ACT Mean and Median: What Changes When a Value Is Extreme?

An extreme value can pull the mean toward it because the mean uses every value in the total. The median depends on the middle position or positions after sorting. To compare a dataset before and after a change, update the total and count for the mean, then sort again for the median. A resistant median can still change.

ACT’s mathematics description includes averages, medians, and describing the center of distributions. The examples below are original Kibo practice. The fictional measurements are supplied for these exercises; they are not official ACT items or student performance data.

Start with the total, count, and middle positions

OpenStax’s measures-of-center lesson defines the arithmetic mean as the total divided by the count. For the median, sort from smallest to largest. Use the single middle value for an odd count, or average the two middle values for an even count. Keep repeated values in the list.

$$\text{mean}=\frac{\text{sum of all values}}{\text{number of values}}$$

For five values, the median is the third sorted value. For six, it is the average of the third and fourth. These are positions, not the numbers to report as your answer.

Worked example: increase the largest value

Five print jobs take $6, 7, 8, 9, 10$ minutes. Suppose the last duration is corrected from $10$ to $40$ minutes. How do the mean and median change?

The total rises by $40-10=30$ minutes. Spread across five jobs, that raises the mean by $30/5=6$ minutes, from $8$ to $14$. The third sorted value remains $8$, so the median stays at $8$ minutes.

In the corrected data, $40$ minutes is far from the other durations of $6$ through $9$ minutes. The mean reflects that long job even though four of the five jobs finish below the mean. The NIST explanation of measures of location describes why extreme values affect the mean more strongly than the median. Here the median describes the middle job, while the mean lets you recover the total time across all five jobs.

A shortcut when one value is replaced

Suppose a dataset contains $n$ values, with $n>0$, and one value $a$ is replaced by $b$. All other values and the count stay fixed. The new total is the old total plus $b-a$, so:

$$\text{new mean}=\text{old mean}+\frac{b-a}{n}$$

For the print jobs, this gives $8+(40-10)/5=14$. If the replacement is smaller, then $b-a$ is negative and the mean falls. The change in one observation is divided among all $n$ observations when calculating the mean.

This shortcut applies to a replacement. Adding or removing an observation also changes the count; rebuild the total and divide by the new count in those cases. Our guide to missing-value average problems works through that total-and-count method.

Why the median can change too

Return to the original durations $6, 7, 8, 9, 10$. This time replace $6$ with $40$, keeping the other four values. The sorted list is now:

$$7, 8, 9, 10, 40$$

The third value is now $9$, so the median rises from $8$ to $9$. The new total is $40-6+40=74$, giving a mean of $74/5=14.8$. The old smallest observation moved to the end, changing which observation occupies the middle position.

Once this moved value is above $10$, increasing it further leaves the median at $9$, while the mean continues to rise. Resistance describes this limited sensitivity to the size of a value in the tail. It does not mean that every possible edit leaves the median unchanged.

Removing an extreme value changes the group

Start with the corrected list $6, 7, 8, 9, 40$. If a problem explicitly removes the $40$-minute job, four observations remain: $6, 7, 8, 9$. The remaining mean is $(70-40)/4=7.5$ minutes. The median now averages the two middle values: $(7+8)/2=7.5$ minutes.

Removing that job changes both statistics: the mean falls from $14$ to $7.5$, and the median falls from $8$ to $7.5$. Replacing $40$ with $10$ instead would restore five observations and the original mean and median of $8$.

For a question about the supplied dataset, include every observation unless the prompt changes the group. An unusually large value is not permission to delete it. Follow the requested operation and report the statistic asked for, even when the mean and median describe the group differently.

Try three fresh problems

1. The values are $4, 7, 9, 11, 14$. Replace $14$ with $39$. Find the original and new mean and median.
2. The values are $3, 6, 10, 12, 15, 20$. Replace $3$ with $33$. Find the new mean and median, and compare each with its original value.
3. Eight observations have an exact mean of $18$. One observation changes from $52$ to $20$, while the other seven remain fixed. Find the new mean. Is there enough information to calculate the new median?

Check these mistakes before choosing an answer

• Using the unsorted middle entry. Sort the entire changed list before locating the middle.
• Forgetting the second middle value. With an even count, average both middle values.
• Applying the whole change to the mean. For a replacement, divide the change in the observation by the unchanged count.
• Assuming the median is fixed. Track the middle positions after the stated change, including any change to the count.

If a practice question went wrong, record whether the issue was the total, the count, or the sorted positions in our practice error log. Then try a different dataset with the answer covered.

Try a free ACT practice question with Kibo.

Prepared with AI assistance. Original examples were checked with separate exact calculations; no human review is claimed. Sources checked November 10, 2026. ACT is a registered trademark of ACT, Inc., which is not affiliated with or endorsing this guide.