Calculate the variance and standard deviation of a frequency distribution.

1. Example 15: Miles Run
Class Boundaries Frequency
5.5 – 10.5 1
10.5 – 15.5 2
15.5 – 20.5 3
20.5 – 25.5 5
25.5 – 30.5 4
30.5 – 35.5 3
35.5 – 40.5 2
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation

2. Example 15: Miles Run
Class Boundaries Frequency
5.5 – 10.5 1
10.5 – 15.5 2
15.5 – 20.5 3
20.5 – 25.5 5
25.5 – 30.5 4
30.5 – 35.5 3
35.5 – 40.5 2
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
= =

3. Example 15: Miles Run
Class Boundaries Frequency
5.5 – 10.5 1
10.5 – 15.5 2
15.5 – 20.5 3
20.5 – 25.5 5
25.5 – 30.5 4
30.5 – 35.5 3
35.5 – 40.5 2
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
= =

4. Example 15: Miles Run
Class Boundaries Frequency
5.5 – 10.5 1
10.5 – 15.5 2
15.5 – 20.5 3
20.5 – 25.5 5
25.5 – 30.5 4
30.5 – 35.5 3
35.5 – 40.5 2
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
=

5. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint
5.5 – 10.5 1
10.5 – 15.5 2
15.5 – 20.5 3
20.5 – 25.5 5
25.5 – 30.5 4
30.5 – 35.5 3
35.5 – 40.5 2
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
=

6. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 8
10.5 – 15.5 2
15.5 – 20.5 3
20.5 – 25.5 5
25.5 – 30.5 4
30.5 – 35.5 3
35.5 – 40.5 2
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
=

7. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13
15.5 – 20.5 3
20.5 – 25.5 5
25.5 – 30.5 4
30.5 – 35.5 3
35.5 – 40.5 2
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
=

8. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13
15.5 – 20.5 3 (15.5+20.5)/2 = 18
20.5 – 25.5 5
25.5 – 30.5 4
30.5 – 35.5 3
35.5 – 40.5 2
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
=

9. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13
15.5 – 20.5 3 (15.5+20.5)/2 = 18
20.5 – 25.5 5 (20.5+25.5)/2 = 23
25.5 – 30.5 4
30.5 – 35.5 3
35.5 – 40.5 2
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
=

10. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13
15.5 – 20.5 3 (15.5+20.5)/2 = 18
20.5 – 25.5 5 (20.5+25.5)/2 = 23
25.5 – 30.5 4 (25.5+30.5)/2 = 28
30.5 – 35.5 3
35.5 – 40.5 2
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
=

11. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13
15.5 – 20.5 3 (15.5+20.5)/2 = 18
20.5 – 25.5 5 (20.5+25.5)/2 = 23
25.5 – 30.5 4 (25.5+30.5)/2 = 28
30.5 – 35.5 3 (30.5+35.5)/2 = 33
35.5 – 40.5 2
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
=

12. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13
15.5 – 20.5 3 (15.5+20.5)/2 = 18
20.5 – 25.5 5 (20.5+25.5)/2 = 23
25.5 – 30.5 4 (25.5+30.5)/2 = 28
30.5 – 35.5 3 (30.5+35.5)/2 = 33
35.5 – 40.5 2 (35.5+40.5)/2 = 38
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
=

13. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13
15.5 – 20.5 3 (15.5+20.5)/2 = 18
20.5 – 25.5 5 (20.5+25.5)/2 = 23
25.5 – 30.5 4 (25.5+30.5)/2 = 28
30.5 – 35.5 3 (30.5+35.5)/2 = 33
35.5 – 40.5 2 (35.5+40.5)/2 = 38
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)

14. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13
15.5 – 20.5 3 (15.5+20.5)/2 = 18
20.5 – 25.5 5 (20.5+25.5)/2 = 23
25.5 – 30.5 4 (25.5+30.5)/2 = 28
30.5 – 35.5 3 (30.5+35.5)/2 = 33
35.5 – 40.5 2 (35.5+40.5)/2 = 38
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)

15. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13
15.5 – 20.5 3 (15.5+20.5)/2 = 18
20.5 – 25.5 5 (20.5+25.5)/2 = 23
25.5 – 30.5 4 (25.5+30.5)/2 = 28
30.5 – 35.5 3 (30.5+35.5)/2 = 33
35.5 – 40.5 2 (35.5+40.5)/2 = 38
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)

16. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26
15.5 – 20.5 3 (15.5+20.5)/2 = 18
20.5 – 25.5 5 (20.5+25.5)/2 = 23
25.5 – 30.5 4 (25.5+30.5)/2 = 28
30.5 – 35.5 3 (30.5+35.5)/2 = 33
35.5 – 40.5 2 (35.5+40.5)/2 = 38
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)

17. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54
20.5 – 25.5 5 (20.5+25.5)/2 = 23
25.5 – 30.5 4 (25.5+30.5)/2 = 28
30.5 – 35.5 3 (30.5+35.5)/2 = 33
35.5 – 40.5 2 (35.5+40.5)/2 = 38
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)

18. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115
25.5 – 30.5 4 (25.5+30.5)/2 = 28
30.5 – 35.5 3 (30.5+35.5)/2 = 33
35.5 – 40.5 2 (35.5+40.5)/2 = 38
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)

19. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112
30.5 – 35.5 3 (30.5+35.5)/2 = 33
35.5 – 40.5 2 (35.5+40.5)/2 = 38
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)

20. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)

21. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76
= ∑
= 20
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
( · )

22. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
( · )

23. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
,

24. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
,

25. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
,

26. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
,

27. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
,

28. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
,

29. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
,

30. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
,

31. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
, ( · )

32. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
, ( · )

33. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
, ( · )

34. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
, ( · )

35. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 972
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
, ( · )

36. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 972
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 2645
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
, ( · )

37. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 972
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 2645
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 3136
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
, ( · )

38. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 972
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 2645
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 3136
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 3267
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
, ( · )

39. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 972
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 2645
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 3136
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 3267
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444 2∙1444 = 2888
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
, ( · )

40. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 972
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 2645
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 3136
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 3267
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444 2∙1444 = 2888
= ∑
= 20 ∑( · )
= 490
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
( · )

41. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 972
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 2645
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 3136
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 3267
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444 2∙1444 = 2888
= ∑
= 20 ∑( · )
= 490 ∑( · 2)
= 13310
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
( · )

42. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 972
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 2645
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 3136
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 3267
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444 2∙1444 = 2888
= ∑
= 20 ∑( · )
= 490 ∑( · 2)
= 13310
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
∑( · )
− ∑( · )
( − 1)
=
20 13310 − (490)
20(19)
=
266200 − 240100
380
=
26100
380
≈ 68.7

43. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 972
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 2645
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 3136
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 3267
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444 2∙1444 = 2888
= ∑
= 20 ∑( · )
= 490 ∑( · 2)
= 13310
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
26100
380
≈ 68.7

44. Example 15: Miles Run
Class Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 64 1∙64 = 64
10.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 338
15.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 972
20.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 2645
25.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 3136
30.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 3267
35.5 – 40.5 2 (35.5+40.5)/2 = 38 2∙38 = 76 382 = 1444 2∙1444 = 2888
= ∑
= 20 ∑( · )
= 490 ∑( · 2)
= 13310
Below is a frequency distribution of the number of miles run per week for a sample
of individuals. Find the variance and standard deviation
: =
26100
380
≈ 68.7 : =
=
26100
380
≈ 8.3