The document provides a frequency distribution of the number of miles individuals run per week, organized into class boundaries. It contains the class boundaries, frequencies, and calculates the midpoints and products of frequency and midpoint (f*x) in order to find the variance and standard deviation of miles run. The goal is to analyze the data and find the variance and standard deviation.

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