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Finding Sample Variance & Standard Deviation

Given: The times, in seconds, required for a sample of students to perform a required task were:

Find: a) The sample variance, s2

6, 10, 13, 11, 12, 8

b) The sample standard deviation, s

Using the Definition Formula

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The calculation of a sample statistic requires the use of a formula. In this case, use:

(x-x)2

n -1Sample variance: s2 =

The Formula - Knowing Its Parts

• s2 is “s-squared”, the sample variance

• (x-x) is the “deviation from mean”

• x is “x-bar”, the sample’s mean

xs2

(x-x)

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Sample variance: s2 = (x-x)2

n -1(x-x)2

The Formula - Knowing Its Parts (Cont’d)

(Do you have your sample data ready to use?)

• n -1 is the “sample size less 1”

• (x-x)2 is the “sum of all squared deviations”

• (x-x)2 is the “squared deviation from the mean”

(x-x)2

n -1

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(11-10)2 (12-10)2 (8-10)2

(-4)2 (0)2 (3)2(-4)2 (0)2 (3)2

1013( - )213-101010( - )210-10( - )2 8-1012-1011-106 106-10

0 9 1 4 416(1)2 (2)2 (-2)2

First, find the numerator:

Finding the Numerator

= (x-x)2

n -1s2 =

+ + + + +=

(x-x)2

n -1s2 =

(1)2 (2)2 (-2)2

=+ + + + +

=+ + + + +

=34

Sample = { 6, 10, 13, 11, 12, 8 } and mean x = 10.0

0 9 1 4 416

5

- 1

Finding the Denominator

Next, find the denominator:

Sample = { 6, 10, 13, 11, 12, 8 }

1 2 3 4 5 6

5

n = 61 2 3 4 5 6

(x-x)2

n -1Sample variance: s2 = = 34

n -1 = 666 = 55

(x-x)2

n -1 = 34s2 =

6

Finding the Answer (a)

Lastly, divide and you have the answer!

6.8

The sample variance is 6.8

Note: Variance has NO unit of measure, it’s a number only

5 (x-x)2

n -1 = 34s2 =

=5 (x-x)2

n -1 = 34s2 =

7

Finding the Standard Deviation (b)

The standard deviation is the square root of variance:

s = s2

Therefore, the standard deviation is:

s = s2 = 6.8 = 2.60768 =

The standard deviation of the times is 2.6 seconds

Note: The unit of measure for the standard deviation is the unit of the data

2.6