# Histograms & Cumulative Frequency

Contents

## Summary

• A histogram show the distribution of numerical data. It is an estimate of the probability distribution of a continuous variable
• For a histogram In order to calculate the frequency density, we use $Frequency\quad density\quad =\quad \frac { frequency }{ class\quad width }$
• Cumulative frequency is accumulation of the frequencies
• First plot the graph and then join up the points to make a cumulative curve

A histogram show the distribution of numerical data. It is an estimate of the probability distribution of a continuous variable.

### Properties of Histograms :

1. It uses quantitative data (numeric data).
2. It doesn’t have any gaps between the bars in the graph, if it does then the data is considered to be missing.
3. Bar width in histogram is constant.
4. The y axis corresponds to the frequency.

#### Example #1

Q. Grades of 15 students are given below, draw a histogram from the given data.

 88 48 60 51 57 85 69 75 97 72 71 79 65 63 73

Solution:

Step #1

We need to classify the range of values into intervals

So we will divide the value into a range of 10 and add the grades into their respected ranges.

This was a simple example, now let’s take a look at a slightly different example, where the all the class widths are not equal, thus for that we need to do a little working.

### Use of Frequency Density Formula

#### Example #2

The table below shows the height of the university students, draw a histogram with the information provided;

Height (h/cm)Frequency
$140\quad \le \quad h\quad \le \quad 145$12
$145\quad \le \quad h\quad \le \quad 150$24
$150\quad \le \quad h\quad \le \quad 160$37
$160\quad \le \quad h\quad \le \quad 175$18
$175\quad \le \quad h\quad \le \quad 190$9

Solution:

We have the frequency and height, we will first write down the class width and then calculate the frequency density.

In order to calculate the frequency density, we use the following formula:

$Frequency\quad density\quad =\quad \frac { frequency }{ class\quad width }$

E.g

$Frequency\quad density\quad =\quad \frac { frequency }{ class\quad width } \quad =\quad \frac { 12 }{ 5 } \quad =\quad 2.4$

Similarly we will calculate the rest of the frequency densities.

Height (h/cm)FrequencyClass widthFrequency density
$140\quad \le \quad h\quad \le \quad 145$1252.4
$145\quad \le \quad h\quad \le \quad 150$2454.8
$150\quad \le \quad h\quad \le \quad 160$37103.7
$160\quad \le \quad h\quad \le \quad 175$18151.2
$175\quad \le \quad h\quad \le \quad 190$9150.6

Next, we will draw the histogram with the frequency density on the y axis and height on the x axis.

Note:

In an exam you can also get a question, where the histogram is drawn and you have to fill in the table, for that the class width is shown on the x axis of the histogram and frequency density is shown on the y axis, to find the frequency from the frequency density, use the frequency density formula:

Frequency = frequency density X class width

### Cumulative Frequency

Cumulative frequency is accumulation of the frequencies. Let’s have a look at one its example, to make it clear on how to calculate cumulative frequency and how to show it on the graph.

#### Example #3

The data below shows the age of toddlers and their weight gain throughout the year. Draw a cumulative frequency graph to represent the data.

Age Weight gain
04
19
22
32
43
54
63

Solution:

We will first make a column of cumulative frequency. In order to do that we will add every value of weight gain with the next one.

E.g:

4 + 9 = 13

13 + 2 = 15

15 + 2 = 17 and so on

We will now plot a graph of the age on x axis and the cumulative weight on the y axis. We will first plot the graph and then join up the points to make a cumulative curve.