a more representative description of the middle appointment than the mean, \(6.667\) minutes. The median paired with the IQR of \(2\) minutes is the suitable main summary. Both are in minutes, and the IQR describes the spread of the middle half rather than the full range of waits.
Choosing Summaries When Comparing Groups
The same decision should be made for each group being described. If the groups have similar shapes and no prominent extremes, means and standard deviations can support a clear comparison. If one group is strongly skewed or has an extreme observation, medians and IQRs are often more informative. When the goal is to compare groups directly, it is usually clearest to report the same type of center and spread for each group, selected with the group shapes in mind.
Worked Example: Choosing Summaries for Two Waiting-Time Groups
Consider two hypothetical sets of appointment waiting times, both in minutes. Group A has values \(12,14,15,16,17,18,19,20\). Group B has values \(5,6,7,8,9,10,11,36\). Choose summaries that describe each group and make their centers comparable.
Group A. Its sum is \(131\), so its mean is
Its median is the average of the fourth and fifth observations: \((16+17)/2=16.5\) minutes. The lower half is \(12,14,15,16\), so \(Q_1=(14+15)/2=14.5\). The upper half is \(17,18,19,20\), so \(Q_3=(18+19)/2=18.5\). Therefore, \(IQR=18.5-14.5=4\) minutes. The values are fairly balanced, so mean and standard deviation would be a reasonable pair for Group A.
Group B. Its sum is \(92\), giving
Its median is \((8+9)/2=8.5\) minutes. The lower half is \(5,6,7,8\), giving \(Q_1=(6+7)/2=6.5\). The upper half is \(9,10,11,36\), giving \(Q_3=(10+11)/2=10.5\). Thus \(IQR=10.5-6.5=4\) minutes. The value 36 creates a long right tail, so the median and IQR are more resistant summaries for Group B.
For a direct comparison between groups, reporting medians and IQRs for both gives a consistent, resistant comparison: the medians are 16.5 and 8.5 minutes, and both IQRs are 4 minutes. Group A’s mean is also close to its median, while Group B’s mean is pulled upward by its extreme wait. Mixing the mean for one group with the median for the other would make the comparison harder to interpret because the two reported centers would be defined differently.
A Practical Decision Process
A reliable choice can be made by following a short sequence. First, inspect the distribution’s shape and look for unusually extreme observations. Next, decide whether those features could distort a mean-based description. Finally, select a center and spread that respond appropriately, and state both in the units of the variable.
Use a suitable graph and note whether the data are roughly symmetric, skewed, or marked by extreme observations.
For a roughly symmetric distribution without prominent extremes, consider the mean. For substantial skew or extreme values, consider the median.
Pair the mean with standard deviation, or pair the median with IQR, according to the distribution’s shape and sensitivity.
Name the distribution feature that matters and report summaries with the variable’s units.
Common Mistakes
- Choosing by habit. Always reporting the mean can hide the fact that an extreme observation pulls it away from where most values lie. Inspect the shape before selecting a main summary.
- Pairing measures with mismatched sensitivity. Reporting a median with a standard deviation, or a mean with an IQR, is not automatically invalid, but it may combine measures that respond differently to extremes. The conventional pairs are mean with standard deviation and median with IQR.
- Treating resistance as immunity. A median or IQR can change if observations near the middle or quartile positions change. Resistance means less sensitivity to many extreme values, not no sensitivity at all.
- Comparing unlike centers. The mean and median answer different summary questions. If comparing groups, use the same kind of center for each group unless there is a clear reason to show both.
Check Your Understanding
For each question, consider both the shape of the data and the sensitivity of the summaries.
- A roughly symmetric distribution of package weights has no prominent outliers. Which center-and-spread pair is a conventional choice, and why?
- A small number of unusually large repair costs make a distribution strongly right-skewed. Which center and spread would usually be more resistant?
- If the largest observation increases while remaining the largest, what happens to the mean when the increase is \(c\) and the sample size is \(n\)? Under what condition might the median remain unchanged?
- Why can comparing one group’s mean with another group’s median be difficult to interpret?
- A data set is roughly symmetric, but its mean and median differ somewhat. Does that fact alone prove that the data are strongly skewed? Explain what else you would inspect.