QUANTITATIVE REASONING · GMAT-Q-AV-01

Averages and Weighted Means

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

Concepts and problem-solving strategy

x = \frac{\sum x _{i}}{n}

x = n x - (n - 1) x_{known}

x = \frac{n _{1} x _{1} + n _{2} x _{2}}{n _{1} + n _{2}}

a x + b = a x + b

Convert each average to a total, then combine sums and counts.

Use total = average times count before removing or adding an observation.

Five numbers have mean 12. Find their sum.

5 \cdot 12 = 60

Multiply count by mean.

Groups of 2 and 3 have means 10 and 20. Find combined mean.

\frac{2 ( 10 ) + 3 ( 20 )}{5} = 16

Weight means by group sizes.

Five values average 14. Four of them average 11. Find the fifth value.

5 (14) - 4 (11) = 26

Recover the two totals before subtracting.

Four observations average 6. Each value is doubled and then increased by 3. What is the new mean?

2 (6) + 3 = 15

Apply the linear change directly to the mean.

A data set of 20 observations has mean 48. Five observations whose mean is 60 are removed. What is the mean of the remaining observations?

\frac{20 ( 48 ) - 5 ( 60 )}{15} = 44

Recover the full total, subtract the removed subgroup total, then divide by the 15 remaining observations.

The mean of 9 test scores is 72. A score of 54 is replaced by 81. What is the new mean?

72 + \frac{81 - 54}{9} = 75

The total increases by 27 while the number of scores remains 9, so the mean increases by 27/9=3.