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Histogram

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Histogram represents data in the form of a diagram and it becomes easy for the reader to analyse and interpret to come to various conclusions and decisions. In histogram the figures are given the size proportionally so that analysis can be done by non qualified and non technical person too.
Histogram consist a series of erect drawn bars in form of the rectangles on the x axis with base equal to the corresponding class intervals and with equal base sections drawn on it.
In drawing the histogram of a given continuous frequency distribution we first mark off along the x-axis all the class intervals on a suitable class. On each class interval intersect rectangles with heights proportional to the frequency of the corresponding class interval so that the area of the rectangle is proportional to the frequency of the class. If, however, the classes are of unequal width then the height of the rectangle will be proportional to the ratio of the frequencies to the width of the classes, the diagram of continuous rectangles so obtained is called histogram.

To draw the histogram for an ungrouped frequency distribution of a variable we shall have to assume that the frequency corresponding to the variate value x is spread over the interval x -$\frac{h}{2}$  to x + $\frac{h}{2}$, where h is the jump from one value to the next.

If the grouped frequency distribution is not continuous first it is to be converted into continuous distribution and then the histogram is drawn. Although the height of each rectangle is proportional to the frequency of the corresponding class, the height of a fraction of the rectangle is not proportional to the frequency of the class so that histogram cannot be directly used to reach frequency over a fraction of a class interval.

Histogram Definition

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A two dimensional frequency density diagram is called a histogram. Histograms are diagrams which represent the class interval and the frequency in the form of a rectangle. There will be as many adjoining rectangles as there are class intervals and is a graphical representation showing a visual impression of data, first introduced by karl pearson.  It can estimate the probability distribution of a continuous variable. Frequency distributions can be represented graphically using histograms. The midpoint occurs when about half the area is to the left and half is to the right. Histogram is a type of graph which is also called as frequency density diagram

Histogram Formula

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If 'd' is the gap between the upper limit of any class and the lower limit of the succeeding class the class boundaries for any class are then given by:
Upper class boundary = Upper class limit + $\frac{d}{2}$
Lower class boundary = Lower class limit + $\frac{d}{2}$

Types of Histograms

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Individual data points are grouped together in classes, so that you can get an idea of how frequently data in each class occur in the data set. A histogram with more number of class intervals is more effective in depicting the structure of the frequency distribution.

Different types of histograms are explained below.
Frequency histogram
Histogram is a graphical display of data using bars of different heights.
The relative frequency table is a compact numerical way to present how the data is distributed. If the frequencies are plotted as columns, the resulting plot is called a histogram.

Double peaked distribution
Looks like the back of a two humped camel. Outcomes of two processes with different distributions are combined in one set of data. In some cases bimodal histogram indicates that the sample can be divided into two sub samples that differ from each other in some way.It will have two peaks.
Biomodal


J shaped distribution
Extreme case of negative skew. The reverse J or ‘ski-jump’ shape is an extreme case of positive skew. Cases are scoring tight up against a fixed end limit of a scale. J is like a heap which has one half missing, because the scale comes to an end so there is nowhere for the expected other half of the distribution to fall.
J Shaped Distribution



Skewed Histogram
A non symmetric histogram is called skewed if it is not symmetric. If the upper tail is longer than the lower tail then it is positively skewed. If the upper tail is shorter than it is negatively skewed. A skewed distribution is one that is not symmetrical, but rather has a long tail in one direction.


Right Skewed Distribution

Uniform Histogram
A uniform distribution often means that the number of classes is too small, each class has about the same number of elements. It may describe a distribution which has several peaks. Uniform histogram have all the bars of same height.

Histogram Example Problems

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Example 1: Distribution of marks of 250 students is obtained as follows.
Draw a histogram for the given data.
     Marks           
 Number of students  
   14.5 - 19.5  9
   19.5 - 24.5  11
  24.5 - 29.5  10
   29.5 - 34.5  44
   34.5 - 39.5  45
   39.5 - 44.5  54
 44.5 - 49.5  37
 49.5 - 54.5  26
 54.5 - 59.5  8
 59.5 -64.5  5
 64.5 - 69.5  1

Solution: Since the grouped frequency distribution is not continuous, we first convert it into a continuous distribution with exclusive type classes as given below. The upper and lower class limits of the new exclusive type classes are known as class boundaries.
Histogram Distribution

Example 2: Draw a histogram for the following data
Class Interval   
  Frequency  
 0 - 10  5
 10 - 20  8
 20 - 30  12
 30  - 40  16
 40 - 50  7
 50 - 60  15
 60 - 70  17
 70 - 80  19
 80 - 90  25
 90 -100  28

Solution:
Histogram Graph