Standard deviation of the mean

Standard deviation of the mean: If the values of x or f are large, the calculation of AM by the direct method is quite tedious and time consuming, because calculations involved are lengthy. In such a case to minimize the time involved in calculation, we take deviations from an arbitrary point as discussed below. Let x1, x2, x3, …., xn be values of a variable x with corresponding frequencies f1, f2, f3,…, fn respectively. Taking deviations about an arbitrary point ‘A’, we have

Taking the deviation d of the mid-point of class interval from the mean, squaring it to get d2, multiplying this by the frequency of the class, i.e., fd2, adding all the items, i.e., Sfd2, taking the average, Sfd2/n, and then taking the square root, we get, Standard deviations from the mean is given by S.D. = s = sqrt(Sfd2/n)

Standard deviation from the mean: To understand the concept of standard deviation from mean, we should know The population standard deviation, or s, is simply the square root of the population variance. Because the variance is the average of the squared distances of the observations from the mean, the standard deviation is the square root of the average of the squared distances of the observations from the mean. While the variance is expressed in the square of the units used in the data, the standard deviation is in the same units as those used in the data.
The standard deviation of the mean formula is
s = sqrt(s2) = sqrt(S(x – µ)2/N) = sqrt((Sx2/n) – µ2)
where x = observation, µ = population mean, N = total number of elements in the population, S = sum of all values (x – µ)2, or all the values x2, s = population standard deviation,s2 = population variance.The square root of a positive number may be either positive or negative because a2 = (-a)2. When taking the square root of the variance to calculate the standard deviation, however, statisticians consider only the positive square root.

Standard deviation of the mean example
Example: Find out the standard deviation of the following items: 8, 10, 12, 14, 16, 18, 20, 22, 24, 26.
Solution: Calculation of standard deviation.
Size of items (x) Deviation from mean = 17(d) (d2)
8 -9 81
10 -7 49
12 -5 25
14 -3 9
16 -1 1
18 +1 1
20 +3 9
22 +5 25
24 +7 49
26 +9 81
Sx = 170 Sd2 = 330

Arithmetic average or  = Sx/n = 170/10 = 17
Standard deviation or s = sqrt(Sd2/n) = sqrt(330/10) = sqrt(33) = 5.74