Frequency Constructions

It is very difficult to lay down any hard & fast rules for making a frequency distribution since much depends on the nature of the given data and the object of classification.

However the following normal considerations may keep in mind for ensuring meaningful classification of the data.

The number of classes must prefer between 5 & 20; however there is no rigidity about it. The classes can be more than 20 depending upon the octal number of items in the series and the details needed but they should not be less than five as in that case the classification may not reveal the essential specifications. The choice of the number of classes mainly depends upon.

(a) The number of figures to be classified.

(b) The magnitude of the figures.

(c) The details needed and

(d) The Ease of calculation for further statistical world.

As far as possible one must avoid values of class -intervals as 37, 11, 26, 39 etc. preferably one should have the class intervals of either 5 or multiples of 5 such as 10, 20, 25, 100, etc. The reason is that the human mind is accustomed more to think in terms of certain multiples of 5, 10 and the same, however where the data necessitate of a class- interval of less than 5 it can be any value between 1 and 4.

The starting point of the lower limit of the first class must either be 0 or 5 or multiple of 5. For e.g. if the lowest value of the data is 63 and we have take class interval of 10. Then the first Class must be 60-70 instead of 63-73. Similarly if the lowest value of the data is 76 and the class interval is 5 then the first class must be 75 to 80 instead of 76 to 81.

To ensure the continuity and to get correct class interval we should adopt exclusive techniques of classification. However where inclusive techniques has been adopted it is necessary to make an adjustment to determine the correct class interval to have continuity. The adjustment involves, finding the difference between the lower limit of the second class and the upper limit of the first class dividing the difference by 2. Subtracting the value which is obtained from all lower limits and adding this value to all upper limits. This can be stated in the form of a formula as follows:

Correction factor = [(lower limit of the 2nd class - upper limit of the 1st class) / 2]

How the adjustment is made when data are given by inclusive method can be seen from the following Examples:

To adjust the class limits we take her the difference between 900 and 899, which is one. By dividing it by two we get ½ or 0.5. This (0.5) is upper limits. The adjusted classed would then be as follows:

It should be noted that before adjustment the class - interval was 99 but after adjustment it is 100. Observe another case:

The correction factor here is = (10 - 9.5) / 2 = 0.25

after adjustment the classes will be

The class - interval now is 5 and not 4.5. Taking a third example, if the class limits are:

The correction factor would be (10-9.99)/2 = 0.01 / 2 = 0.005.

After adjustment classes will become:

Wherever possible it is desirable to use class intervals of equal sizes as comparison of frequencies among classes is facilitated and subsequent calculations from the distribution are simplified. However this is not always a practical process. For e.g. in case of data on monthly income of families in order to represent the details for the portion of frequency distribution where the majority of incomes lie. The class intervals of 100 or 200 may be used starting say from 600-700 onwards up to about 1.200 then intervals of 400 to 500 may be used up to 2.500  and a final class of 2.500 and above may be shown for  relatively small number of families having theses highest incomes. It is obvious that if we have a similar class interval of 100 there would be too many classes.

On the other hand if much bigger class intervals were used say 500, too many families would be lumped together in the first one or two classes and the information how these incomes were distributed would be lost. To resolve this dilemma at times, we use open-end interval along with equal or unequal class intervals. The use of unequal class sizes and open-end intervals generally becomes necessary in case where most of the data are concentrated within a certain range where gaps appear in which relatively few items are observed and where there are very few extremely small or large values.

Illustration:-

Prepare a frequency distribution of the marks obtained by 50 students in an examination with width of each class interval as 10 use exclusive method of classification:

Solution:-

Preparation of frequency distribution

Frequency distribution of the marks