QUANTITATIVE REASONING · GMAT-Q-STAT

Statistics: Median, Range and Spread

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

Core mathematical relationship

x ˉ = \frac{\sum x _{i}}{n}

Translate the condition

A mean is a total divided by a count. Convert subgroup means back to totals before combining groups; sort observations before finding a median.

Test the result

Check the combined total and combined count. A combined mean must lie between the subgroup means when all subgroup sizes are positive.

Think Like a GMAT Test-Taker

Convert each group mean to a total, or sort the values, before combining information.

Faster Methods

Recover group totals with mean × count.

For an even-sized data set, average the two middle ordered values.

Worked example 1

The three observations are 10, two greater, and four greater. What is their mean?

x ˉ = \frac{10 + 12 + 14}{3} = 12

The observations 10, 12, and 14 are symmetric around 12, so their mean is (10 + 12 + 14) divided by 3 = 12.

Worked example 2

Four observations have mean 18. After a fifth observation is added, the mean is 20. Find the added value.

x = 5 (20) - 4 (18) = 100 - 72 = 28

Convert each mean to a total before comparing. The difference between the new and original totals is the added observation. Check by including twenty-eight in the original total and dividing by five.

Worked example 3

A set has mean 8 and range 5. Every value is doubled and then increased by 3. Find the new mean and range.

y ˉ = 2 (8) + 3 = 19, R_{y} = 2 (5) = 10

Doubling multiplies all distances between values by two. Adding three shifts every value equally, so it changes the mean but leaves the range unchanged. Keep the transformation of location separate from the transformation of spread.

Common Traps

Do not take the unweighted mean of group means when group sizes differ.
Do not use the mean position to find a median.