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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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