Find a Weighted Average

Understand the concept

The arithmetic mean is total value divided by the number of observations. If five observations total 90, their mean is 90/5 = 18. The mean is the equal share each observation would receive if the total were redistributed evenly.

When combining groups, rebuild their totals before calculating the new mean. A group of n observations with mean m has total n × m. Add all group totals and divide by the sum of the group sizes. For two groups, the combined mean is (n1 × m1 + n2 × m2)/(n1+n2).

The group sizes act as weights. A mean calculated from thirty observations should influence the combined result three times as much as a mean calculated from ten observations. Taking the simple average of group means silently gives the groups equal weight and is generally wrong when their sizes differ.

An equivalent viewpoint is a balance. The combined mean lies closer to the larger group's mean because more observations support that side. If all weights are positive, the result must lie between the smallest and largest group means. This gives a quick error check, although it does not by itself prove the arithmetic is correct.

Use the appropriate weight for the situation. When averaging scores across students, use student counts. When averaging a price paid per item, use item counts. An average speed problem needs total distance divided by total time; averaging the speed numbers without considering the time spent at each speed can be misleading. Define the total and count, or total and measure, before applying an averaging rule.

Worked example

A group of 12 learners has an average score of 70. Another group of 18 learners has an average score of 80. Find the average score for all 30 learners.

The first group's total score is 12 × 70 = 840. The second group's total is 18 × 80 = 1,440. The combined total is 2,280, so the combined average is 2,280/30 = 76.

The simple average of 70 and 80 would be 75, which gives equal weight to the two groups. The correct result is 76 because the higher-scoring group contains more learners. It lies between 70 and 80 and is closer to 80, as expected.

Practice and answer

1. Four items have an average mass of 15 grams and six items have an average mass of 20 grams. Find the combined average.
Answer: (4 × 15 + 6 × 20)/10 = 180/10 = 18 grams.

2. Ten observations have a mean of 24. One observation equal to 35 is added. Find the new mean.
Answer: the original total is 240. The new total is 275 across 11 observations, so the new mean is 25.

3. When is the simple average of two group means valid?
Answer: when the groups have equal counts, assuming each observation has equal weight.

What you will learn

• Reconstruct a total from a count and an average.
• Choose and apply the correct group weights.
• Check a weighted average against its input values.