Question type

How mean and median change on the SAT

The mean uses every value, so an outlier pulls it strongly; the median is the middle value, so it barely moves. To find a missing value from a mean, use totals: mean × count = sum. If 6 values average 14 and 7 values average 15, the new value is $105 - 84 = 21$ .

These questions change a data set, by adding, removing or replacing a value, and ask what happens to the mean and the median, or give a mean and ask for a missing value.

It appeared on three of the five official practice tests we analysed. Most of these questions can be answered without computing anything exactly.

On 3 of 5 official practice tests
24 practice questions of this type
Desmos helps

Why students miss it

Assuming the median never moves. Removing a value always shifts which value is in the middle, even if only slightly.

Averaging averages. The new value is found from totals, not by comparing 14 and 15.

Overthinking. "The mean decreases more than the median" is often visible without a calculation.

How to spot it

Two data sets that differ in one value, often an extreme one.
"A value is added", "removed" or "changed".
A given mean and a question about one missing value.

The method

Mean and median

Ask what the change does to the total, and to the middle position.

An extreme value changes: the mean moves a lot, the median little or not at all.
A value is removed: the middle shifts by half a position.
A missing value: new total − old total.

With Desmos

Desmos computes both statistics from a list in one line each.

Type L=[11,14,14,17,20,23,45], then mean(L) and median(L).
Delete the 45 from the list and watch both values change.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#437Data

The mean of a data set of 6 values is 14. When a seventh value is added to the data set, the mean of the 7 values is 15. What is the seventh value?

Pick an answer, then check it.

Show the answer and explanation

Work with sums. The first six values total

6 \times 14 = 84

. After adding the seventh, the total must be

7 \times 15 = 105

. So the seventh value is

105 - 84 = 21

2Question 2 of 2#448Data

A data set consists of the values 11, 14, 14, 17, 20, 23, and 45. The value 45 is removed from the data set. Which statement correctly describes the resulting changes?

Pick an answer, then check it.

Show the answer and explanation

Removing an outlier hits the mean much harder than the median. Mean goes from

144/7 \approx 20.6

99/6 = 16.5

, a drop of about

4.1

. Median goes from

17

(middle of seven values) to

(14 + 17) /2 = 15.5

, a drop of

1.5

. So the mean falls by more.

Questions students ask

Does an outlier affect the mean or the median more?

The mean. It adds the whole outlier into the total, while the median only depends on which value is in the middle.

How do you find a missing value from the mean?

Multiply the mean by the number of values to get the total, do the same for the other set, and subtract.

Practise this type until it is automatic

Five more mean-and-median questions, checked instantly, with every wrong choice explained. Free, no card.

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