Business Statistics - Measures of Central Tendency (Formulas)

 

Measures of Central Tendency

–        Formulas

 

There are 3 measures of central tendency – Mean, Median and Mode. Again, Mean is of 3 types – Arithmetic Mean (A.M.), Geometric Mean (G.M.) and Harmonic Mean (H.M.)

 

Note: The words ‘mean’ and ‘average’ refer to Arithmetic Mean only.

 

Formulas of Mean (AM)

 

A.         Direct Method:

For simple distribution, Mean = (∑x)/n

For frequency distribution, Mean = (∑fx)/ (∑f)

 

B.         Short-cut Method:

For simple distribution,

Mean = A + (∑d)/n

[Here, A = Assumed Mean (as near as possible to the true mean); d = x – A]

 

For frequency distribution,

Mean = A + (∑fd)/ (∑f)

[Here, A = Assumed Mean (value of that observation or that mid-value which has the highest frequency); d = x – A]

 

C.         Step-deviation Method:

For simple distribution,

Mean = A + [(∑d)/n]*c

[Here, A = Assumed Mean (as near as possible to the true mean); d = (x – A)/c; c = Common factor]

 

For frequency distribution,

Mean = A + [(∑fd)/ (∑f)]*c

[Here, A = Assumed Mean (value of that observation or that mid-value which has the highest frequency); d = (x – A)/c; c = Common factor or common width]

 

Important note:

Formula of mean for simple frequency distribution and grouped frequency distribution under all the above three methods are same. Only thing to be remembered is that in case of grouped frequency distribution x_i will be the mid-values of the class intervals.

 

Important Properties of Arithmetic Mean

 

1.       The total of a set of observations is equal to the product of their number and the A.M. Symbolically,

(i)              ∑x_i = n(Mean)

(ii)            ∑f_ix_i = N(Mean)     [N = ∑f_i]

2.       The sum of the deviations of a set of observations from their A.M. is always zero.

Symbolically,

(i)              ∑(x_i – Mean) = 0

[When Mean = (∑x_i)/n]

(ii)            ∑f_i(x_i – Mean) = 0   

[When Mean = (∑f_ix_i)/N]

3.       If two variables x and y are so related that

x = c + dy, where c and d are constants, then Mean of x = c + d(Mean of y)

4.       The sum of the squares of deviations of a set of observations has the smallest value, when deviations are taken from their A.M.

Symbolically,

(i)              ∑ (x_i – A) ^2 is minimum, when A = simple A.M.

(ii)            ∑f_i (x_i – A) ^2 is minimum, when A = weighted A.M.

Mean of Composite Group

For simple distribution,

Composite mean = [(∑x) + (∑y)]/ (n1 + n2)

= [(n1*Mean of x) + (n2*Mean of y)]/ (n1 + n2)

 

For frequency distribution,

Composite mean = [(∑fx) + (∑fy)]/ (N1 + N2)

= [(N1*Mean of x) + (N2*Mean of y)]/ (N1 + N2)

 

[Here,

x_i = Observations of Group- 1

y_i = Observations of Group- 2

n1 = Total number of observations of Group- 1

n2 = Total number of observations of Group- 2

N1 = Total frequency of Group- 1

N2 = Total frequency of Group- 2]

 

Formulas of Geometric Mean (GM)

 

For simple distribution,

G = (x1*x2*x3*..................*xn) ^ (1/n)

G = antilog [(1/n)*∑logx]

 

For frequency distribution,

G = [(x1^f1)*(x2^f2)*...........*(xn^fn)] ^ (1/∑f)

G = antilog [(1/∑f)*∑f (logx)]

 

Geometric Mean of Composite Group

G = [(G1^N1)*(G2^N2)] ^ (1/N)

 

[Here,

G1 = Geometric mean of Group- 1

G2 = Geometric mean of Group- 2

N1 = Total frequency of Group- 1

N2 = Total frequency of Group- 2

N = N1 + N2]

 

Formulas of Harmonic Mean (HM)

 

For simple distribution,

H = n/∑ (1/x)

 

For frequency distribution,

H = ∑f/∑ (f/x)

 

Harmonic Mean of Composite Group

H = [N1 + N2]/ [(N1/H1) + (N2/H2)]

 

[Here,

H1 = Harmonic mean of Group- 1

H2 = Harmonic mean of Group- 2

N1 = Total frequency of Group- 1

N2 = Total frequency of Group- 2

N = N1 + N2]

 

Relation between AM, GM and HM

GM = (AM * HM) ^ (1/2)

 

 

 

Formulas of Median

 

For simple distribution

When n is odd −

Step 1:

Arrange the data in ascending order.

 

Step 2:

Median = Value of [(n +1)/2]^th observation.

 

When n is even –

Step 1:

Arrange the data in ascending order.

 

Step 2:

Median = ½ of [Value of (n/2)^th observation +

              Value of ((n + 2)/2)^th observation]

 

For simple frequency distribution

Step 1:

Construct a cumulative frequency distribution table (“less than” type)

 

Step 2:

Median = Value of observation corresponding to cumulative frequency [(N + 1)/2]

 

For grouped frequency distribution

Step 1:

If the class intervals are given in ‘limits’, convert the class limits into class boundaries

 

Step 2:

Construct a cumulative frequency distribution table (“less than” type)

 

Step 3:

Median = L1 + [(L2 – L1)/f]*(m – c)

 

[Here,

L1 = Lower boundary of the median class

L2 = Upper boundary of the median class

f = Frequency of the median class

m = N/2

N = Total frequency

c = Cumulative frequency of the class interval immediately preceding the median class

Median class = The class interval in which the (N/2)^th observation lies

 

 

 

 

Formulas of Mode

 

For simple distribution

Mode is the value of that observation which occurs for maximum number of times in the given data.

 

For simple frequency distribution

Mode is the value of that observation which corresponds to the largest frequency.

 

For grouped frequency distribution

Step 1:

If the class intervals are given in ‘limits’, convert the class limits into class boundaries

 

Step 2:

Mode = L1 + [(f_m – f₁)/(2f_m – f₁ – f₂)]*c

 

[Here,

L1 = Lower boundary of the modal class

f_m = Frequency of the modal class

f₁ = Frequency of the class interval immediately preceding the modal class

f₂ = Frequency of the class interval immediately following the modal class

c = Width of the modal class

Modal class = The class interval having the largest frequency

 

Relation between Mean, Median and Mode (Empirical Relation)

 

Mean – Mode = 3(Mean – Median)

Or, Mode = 3Median – 2Mean