This is one of the Standard Deviation Examples. In this Given data Set is S = (1, 2, 3, 2, 2, 4, 4, 4, 5, 3) and total number of data values N = 10, now to find standard deviation of this we first need to calculate the average or Mean value of this data set as:
Mean (Y’) = Sum of all observations/total number of observation
Mean (Y’) = ∑Y / n,
Mean (Y’) = (1 + 2 + 3 + 2 + 2 + 4 + 4 + 4 + 5 + 3) / 10,
Mean (Y’) = 30/10,
Mean (Y’) = 3,
Now to calculate standard deviation we need to calculate the variance of each observation as: (Y – Y’)
(1 - 3)2 = (-2)2 = 4,
(2 - 3)2 = (-1)2 = 1,
(3 - 3)2 = (0)2 = 0,
(2 - 3)2 = (-1)2 = 1,
(2 - 3)2= (-1)2 = 1,
(4 - 3)2 = (1)2 = 1,
(4 - 3)2 = (1)2 = 1,
(4 - 3)2 = (1)2 = 1,
(5 - 3)2 = (2)2 = 4,
(3 - 3)2 = (0)2 = 0,
So here we found the variance of each observation now we need to calculate the average of these variances as:
Average of variance =∑ (Y –Y’)2/n,
= (4 + 1 + 0 + 1 + 1 + 1 + 1 + 1 + 4 + 0) / 10,
= 14/10.
Now we will take Square root of the average value of the above:
Standard Deviation (S) = Square root of the average of variance = sqrt [∑ (Y –Y’)2/n],
Standard Deviation   (S) = √ (14/10),
Standard Deviation   (S) = √ (1.4),
So here standard deviation is the √1.4.