# 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

## Example 3

Recall that
the quartiles divide the data into four parts.

Quartiles

So the percent of the quartiles are
Q1 = 25%, Q2 = 50%, and Q3 = 75%.

Then set the number of students
of 25% as x1, 50% as x2, and 75% as x3.

The goal is to find x1, x2, and x3,
then find the intervals that have these values.

The cumulative frequency of 25% is x1.
n = 20

So [x1/20]⋅100 = 25

Then x1 = 5.

The cumulative frequency of 50% is x2.
n = 20

So [x2/20]⋅100 = 50.

Then x2 = 10.

The cumulative frequency of 75% is x3.
n = 20

So [x3/20]⋅100 = 75

Then x3 = 15.

The cumulative frequency of 25% is 5.

And the cumulative frequency 5
is in 60-69 interval.

So Q1 is in 60-69 interval.

The cumulative frequency of 50% is 10.

And the cumulative frequency 10
is in 70-79 interval.

So Q2 is in 70-79 interval.

The cumulative frequency of 75% is 15.

And the cumulative frequency 15
is in 90-99 interval.

So Q3 is in 90-99 interval.

So Q1 is in 60-69 interval.
Q2 is in 70-79 interval.
And Q3 is in 90-99 interval.