Definition

Cumulative frequency

Statistics • Frequency Distribution

Statement

The cumulative frequency at a value or class is the total of all frequencies up to and including it:

C_{k} = f_{1} + f_{2} + \dots + f_{k}

. Reversing this,

f_{k} = C_{k} - C_{k - 1}

, with

f_{1} = C_{1}

, and the final cumulative frequency is the total number of observations.

How to use it

1To recover a missing frequency, subtract the previous cumulative entry from its own.
2To check a completed table, rebuild the cumulative column by running totals and compare it with the printed one.
3The last cumulative entry is $\sum f_{i}$ , which is the divisor in any mean or median calculation.

Worked example

A table gives cumulative frequencies

6

15

15

23

. What are the four frequencies?

Step-by-step

1
$f_{1} = 6, f_{2} = 15 - 6 = 9$
Why: Each frequency is the jump in the running total.
2
$f_{3} = 15 - 15 = 0, f_{4} = 23 - 15 = 8$
Why: A repeated cumulative entry means that class is empty.

Answer

6, 9, 0, 8

Watch out

A cumulative frequency never decreases; a fall down the column is a misread table. This is the 'less than' form - a 'more than' column runs the other way, and subtracting in the wrong direction there gives negative frequencies. Reporting the two recovered frequencies in the wrong order is the usual slip in multiple choice.

Also known as

cumulative frequency, running total of frequencies, less than ogive, recover frequency from cumulative frequency, cf column

Appears in past papers

Statement

Worked example

Appears in past papers

2019 (I)

2018 (II)

2017 (I)