# Cumulative Frequency Histogram

How to make and solve a cumulative frequency histogram: 4 examples and their solutions.

## Example 1

### Solution

First, make a cumulative frequency table.

Draw a 3 column table like this.

Name the titles
Interval, Frequency, and Cumulative Frequency.

Frequency Table

Fill in the Cumulative Frequency column.

For the first row,
the frequency is 2.

So the cumulative frequency is
2.

The previous cumulative frequency is 2.
The frequency of this row is 3.

Then the cumulative frequency is
2 + 3 = 5.

The previous cumulative frequency is 7.
The frequency of this row is 5.

Then the cumulative frequency is
7 + 5 = 12.

The previous cumulative frequency is 12.
The frequency of this row is 6.

Then the cumulative frequency is
12 + 6 = 18.

The previous cumulative frequency is 18.
The frequency of this row is 2.

Then the cumulative frequency is
18 + 2 = 20.

Use the Interval column
and the Cumulative Frequency column
to make a cumulative frequency histogram.

Frequency Histogram

Draw a rectangular form like this.

Write Test Scores
at the bottom of the form.

Write Cumulative Frequency
on the left side of the form.

Write the intervals of the test scores
at the bottom of the form.

The cumulative frequencies are from 0 to 20.

So write 0 to 20
on the left side of the form.

And inside the rectangle,
lightly draw horizontal lines
that show each frequency.

The cumulative frequency of 50-59 is 2.

So, for 50-59,
draw a vertical bar
whose height is 2.

The cumulative frequency of 60-69 is 5.

So, for 60-69,
draw a vertical bar
whose height is 5.
(in the middle of 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, for 70-79,
draw a vertical bar
whose height is 12.

The cumulative frequency of 80-89 is 18.

So, for 80-89,
draw a vertical bar
whose height is 18.

The cumulative frequency of 90-99 is 20.

So, for 90-99,
draw a vertical bar
whose height is 20.

So this is the cumulative frequency histogram
of the given test scores.

## Example 2

### Solution

This bar shows
the number of students
whose scores are less than 80 points.

Its height is 12.

So the percent of students
whose scores are less than 80 points is,

the height of this bar,
12

over,
the total number of students,
20

times 100.

So the percent of students
whose scores are less than 80 points is
[12/20]⋅100.

Cancel the denominator 20
and reduce 100 to, 100/20, 5.

12⋅5 = 60

This is the percent of students.
So write %.

## Example 3

### Solution

This bar shows
the number of students
whose scores are less than 70 points.

Its height is 6.

Then the number of students
whose scores are
more than or equal to 70 points
is this blue height:
20 - 6.

So the percent of students
whose scores are
more than or equal to 70 points is,

the length of the blue height,
20 - 6

over,
the total number of students,
20

times 100.

So the percent of students
whose scores are
more than or equal to 70 points is
[(20 - 6)/20]⋅100.

20 - 6 = 14

14/20 = 7/10

Cancel the denominator 10
and reduce 100 to, 100/10, 10.

7⋅10 = 70

This is the percent of students.
So write %.

## Example 4

### Solution

To find the quartiles Q1, Q2, and Q3,
find the related cumulaitve frequencies
n1, n2, and n3.

n1 is the related cumulative frequency of Q1.

So n1 is equal to,
the total number of the values, 20
times,
Q1 is the 1/4 of the data, 1/4.

20⋅[1/4] = 5

So n1 = 5.

This means
there are n1 = 5 values
between 0 and Q1.

n2 is the related cumulative frequency of Q2.

So n2 is equal to,
the total number of the values, 20
times,
Q2 is the 1/2 of the data, 1/2.

20⋅[1/2] = 10

So n2 = 10.

This means
there are n2 = 10 values
between 0 and Q2.

n3 is the related cumulative frequency of Q3.

So n3 is equal to,
the total number of the values, 20
times,
Q3 is the 3/4 of the data, 3/4.

Cancel the denominator 4
and reduce 20 to, 20/4, 5.

3⋅5 = 15

So n3 = 15.

This means
there are n3 = 15 values
between 0 and Q3.

n1 = 5
n2 = 10
n3 = 15

Write these values
on the Cumulative Frequency axis.

Find the interval that have Q1.

Find the nearest bar
that covers n1 = 5:
the interval 60-69.

So the interval 60-69 has Q1.

By the same way,
find the interval that have Q2.

Find the nearest bar
that covers n2 = 10:
the interval 70-79.

So the interval 70-79 has Q2.

Find the interval that have Q3.

Find the nearest bar
that covers n3 = 15:
the interval 90-99.

So the interval 90-99 has Q3.

So
Q1: 60-69
Q2: 70-79
Q3: 90-99