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### Graphing Distribution : Histograms

#### The Objectives:

• Construction of grouped frequency distribution.

• Histogram creation of a grouped frequency distribution

• Determination of appropriate bin width #### What is Histogram?

Graphical representation of data where bars of different heights are used. Through histogram the shape of the distribution can be understood. It can be used even when the number of observations is large.

#### Example:

We consider an example of the heights of 177 adults.

#### Construction of a Grouped Frequency Distribution:

In order to construct the grouped frequency distribution of the heights, we make use of the frequency table with class intervals and frequencies. Class intervals or grouping of heights is used to simplify the frequency table. The following is the frequency table for frequency distribution of the heights of 177 adults.

Table 1. Grouped Frequency Distribution of Heights of 177 adults:

#### Frequency

144.55-149.55

1

149.55-154.55

3

154.55-159.55

24

159.55-164.55

58

164.55-169.55

50

169.55-174.55

27

174.55-179.55

2

179.55-184.55

2

Total

177

If we observe the above table we will find that the first interval is 144.55-149.55. 144.55 is the lower class limit and 149.55 is the upper class limit. The frequency corresponding to this interval is 1, so the number of adults who have height between the classes 144.55-149.55 is 1. In the similar manner for the next interval the number of adults having height 149.55-154.55 is 3.

Notice that the width of the class interval i.e., the difference between upper class limit and lower class limit is 5, which provide good detail about the distribution. Also, the width of each class is equal. Details of the width are given in the latter part of this section. #### Creation of Histogram:

Generally for a continuous variable we use histograms. On horizontal axis we locate class boundaries, and over each class interval we erect a rectangle whose area represents the corresponding frequencies

Table 2: The marks obtained by 50 students of a class

 43 34 43 32 87 35 71 65 12 52 19 48 17 24 52 65 40 54 62 45 2 13 18 49 57 21 64 71 45 81 52 40 35 78 43 45 44 55 79 37 19 14 31 71 51 35 27 74 22 8

Fig 1: Histogram of the marks obtained by 50 students of a class ### Shape of the Distribution

The shape of the distribution can be examined through the histogram. From histogram we can see whether the data is symmetric, or it skewed to the right or left. From the above we can see scores at the middle are the highest and decreases at the extremes. The distribution is more or less symmetric. If the scores are more on the right than on the left the distribution can be concluded as right skewed distribution , and if the scores are more on the left than on the right the distribution can be concluded as left skewed distribution. The above two examples have observations that are whole numbers. However, if the score were decimal or continuous ones histogram goes better with that. Class boundaries are used to make the class intervals continuous.

#### Relative Frequency and Actual Frequency:

For creation of histogram we can use relative frequencies in place of actual frequencies. In such cases the histogram presents the proportion rather than the actual number for each class. Thus relative frequencies lie between 0 and 1. Actual frequencies are transformed to relative frequency by simply dividing the actual frequency for each class by the total frequency, or total number of observations.

#### Bin Width

Width of class interval is an important aspect in a histogram. Width is also known as bin widths. Bin width determines the number of class interval. That is to say the number of class intervals depends on the choice of bin width. The choice of bin width and starting point affect the shape of the distribution. There are some rules of thumb that can be used to determine bin width.

1. Sturgis rule: In this rule one needs to choose the number of intervals close to 1+log2(N), N denotes number of observations and 2 denotes the base of the log of N. It can also be written as 1+3.3log10(N).

2. Rice Rule: This rule is preferred to Sturgi's rule. Here we choose the number of intervals by multiplying 2 to the cube root of the number of observations.

However it will be best if one experiments with different choices of the width and see which histogram communicates well with the shape of the distribution.

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