# CUMULATIVE FREQUENCY ANALYSIS

In a data set, the cumulative frequency for a value ‘x’ is the total number of scores that are equal to or less than ‘x’.

For example if we want to know the cumulative frequency regarding the scores of students in a class and assignment. The cumulative frequency suggests that thirty students received a test score of at most sixty, sixty students received a test score of at most sixty and hundred students received a test score of at most seventy.

CUMULATIVE FREQUENCY TABLE

Example

A set of data given below shows the ages of participants in camp or in a summer camp. Draw a cumulative frequency table for the data.

 Age (years) Frequency Ten Seven Eleven Twelve Twelve Twenty seven Thirteen Ten Fourteen Twenty

Solution – The cumulative frequency of the given data can be found by adding the current frequency with the previous frequency.

The cumulative frequency of the first data will be same as the frequency because there is no cumulative frequency before it.

 Age (years) Frequency Cumulative Frequency Ten Seven Seven Eleven Twelve Seven + twelve = Nineteen Twelve Twenty seven Nineteen + twenty seven = Forty six Thirteen Ten Forty six + ten = fifty six Fourteen Twenty Fifty six + twenty = seventy six

CALCULATION OF QUARTILES FROM CUMULATIVE FREQUENCY

Quartile – The word quartile has been taken from the word quarter which means one fourth of something. Thus we can say that quartile means a certain fourth of a data set. When a certain set of data is arranged from the lowest to the highest, then it can be divided into four parts or quartiles.

First quartile – This refers to the data set arranged from the lowest to the highest in one fourth parts i.e. (1 /4) parts.

Second quartile – This refers to the data set arranged from the lowest to the highest in the second fourth parts i.e. (2 / 4) parts.

Third quartile – This refers to the data set arranged from the lowest to the highest in the third fourth parts i.e. (3 /4) parts.

Calculating quartiles from cumulative frequency

Example – Calculate the first, second and third quartiles of the data set using the cumulative frequency curve.

 Age (years) Frequency ten six eleven ten twelve eighteen thirteen twenty fourteen nine fifteen five

Solution -

 Age (years) Frequency Cumulative frequency ten six six eleven ten six + ten = sixteen twelve eighteen sixteen + eighteen = thirty four thirteen twenty thirty four + twenty = fifty four fourteen nine fifty four + nine = sixty three fifteen five sixty three + five = sixty eight sixteen two sixty eight + two = seventy

Calculating quartiles –

The cumulative frequency for the last element in the data set is seventy.

Quartile one position = (seventy + one) / four = 17.75

Quartile second position = 2(seventy + one) / four = 35.5

Quartile third position = 3(seventy + one) / four = 53.25

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