Standard Deviation of a probability distribution measures the distribution of data from the mean value. Probability distribution maps the probability of a variable within a particular range of values.
For Example: Suppose we toss a coin two times
All possible outcomes: HH, HT, TH, TT
Total number of outcomes = 4
Let a variable, X measures the number of heads in this experiment
Probability of n heads = (Outcomess with n heads)/ (Total number of outcomes)
Number of Heads Probability
(X) (Probability of n heads)
0 0.25
1 0.5
2 0.25
For a probability distribution, plot number of Heads(X) on the x-axis and the probability of n heads on the y-axis.
Range of values in this case = 0-2
Consider a data set with following value
2, 2, 4, 5, 3, 6, 13
Step 1: Count the number of elements in the data set
N = 7
Step 2: Find the mean or average of the data set
Arithmetic Mean (average) = (2 + 2 + 4 + 5 + 3 + 6 + 13)/7
= 35/7 = 5
Step 3: Subtract Mean from each element in the data set and square the result
(2 – 5) = -3 : (-3) ^ 2 = 9
(2 – 5) = -3 : (-3) ^ 2 = 9
(4 – 5) = -1 : (-1) ^ 2 = 1
(5 – 5) = 0 : (0) ^ 2 = 0
(3 – 5) = -2 : (-2) ^ 2 = 4
(6 – 5) = 1 : (1) ^ 2 = 1
(13 –5) = 8 : (8) ^ 2 = 64
For Sample Set
Step 4: Divide the sum of the resulting values with one less than the number of elements in the data set and find the square root
SD(Standard Deviation) = Sqrt ((9 + 9 + 1 + 0 + 4 + 1 + 64)/6)
= 3.82
Variance is the square of standard deviation = SD ^ 2 = 14.59
For Complete Set
Step 4: Divide the sum of the resulting values with the number of elements in the data set and find the square root
SD(Population Standard Deviation) = Sqrt ((9 + 9 + 1 + 0 + 4 + 1 + 64)/7)
= 3.54
Population variance is the square of standard deviation = SD ^ 2
= 12.25
Example of Sample Set: Sample voters in an exit poll
Example of Complete Set: Set of all the voters in an election
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