Choosing Appropriate Summary Statistics

Start With the Distribution, Not a Favorite Statistic

A summary statistic is useful when it represents the feature of the data you want to describe. A calculator can report a mean, median, and standard deviation for almost any numerical list, but getting a number does not guarantee that it is a good choice. In this tutorial, you will use a distribution’s shape and unusual values to decide which summaries best represent its center and spread.

In Choosing Mean or Median to Describe Center and Choosing IQR or Standard Deviation to Describe Spread, you learned the basic matching rule. This tutorial puts those choices together and practices explaining them. The key is to inspect the distribution first, as in The SOCS Framework for Describing Distributions, then select a center-and-spread pair that suits its features.

Key idea: For a roughly symmetric distribution without strong outliers, the mean and standard deviation are often an appropriate pair. For a skewed distribution or one with outliers, the median and IQR are often more appropriate. If the distribution has distinct clusters, check whether one overall center and spread meaningfully describe the data at all.

Match Center and Spread as a Pair

The mean and standard deviation work together because both respond to how far observations are from the mean. The mean uses every value, and the standard deviation describes typical distance from that mean. When the distribution is approximately symmetric and has no strong outliers, these summaries can give a useful picture of its center and variability.

The median and IQR work together because both are resistant to extreme values. The median marks the middle of the ordered observations, and the IQR measures the width of the middle half. When a distribution is skewed or has an outlier, these summaries usually give a more representative account of the typical observation and the spread of the central observations.

Selection guide:

Approximately symmetric, no strong outliers: Report the mean and standard deviation.
Skewed or with outliers: Report the median and IQR.
Distinct clusters or groups: Describe the distribution’s shape and consider whether separate summaries for meaningful subgroups are needed. A single overall center may not describe either cluster.

This is a guide, not a rule that replaces judgment. “Approximately symmetric” does not require a perfectly mirrored graph, and an observation that looks unusual should be considered in context, as discussed in What Outliers Mean in Context. Also ask what the question means by “typical.” If it asks for an arithmetic average, the mean may directly answer that request, but you should still explain if skew or outliers make it unrepresentative of most observations.

Whatever pair you choose, report the variable, group, and units. A statistic without context is difficult to interpret. For example, “the IQR is 2.5” does not say what varies or what the 2.5 measures; “the middle half of battery lifetimes spans 2.5 months” is more informative.

Worked Examples: Choose and Justify the Summaries

Worked Example: Battery Lifetimes With a Long Right Tail

A fictional sample of eight rechargeable batteries lasted \(4,5,5,6,6,7,8,\) and \(23\) months. The distribution is right-skewed: most lifetimes are between 4 and 8 months, while 23 months is much higher. Choose and calculate a suitable pair of summaries for describing a typical lifetime and spread.

Choice. The unusually long lifetime pulls the mean upward and affects the standard deviation. Because the distribution is right-skewed, choose the median and IQR, which are more resistant. This matches the guidance in the earlier tutorials on choosing center and spread.

Median. There are eight ordered values, so the median is the average of the fourth and fifth values:

\text{median}=\frac{6+6}{2}=6\text{ months}

IQR. The lower half is \(4,5,5,6\), so \(Q_1=(5+5)/2=5\) months. The upper half is \(6,7,8,23\), so \(Q_3=(7+8)/2=7.5\) months. Therefore,

\text{IQR}=Q_3-Q_1=7.5-5=2.5\text{ months}

Check why the choice matters. The mean is \(64/8=8\) months, since the sum is \(4+5+5+6+6+7+8+23=64\). That mean is higher than seven of the eight observations. The sample standard deviation is also influenced by the distant value:

s=\sqrt{\frac{780-\frac{64^2}{8}}{8-1}} =\sqrt{\frac{780-512}{7}} =\sqrt{\frac{268}{7}} \approx6.1875\text{ months}

Here, \(780\) is the sum of squared observations: \(16+25+25+36+36+49+64+529=780\). The calculation illustrates why the mean and standard deviation can be pulled upward by a high observation; it is not necessary to report them as the primary summaries.

Conclusion in context. The median battery lifetime is 6 months, and the middle half of battery lifetimes extends from 5 to 7.5 months, a width of 2.5 months. The median and IQR are appropriate because the distribution is right-skewed and includes a high observation.

Worked Example: Fill Volumes That Are Approximately Symmetric

A fictional sample of eight drink bottles has fill volumes, in milliliters, of \(18,20,22,24,26,28,30,\) and \(32\) milliliters above a target mark. A dotplot shows a roughly balanced pattern around the center, with no strong outliers. Choose and calculate appropriate summaries.

Choice. The distribution is approximately symmetric and has no strong outliers, so use the mean and sample standard deviation. These summaries describe the arithmetic center and typical distance from that center.

Mean. The values sum to \(200\), so

\bar{x}=\frac{200}{8}=25\text{ mL}

Sample standard deviation. The deviations from 25 are \(-7,-5,-3,-1,1,3,5,7\) milliliters. Their squared values sum to \(49+25+9+1+1+9+25+49=168\). Thus,

s=\sqrt{\frac{168}{8-1}} =\sqrt{24} \approx4.8990\text{ mL}

As a check, the sample variance is \(168/7=24\) square milliliters, and its square root is about \(4.8990\) milliliters. The standard deviation is in the same units as the measurements.

