Statistical Mean a stepwise approach

Mean Statistics
In Statistics a branch of Mathematics, the expression for the mathematical mean of a statistical distribution is the mathematical average of all the terms in the data. Here we add up the values of the terms given and divide the sum by the number of terms in the data. This expression is also called the Arithmetic Mean. For example, let us find the Statistical Mean of the following data 6, 8, 5, 10, 10, 12, 8, 6. Here first we need to find the summation of the data values which would be 6 + 8 + 5 +10 + 10 +12+8 +6 = 65. The Arithmetic Mean is got by dividing this sum with the total number of data values. The total number of data values is 8 and so Mean = 65/8= 8.125

Mean Math of the given data is the average of the total number of given data values. It is very simple to calculate we just add up the data values and divide by the count of data values. When the give data values are positive, we just add them and divide by the count to get the mean. The mean of 5, 8, 10, 12, 15 would be, (5+8+10+12+15)/5 = 50/5= 10. When the given data has negative data values, the method would be the same except that we need to combine the like terms. For example, The mean of 5, -3, 7,       12, - 2 would be (5 +(-3) +7 + 12- 2). Combining the like terms, [5+7+12 +(-3-2)] = [24 – 5] = 19. The Mean would be, sum of the data values/total number of data values = 19/5 = 3.8

Short cut Method
A short cut method of calculating the arithmetic mean is based on the property of arithmetic average. In this method the deviations (D) of the items from an assumed mean are first calculated and then multiplied with their respective frequencies (f). Then, the total of these products [summation(fD)] is divided by the total frequencies [summation(f)]and added to the assumed mean(A). The figure we get is the actual arithmetic average or the Arithmetic Mean.

Formula used in the short-cut method of calculating the arithmetic mean:
X(bar) = A + summation(fD)/summation(f)
Given the following data, calculate the Mean using the short cut Method
Weight(Kg) 68 70 72 74 76
Number of students 3 4 2 1 2
Summation of f(i) = 3+4+2+1+2 = 12

Let us assumed mean A= 72, let us tabulate the deviation
X(i) – A f(i).x(i)
(68 – 72) = -4 -4x 3 = -12
(70-72) = -2 -2x4  = -8
(72 – 72) = 0 0 x 2 = 0
(74 – 72) = 2 2 x 1 = 2
(76 – 72) = 4 4 x 2 = 8

Summation(f(i)x(i) = -10
So, we have, summation f(i) = 12, Summation(f(i)x(i) = -10 and A = 72
X(bar) = 72 + (1/12)(-10) = 72 – 0.833 =  71.17 kg