Find the sample variance and standard deviation. 20, 16​, 2​, 7​, 12

skilpadw3 2022-07-26 Answered
Find the sample variance and standard deviation. 20, 16​, 2​, 7​, 12
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Answers (2)

constanzma
Answered 2022-07-27 Author has 8 answers
Step 1
Given data set is 20,16,2,7,12
Let the given data set be x 1 , x 2 , x 3 , x 4 , x 5
x 1 = 20 x 2 = 16 x 3 = 2 x 4 = 7 x 5 = 12
It is clear that, the total number of data points ( x ) = 5
The formula for the sample variance is given by
Sample variance ( σ 2 ) = ( x i x ¯ ) 2 n 1 Formula
where x ¯ = m e a n.
which is given by x ¯ = Sum of all data points Total number of datapoints Formula.
Calculation of mean ( x ¯ )
We know m e a n ( x ¯ ) = Sum of all data points Total number of data points
x ¯ = x 1 + x 2 + x 3 + x 4 + x 5 n
Step 2
x ¯ = 20 + 16 + 2 + 7 + 12 5
x ¯ = 57 5
x ¯ = 11.4
mean ( x ¯ ) of the given data set is x ¯ = 11.4
We know σ 2 = ( x 1 x ¯ ) 2 + ( x 2 x ¯ ) 2 + ( x 3 x ¯ ) 2 + ( x 4 x ¯ ) 2 + ( x 5 x ¯ ) 2 5 1
σ 2 = ( 20 11.4 ) 2 + ( 16 11.4 ) 2 + ( 2 11.4 ) 2 + ( 7 11.4 ) 2 + ( 12 11.4 ) 2 4
σ 2 = ( 8.6 ) 2 + ( 4.6 ) 2 + ( 9.4 ) 2 + ( 4.4 ) 2 + ( 0.6 ) 2 4
σ 2 = 73.96 + 21.16 + 88.36 + 19.36 + 0.36 4
σ 2 = 203.2 4
σ 2 = 50.8
Step 3
We obtained
Variance ( σ 2 ) = 50.8
Standard Deviation ( σ ) = V a r i a n c e
σ = 50.8
σ = 7.127
Standard deviation ( σ ) = 7.127
We obtained
Sample variance ( σ 2 ) = 50.8
Sample standard deviation ( σ ) = 7.127

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Elsa Brewer
Answered 2022-07-28 Author has 3 answers
Step 1:
Add the data
20 + 16 + 2 + 7 + 12 = 57
Divide this by number of data = 5
Mean = 57 / 5 = 11.4
Step 2:
20 11.4 = 8.6 ( 8.6 ) 2 .   73.96
16 11.4 = 4.6. ( 4.6 ) 2 .   21.16
2 11.4 = 9.4 ( 9.4 ) 2 .   88.36
7 11.4 = 4.4. ( 4.4 ) 2 .   19.36
12 11.4 = 0.6. ( 0.6 ) 2 .   0.36
Step 3
Sum of all the squares in the third column = 203.20
Step 4
Sample Variance = 203.20 / 5 1 = 203.20 / 4 = 50.80
Step 5
Standard deviation = sample variance 1 / 2 = 50.80 1 / 2 = 7.1274
So,the standard deviation is 7.1274.

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