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Frequency Distribution

Statistics is the science that deals with a large amount of data. The raw data has to be organized and tabulated in a meaningful fashion. The graphs and plots play an important role in the world of statistics since the pictorial representation has an ability to display a comparison or a trend in a clearer manner. The graphs and diagrams are the most effective ways to deal with the statistical data. There are several interesting graphs through which the data is displayed and interpreted. The graphs that are most commonly used, are:

 Pie Chart

Line Graphs

Scatter Plot

Box and Whisker Plot

Stem and Leaf Plot

Bar Graph



Cumulative Frequency Curve

Let us study about histogram in this lesson.

A histogram accurately pictures the distribution of given numerical data. It is considered to be an estimation of continuous variable probability distribution. The histogram was discovered by Karl Pearson. Histogram uses rectangles or bars to represent the data. It is a sort of bar graph. The only difference is that a bar graph has gaps between two bars, while a histogram doesnt. A general example of a histogram is:

frequency distribution histogram


Let us understand how a histogram is drawn.

For Example: Let us try drawing a histogram from the following data.

$6, 34, 5, 35, 19, 47, 50, 12, 10, 17, 31, 29, 52, 23, 16, 8, 48, 41$

The following steps are to be followed.

Step 1: Determine the Range

It is important to figure out the amount of space required. For this, the range has to be calculated. The range is determined by subtracting the smallest value from the largest value in the distribution. In above example:

Range = $52 - 5$ = $47$

Step 2: Find the Class Width

The term is used to for an interval or a range to be shown through a histogram. There are no specific rules to find the numbers of bins to be created. Normally, it should not be fewer than $5$ and more than $20$ because in case of less than $5$ bins, the graph would be meaningless to draw and if bins are more than $20$, then it would be too difficult to interpret. So mostly, it is ideal to choose $5$ to $20$ bins.

In above example, lets choose $6$ bins.

$\frac{47}{6}$ = $7.8 \approx 8$

This determines class width.

Step 3: Creating Bins or Groups

$5 - 12$

$13 - 20$

$21 - 28$

$29 - 36$

$37 - 44$

$45 - 52$

Step 4: Construct Frequency Table

Frequency for a particular group is the number of value that fall within that particular group.

 Score   Frequency 
 5 - 12  5
 13 - 20   3 
 21 - 28   2
 29 - 36   3 
 37 - 44   1
 45 - 52   4

Make sure that the sum of frequency must be equal to the total number of values.

Step 5: Labeling

Label $X$ and $Y$ axis on the graph. In this example, label score on $X$ axis and frequency on $Y$ axis.

Step 6: Draw the histogram

Following histogram is obtained.

frequency distribution

Is a histogram the same thing as a frequency polygon?

The histograms and frequency polygons are two different graphs. However, they are quite similar when we talk about the method of their construction. A frequency polygon may be referred as an extended graph of histogram. In frequency polygon, the class marks are considered instead of class intervals. A class mark is a mid-value of a particular class interval.

Class mark = $\frac{lower \ limit + upper \ limit}{2}$

In order to draw frequency polygon, a histogram is drawn first and then the mid points of each bin is joined. An example is shown as under:


A frequency polygon without a histogram can be drawn when the frequencies are plotted against each class mark. For instance:


When can you use a frequency polygon?

A frequency polygon can be used to analyze or measure the way of distribution of data in a particular data set. Usually, it is utilized for the comparison among two or more distributions over some unit such as a period of time.

For example: marks obtained by the students each year for few years, runs scored in each over in a particular cricket match or performance (runs scored) of a cricketer per match for few matches, value of stock-market shares with respect to days etc.

In other words, the basic use of frequency polygons over a histogram is comparing distributions. This comparison can be performed by overlaying two or more frequency polygons together.

For instance:


Here by just seeing, it can be concluded that the values of first distribution (large target) are higher than second one (small target).

Related Calculators
Frequency Distribution Calculator Frequency Calculator
Calculate Relative Frequency Frequency and Wavelength Calculator

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Frequency Distribution Histogram
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