Counting to the median

A grouped table has no single middle reading. You count up to it.

The running total finds the class

Here are 40 made-up readings. A running total goes down the rows. It is the cumulative frequency.

The median is the middle reading. Half the readings lie below it. Half of 40 is 20. The totals 9 and 16 are too small. 26 is the first total to reach 20. So 30-40 is the median class.

The formula

Here n = 40 is the number of readings. The median sits at position n ÷ 2 = 20. 16 readings lie before the median class. We need 4 more, from the 10 inside it. Assume the 10 are spread evenly across the class. Then 4 of the 10 cover 4/10 of its width.

The formula has four letters for the median class. l = 30 is its lower bound. cf = 16 is the total before that class. f = 10 is its frequency. h = 10 is its width. The classes must touch, with no gaps.

Median = l + ((n ÷ 2 − cf) ÷ f) × h.

Median = 30 + ((20 − 16) ÷ 10) × 10 = 34.

Step by step: 20 − 16 = 4. Then 4 ÷ 10 = 0.4. Then 0.4 × 10 = 4. So 30 + 4 = 34.

Add frequencies down the table until the total reaches half of n. That class is the median class. Then use the formula inside that class.

Add rows until the total reaches half.