# Cumulative Frequency Histogram

How to make a cumulative frequency histogram from the given data and solve the related problems: examples and their solutions.

## Example 1: Making the Cumulative Frequency Table

To draw the cumulative frequency histogram,

first make a column for the Cumulative Frequency

right next to the given table.

For 50-59 interval,

the frequency is 2.

So the cumulative frequency is 2.

For 60-69 interval,

the frequency is 3.

So the cumulative frequency is

2 + 3 = 5.

For 70-79 interval,

the frequency is 7.

So the cumulative frequency is

5 + 7 = 12.

For 80-89 interval,

the frequency is 6.

So the cumulative frequency is

12 + 6 = 18.

For 90-99 interval,

the frequency is 2.

So the cumulative frequency is

18 + 2 = 20.

Use the Interval column

and the Cumulative Frequency column

to draw a cumulative histogram.

Draw the histogram axes like this.

Set the vertical axis

as Cumulative Frequency:

from 0 to 20.

And set the horizontal axis

as Test Scores:

from 50-59 to 90-99.

Frequency histogram

The cumulative frequency of 50-59 is 2.

So draw a vertical bar in the histogram

whose height is 2.

The cumulative frequency of 60-69 is 5.

So draw a vertical bar

whose height is 5.

(between 4 and 6)

Draw the bar right next to the previous bar

so that there would be no space between the bars.

The cumulative frequency of 70-79 is 12.

So draw a vertical bar

whose height is 12.

Draw the bar right next to the previous bar.

The cumulative frequency of 80-89 is 18.

So draw a vertical bar

whose height is 18.

Draw the bar right next to the previous bar.

The cumulative frequency of 90-99 is 20.

So draw a vertical bar

whose height is 20.

Draw the bar right next to the previous bar.

So this cumulative histogram is the answer.

## Formula: Cumulative Frequency and Percent

By using the cumulative histogram,

it's easy to find the percent of the parts.

If the cumulative frequency is the number of the parts,

use this formula.

(percent) = [*x*/*n*]⋅100*x*: Cumulative frequency*n*: Number of data

## Example 2

The number of students that score less than 80

is the height of the interval 70-79:

12.

This 12 is the number of students

that score from minimum to 79.

*x* = 12*n* = 20

(percent) = [12/20]⋅100

Reduce 12 to 6

and reduce 20 to 10.

Cancel 10

and reduce 100 to 10.

6⋅10 = 60

So 60% is the answer.

## Example 3

Recall that

the quartiles divide the data into four parts.

Quartiles

So the percent of the quartiles are*Q*_{1} = 25%, *Q*_{2} = 50%, and *Q*_{3} = 75%.

Then set the number of students

of 25% as *x*_{1}, 50% as *x*_{2}, and 75% as *x*_{3}.

The goal is to find *x*_{1}, *x*_{2}, and *x*_{3},

then find the intervals that have these values.

The cumulative frequency of 25% is *x*_{1}.*n* = 20

So [*x*_{1}/20]⋅100 = 25

Then *x*_{1} = 5.

The cumulative frequency of 50% is *x*_{2}.*n* = 20

So [*x*_{2}/20]⋅100 = 50.

Then *x*_{2} = 10.

The cumulative frequency of 75% is *x*_{3}.*n* = 20

So [*x*_{3}/20]⋅100 = 75

Then *x*_{3} = 15.

The cumulative frequency of 25% is 5.

And the cumulative frequency 5

is in 60-69 interval.

So *Q*_{1} is in 60-69 interval.

The cumulative frequency of 50% is 10.

And the cumulative frequency 10

is in 70-79 interval.

So *Q*_{2} is in 70-79 interval.

The cumulative frequency of 75% is 15.

And the cumulative frequency 15

is in 90-99 interval.

So *Q*_{3} is in 90-99 interval.

So *Q*_{1} is in 60-69 interval.*Q*_{2} is in 70-79 interval.

And *Q*_{3} is in 90-99 interval.