In these series all the items are divided in certain groups, but these groups are not continuous, therefore these series are known as discrete series. The numbered item that fall in every group are shown in each group which are known as frequencies. The following examples will illustrate this :

Examples :

(i) Discrete Series in Ascending Order :

(ii) Discrete Series in Descending Order :

There is no hard and fast rule about how many class we choose; but as a working rule the number of classes should lie between 5 and 15. It should be noted that the number of classes will be large if we choose small size class intervals and it will be small if the size of class intervals is large.

As an illustration, suppose the range is 70, and we choose classes of width 2 each. We would require 70 ÷ 2 = 35 classes. However, the number of classes would be 14 if the width of each class was 5.

Size of Class Intervals : We may choose all classes of the same width or of different width. In the case of equal class intervals the size of the class interval is determined as soon as we have decided about the number of classes.

Suppose n is the number of classes and all classes are of width h, then n × h = R.

Knowing the range R and number of classes

n we can abtain h = R/n as the width of class interval. If the range is 70 and we choose 10 classes, the width is 7.

Choice of Class Limits : Suppose x is a continuous variable, such that it can take any value in a given range. In that case, it is possible to choose class limits which are not equal to any of the observed values, For example, height of individuals is a continuous variable, even though, in practice, one can measure height to the nearest of the unit value (in centimetres) as 165, 170, 169, 171 .........; or to the nearest of tenth place of decimals as 165.3, 170.4, 168.9, 170.8, ........ We may specify class intervals as 160.55 165.55,165.55 ....... so that none of the observed values of x is equal to any of the class limits.

Under such series all the variables are divided in certain continuous groups and their respective frequencies will be written with them. The following example will clear the form of such series:

Example:

Following are the elements of a continuous series:

(i) Class Intervals : These are the measurements in which some problems is measured and written in continuous group. In the above example, (0 – 5), (5 – 10) etc. are the class intervals of the series.

(ii) Limits of Class Intervals : Each class interval figure is known as limits of class interval. Small figures class intervals are known as lower limit class interval. In class interval (0 – 5) 0 is lower limit and 5 is a upper limit of this class interval.

(iii) Magnitude of Class Intervals The difference between upper limit and lower limit of a class interval is known as its magnitude. In class interval (0 – 5) 5 is the magnitude.

(iv) Mid Value The average of two limits of the class interval is known as mid value e.g., the mid value of class interval

(v) Frequencies : Number of repetition of items of various class intervals in the universe is known as frequencies which will be written with them.

Exclusive and Inclusive Continuous Series:

(a) Exclusive Series : Where the value of upper limit is not included in the same group, but will be included in next group, it is known as exclusive series e.g.

In the above series, 10, 20, 30,40, 50, and 60 will not be included in first, second third, fourth, fifth and sixth group respectively.

(b) Inclusive Series : Where value of upper limit is included in the same group, it is known as inclusive series e.g.,

Individual series : Under this method, the value of all the units are shown separately The following example will illustrate this:

Example : The marks obtained by 10 students in statistics are following :

The individual series may be arranged in following two orders :

(a) Ascending Order : When data are arranged in ascending order i.e., a small value to a big value it is known as arranging them in ascending order. The figures of above example may be arranged in ascending order as follows :

(b) Descending Order : When data are arranged serially starting from a big value to small value it is known as arrangement of data in descending order.

Frequency Array : We obtain a frequency array if the variable x is discrete and we have frequencies corresponding to each value (there are no class intervals). Let us illustrate with the following example.

Example : A survey of 100 households was carried out to obtain information on their size, i.e., the number of members of households. The results of the survey are classified as a frequency array in table below :

Frequency Array of Size of Households

The column (1) of the table gives the values which the variable x (size of the households) takes; and column (2) gives the corresponding frequencies (number of households). Thus, there are 5 households whose size is 1, there are 15 households of size 2, and so on.

Frequency Distribution : The largest value of X is B and smallest value is A. Then X = B – A is the total range of X. A large range indicates that the values of X are spread over a large interval or the variation of value of X is large. A small range indicates smaller variation in the values of X. Thus, the range is measure of variation (or dispersion) of X.

For example : Suppose we have data on monthly income of 10,000 individuals, the maximum of which is Rs.50,000 and minimum is Rs. 1,000. Thus, the range is Rs.49,000. We observe that majority of individuals say, 70% have small incomes close to Rs. 5,000 and minority, say 2% have income close to Rs.30,000.

In order to get a better idea about the distribution of values within the range, we should subdivide the total range into a number of class intervals and find out the number of values in different classes.