Mean and median with frequencies

What is the average family size in a class? Ask everyone, add up, and divide. But data often comes as a frequency table (a table that says how many times each value occurs), and that is where a common mistake happens.

Here is one:

The tempting answer is (3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)/8 = 52/8 = 6.5. It is wrong, and by a lot.

Why? An average adds all the values in the data, and there are not 8 values here but 36 families. The eleven families of size 4 must be counted eleven times, not once.

Do it properly. Multiply each size by its frequency and add:

(3 × 3) + (4 × 11) + (5 × 9) + (6 × 7) + (7 × 3) + (8 × 1) + (9 × 1) + (10 × 1) = 9 + 44 + 45 + 42 + 21 + 8 + 9 + 10 = 188.

The count is 3 + 11 + 9 + 7 + 3 + 1 + 1 + 1 = 36. So the average is 188/36 = 5.22. That is well below 6.5, because the wrong method gave the few large families as much weight as the many small ones.

The median can also be read from the table. With 36 values, the median is the average of the 18th and 19th values in sorted order. Do we have to write out all 36? No.

Add up the frequencies from the smallest value, and see which positions each value fills:

• Size 3 fills positions 1 to 3.
• Size 4 fills positions 4 to 14 (3 + 11 = 14).
• Size 5 fills positions 15 to 23 (3 + 11 + 9 = 23).

Positions 18 and 19 both fall inside the 5s, so the median is 5. (Always find the run that contains the middle position; do not just pick the nearest running total.)