How an outlier affects the mean and median

An outlier is an observation unusually far from the rest of a data set. It can change a summary statistic, but not every statistic changes in the same way. To predict the effect, ask whether the measure uses the value's size or only its position in the ordered data.

Why the mean moves

The mean is the sum of all observations divided by the number of observations. A very high or low value changes the total, so it can shift the mean substantially. The more extreme the value and the smaller the data set, the larger its possible effect.

Why the median is more resistant

The median is determined by the center position after the values are ordered. An extreme observation at one end may leave that center position unchanged. The median can still change if enough observations move or if a data set is small enough that an added or replaced value changes the middle positions. “Resistant” means less affected by extremes, not immune to every change.

What happens to other measures

Choose a summary that fits the data

For a roughly symmetric distribution without influential outliers, the mean and standard deviation can summarize center and spread well. For a skewed distribution or one with extreme values, the median and interquartile range are often more representative. The choice depends on what the question asks and what the data represent; an outlier should not be removed just because it changes the answer.

Investigate before discarding an observation

An unusual value may be a recording error, a measurement problem, a member of a distinct subgroup, or a valid rare event. Check the original data and context. If the value is valid, explain how it affects the summary. If a documented error is corrected or a value is excluded under a stated method, report that decision rather than silently deleting it.

Do not confuse detection rules

Different disciplines use different rules to flag potential outliers, such as plots, standardized scores, or an interquartile-range rule. A rule marks a value for examination; it does not prove the observation is invalid. Use the criterion supplied by the question. A box plot can show potential outliers as separate points under a particular convention, but another graph may use different labeling.

How to write an interpretation

Name the statistic and its sensitivity: “The high observation raises the mean much more than the median, so the median better represents the center of this skewed sample.” If you compare two data sets, note whether their means differ because of a change in most values or because of one extreme point. Avoid saying an outlier always makes the mean larger; a low outlier can pull it downward.

For FTCE questions, calculate the requested measure, then describe how an extreme value affects it. The mean responds to magnitude; the median responds to rank. Range and standard deviation usually reflect extremes more than the interquartile range does.

An extreme value pulls the mean

Because the mean includes every observation, an outlier can shift it substantially. For 4, 5, 5, 6, 7, the mean is 5.4 and median is 5. Replace 7 with 47 and the mean becomes 13.4, while the median remains 5. The one extreme value changes the mean much more because it adds a large amount to the total without changing the middle position.

An outlier does not always move the mean in the same direction: a very low value pulls it downward and a very high value pulls it upward. The size of the effect depends on the value’s distance from the rest of the data and on how many observations are present. Check whether the outlier is an error or a valid extreme before deciding how to treat it.

The median depends on position

The median is based on the ordered middle value or middle pair, so an extreme observation often has little effect. If a new outlier is added, however, the number of observations changes and the middle position may shift. With an even-sized data set, adding one value makes the sample odd-sized, so the median becomes a single central observation. Recalculate rather than assuming it is unchanged.

The range is especially sensitive to an extreme value because it uses only the minimum and maximum. In the example, the range changes from 7 − 4 = 3 to 47 − 4 = 43. The interquartile range, based on the middle half of the ordered data, is generally less affected by an isolated extreme.

Choose a summary that reflects the distribution

For a roughly symmetric distribution without extreme values, the mean and median may be similar. For a skewed distribution or one with outliers, the median often better represents a typical observation. The mean remains useful when every value should contribute to a total or when later statistical calculations require it. Report the measure that answers the question, not simply the one least affected by an unusual value.

Do not remove an outlier solely because it changes the result. Verify the measurement, check the population and collection method, and explain any exclusion. A valid extreme case may be important to the question being studied. If both mean and median tell different stories, reporting both can make the distribution’s shape visible.

• The mean moves toward an extreme value.
• The median depends on the middle position and may be more resistant.
• The range changes directly when a minimum or maximum changes.
• Recalculate the median if the sample size changes.
• Investigate an outlier before excluding it; it may be a valid case.