Conclusion in context. The mean fill volume is 25 milliliters above the target mark, and a typical bottle’s fill volume differs from that mean by about 4.90 milliliters. The mean and standard deviation are suitable summaries because the distribution is roughly symmetric without strong outliers.

Worked Example: Two Clusters and a Misleading Overall Center

A fictional group of ten students reports travel times to campus, in minutes: \(10,11,12,13,14,30,31,32,33,\) and \(34\). The first five students live nearby; the other five travel from a farther neighborhood. The combined distribution has two distinct clusters and a gap from 15 through 29 minutes. Decide whether one overall center and spread adequately describe the group.

Calculate the overall center and spread. The total is \(220\), so the overall mean is \(220/10=22\) minutes. With ten observations, the median is the average of the fifth and sixth values:

\text{median}=\frac{14+30}{2}=22\text{ minutes}

The lower half is \(10,11,12,13,14\), with median \(Q_1=12\) minutes. The upper half is \(30,31,32,33,34\), with median \(Q_3=32\) minutes. Therefore, the overall IQR is \(32-12=20\) minutes.

Interpret the result. Both the mean and median equal 22 minutes, but no student in this sample has a travel time near 22 minutes. The IQR of 20 minutes indicates substantial spread, yet it does not reveal that the values fall into two separate clusters. An overall center and spread alone would hide an important feature of the distribution.

Use the context. Because the neighborhood groups are known and meaningful, report their summaries separately as well. The nearby students have a median travel time of 12 minutes; the farther-neighborhood students have a median of 32 minutes. These medians come from the middle values of their respective five-student groups.

Conclusion in context. A single overall median of 22 minutes is not a useful description of a typical student’s travel time in this sample, because it falls in the gap between the clusters. Describe the two-cluster shape and report the subgroup medians. If the reason for the clusters were not known, describe the pattern without guessing at its cause.

How to Justify Your Choice

A strong justification names the feature of the distribution and connects that feature to the behavior of the chosen statistics. Avoid giving only a label such as “skewed” or “symmetric.” State what the graph shows and why that matters for the summaries.

Inspect the graph.
Describe the overall shape and check for outliers, gaps, and distinct clusters.

Choose a compatible pair.
Use mean and standard deviation for approximate symmetry without strong outliers; use median and IQR for skew or outliers.

Check whether one pair is enough.
If there are distinct clusters, consider whether a single center falls in a gap or obscures meaningful subgroups.

Report in context.
Name the group and variable, give the chosen summaries and units, and connect the choice to the distribution’s features.

For example, a complete justification could say: “The distribution of battery lifetimes is right-skewed, with most lifetimes between 4 and 8 months and one much higher value. The median and IQR are appropriate because they are resistant to extreme observations.” This gives evidence for the shape, identifies the relevant feature, and explains the choice.

Common Mistakes and AP Exam Tips

Choosing summaries before inspecting shape. A familiar statistic is not automatically the best one. Look at the graph first, then justify the choice with visible evidence.
Pairing statistics that answer different questions. Mean and standard deviation naturally describe data relative to the mean; median and IQR describe the center and width of the middle half. Choose a coherent pair for the distribution.
Claiming a statistic is “wrong.” A mean can be calculated for skewed data, but it may not represent a typical observation well. Explain why another summary is more informative rather than saying the mean is invalid.
Assuming symmetry means exactly mirrored data. AP Statistics uses approximate descriptions of shape. A few small irregularities do not necessarily make mean and standard deviation inappropriate; consider whether there is meaningful skew or a strong outlier.
Reporting a center that hides clusters. A mean or median can land in an empty interval. Describe the clusters and, when group membership is known and relevant, summarize the groups separately rather than treating the pooled center as typical.
Leaving out context and units. “Median 6, IQR 2.5” is incomplete. Identify what is being measured, which group is described, and the units.

For full credit, make the connection explicit: “Because the distribution is [feature], [chosen summaries] are appropriate because [statistical reason].” Then report the summaries in the context of the variable, group, and units. When clusters are present, explain whether one overall pair captures the pattern adequately.

Key takeaway: Inspect the distribution before choosing summaries. Use mean and standard deviation for approximately symmetric data without strong outliers, and median and IQR for skewed data or data with outliers. If distinct clusters make an overall center unrepresentative, describe the clusters and consider meaningful subgroup summaries.

Check Your Understanding

For each situation, choose suitable summaries and justify your choice using the distribution’s features.

A distribution of monthly household water use is strongly right-skewed, with one unusually high observation. Which center-and-spread pair would you report, and why?
A roughly symmetric distribution of package weights has no strong outliers. Which pair is appropriate? Explain what each measure describes.
A group’s values form two clear clusters with a wide empty interval between them. Why might the overall median fail to describe a typical observation?
In the battery example, explain why a median of 6 months and an IQR of 2.5 months are more useful as the main summaries than the mean of 8 months and sample standard deviation of about 6.19 months.
Write one complete sentence justifying the use of mean and standard deviation for a roughly symmetric distribution without strong outliers. Include the variable, group, and units in your sentence.