Chapter 14

Statistics

Class Mark:

Mean (Arithmetic mean) of ungrouped data:

Mean of n observations x₁, x₂, x₃, - - - - - x_n is given by

Mean (Arithmetic mean) of grouped data:

In a discrete frequency distribution the arithmetic mean may be computed by any one of the following methods.

(A) Direct Method: Mean of observations x₁, x₂, x₃… x_n with frequency f₁, f₂, f₃… f_n respectively is given by

(B) Assumed Mean Method:

Mean of a grouped frequency distribution is given by

Where a = Assumed mean

f = frequency

d_i = x_i – a (deviation)

(C) Step-deviation Method:

Where

h = class size

a = assumed mean

The step-deviation method will be convenient to apply, if all the d_i’s have a common factor.

(D) Weighted Arithmetic mean:

If w₁, w₂, w₃ - - - - w_n are weights respectively to x₁, x₂, x₃ - - - x_n, then weighted arithmetic mean =

Mode: Mode is that value in the observations which occurred most.

Mode of Grouped Data: The mode or modal value of a distribution is that value of the variable for which the frequency is maximum.

Where, l = lower limit of the modal class.

h= size of the class interval (assuming all class sizes to be equal)

f_i= frequency of the modal class

f_o= frequency of the class preceding the modal class.

f₂= frequency of the class succeeding the modal class.

Median:

(a) Median of ungrouped data:

Median is a measure of central tendency which give the value of the middle-most observation the data.

When we have to find the median of ungrouped data first arrange the data values of the observations in ascending or descending order.

(i) If n is odd;

Median = observation

(ii) If n is even

Median =

(b) Median of Grouped Data:

Median Class: The class whose cumulative frequency is greater than (and nearest to) n/2, where n = number of observations.

Where l = lower limit of median class

n = number of observations

cf = cumulative frequency of class preceding the median class

f = frequency of median class

h = class size (assuming class size to be equal)

Relationship among Mean, Mode, and Median

Mode = 3 Median – 2 Mean

(4) Cumulative frequency Polygon, curve (An Ogive).

In a cumulative frequency polygon or curves the cumulative frequencies are plotted against the lower or upper limits of the class-intervals depending upon the manner in which the series has been cumulated.