EQN-05-01

Mean, Median, Mode & Range

The four summary statistics, what each one really measures, and the reverse problems the GAT-E loves.

~5 min read•Difficulty 2/5

Mean (average) $=$ sum of values $\div$ number of values.
Median $=$ the middle value when the data are sorted (average of the two middle values if the count is even).
Mode $=$ the value that appears most often.
Range $=$ largest $-$ smallest.

sum = mean \times count

This one rearrangement solves almost every "average" word problem — including missing-value and new-average questions.

The median is only meaningful on sorted data. A frequent error is reading the middle of the list as printed.

Adding a value equal to the current mean leaves the mean unchanged.
The median resists outliers; the mean is pulled toward them. A single huge value can make mean ≫ median.

Diagram

The median is the middle value once the data is in order.

Worked examples

Example 1 (difficulty 2/5)

The mean of 4 numbers is 15. Three of them are 10, 12, and 20. What is the fourth?

◎ Predict the answer first:

Example 2 (difficulty 1/5)

Find the median of: 7, 2, 9, 4, 5.

◎ Predict the answer first:

Write "sum $=$ mean $\times$ count" at the top — it converts average problems into simple arithmetic.
Sort the list before finding the median, even if it looks sorted.
If the mean is much larger than the median, suspect a large outlier in the data.

Taking the median from the unsorted list

Why: The "middle of the list as written" feels like the middle value.

Fix: Sort the values first, then pick the middle.

Confusing range with the largest value

Why: "Range" sounds like the extent or biggest number.

Fix: Range

=

maximum

-

minimum, a single difference.