Elementary Statistics (1+1)

5.2.5.1. Histogram

Out of several methods of presenting a frequency distribution graphically, histogram is the most popular and widely used in practice. A histogram is a set of vertical bars whose areas are proportional to the frequencies represented.

While constructing histogram the variable is always taken on the X-axis and the frequencies depending on it on the Y-axis. Each class is then represented by a distance on the scale that is proportional to its class-interval. The distance for each rectangle on the X-axis shall remain the same in case the class-intervals are uniform throughout. If they are different the width of the rectangles shall also vary. The Y-axis represents the frequencies of each class which constitute the height of its rectangle. In this manner we get a series of rectangles each having a class-interval distance as its width and the frequency distance is its height. The area of the histogram represents the total frequency as distributed throughout the classes.

The histogram should be clearly distinguished from a bar diagram. The distinction lies in the fact that wheres a bar diagram is one dimensional. Ie. Only the length of the bar is material and not the width, a histogram is two-dimensional that is, in a histogram both the length as well as the width are important.

The histogram is most widely used for graphical presentation of a frequency distribution. However, we cannot construct a histogram for distribution with open-end classes. Moreover, a histogram can be quite misleading if the distribution has unequal class-intervals and suitable adjustments in frequencies are not made.

The technique of constructing histogram is given below (i) for distributions have equal class-intervals, and (ii) for distributions having unequal class-intervals.

When class-intervals are equal, take frequency on the Y-axis, the variable on the X-axis and construct adjacent rectangles. In such a case the height of the rectangles will be proportional to the frequencies.

When class-intervals are unequal, a correction for unequal class intervals must be made. The correction consists of finding for each class the frequency density or the relative frequency density. The frequency density is the frequency for that class divided by the width of that class. A histogram or frequency density polygon constructed from these density values would have the same general appearance as the corresponding graphical display developed from equal class-intervals.

For making the adjustment we take that class which has lowest class-interval and adjust the frequencies of other classes in the following manner. If one class-interval is twice as wide as the one having lowest class-interval we divided the height of its rectangle by two, if it is three times more we divide the height of its rectangle by three, etc. i.e. the heights will be proportional to the ratio of the frequencies of the width of the class.

Construction of Histogram when only Mid-points are given. When only mid-points are given, ascertain the upper and lower limits of the various classes and then construct the histogram in the same manner.