Example 15: Miles RunClass Boundaries Frequency
5.5 – 10.5 110.5 – 15.5 215.5 – 20.5 320.5 – 25.5 525.5 – 30.5 430.5 – 35.5 335.5 – 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
Example 15: Miles RunClass Boundaries Frequency
5.5 – 10.5 110.5 – 15.5 215.5 – 20.5 320.5 – 25.5 525.5 – 30.5 430.5 – 35.5 335.5 – 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) = =
Example 15: Miles RunClass Boundaries Frequency
5.5 – 10.5 110.5 – 15.5 215.5 – 20.5 320.5 – 25.5 525.5 – 30.5 430.5 – 35.5 335.5 – 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) = =
Example 15: Miles RunClass Boundaries Frequency
5.5 – 10.5 110.5 – 15.5 215.5 – 20.5 320.5 – 25.5 525.5 – 30.5 430.5 – 35.5 335.5 – 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) =
Example 15: Miles RunClass Boundaries Frequency x, Midpoint
5.5 – 10.5 110.5 – 15.5 215.5 – 20.5 320.5 – 25.5 525.5 – 30.5 430.5 – 35.5 335.5 – 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) =
Example 15: Miles RunClass Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 810.5 – 15.5 215.5 – 20.5 320.5 – 25.5 525.5 – 30.5 430.5 – 35.5 335.5 – 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) =
Example 15: Miles RunClass Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 810.5 – 15.5 2 (10.5+15.5)/2 = 1315.5 – 20.5 320.5 – 25.5 525.5 – 30.5 430.5 – 35.5 335.5 – 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) =
Example 15: Miles RunClass Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 810.5 – 15.5 2 (10.5+15.5)/2 = 1315.5 – 20.5 3 (15.5+20.5)/2 = 1820.5 – 25.5 525.5 – 30.5 430.5 – 35.5 335.5 – 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) =
Example 15: Miles RunClass Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 810.5 – 15.5 2 (10.5+15.5)/2 = 1315.5 – 20.5 3 (15.5+20.5)/2 = 1820.5 – 25.5 5 (20.5+25.5)/2 = 2325.5 – 30.5 430.5 – 35.5 335.5 – 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) =
Example 15: Miles RunClass Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 810.5 – 15.5 2 (10.5+15.5)/2 = 1315.5 – 20.5 3 (15.5+20.5)/2 = 1820.5 – 25.5 5 (20.5+25.5)/2 = 2325.5 – 30.5 4 (25.5+30.5)/2 = 2830.5 – 35.5 335.5 – 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) =
Example 15: Miles RunClass Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 810.5 – 15.5 2 (10.5+15.5)/2 = 1315.5 – 20.5 3 (15.5+20.5)/2 = 1820.5 – 25.5 5 (20.5+25.5)/2 = 2325.5 – 30.5 4 (25.5+30.5)/2 = 2830.5 – 35.5 3 (30.5+35.5)/2 = 3335.5 – 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) =
Example 15: Miles RunClass Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 810.5 – 15.5 2 (10.5+15.5)/2 = 1315.5 – 20.5 3 (15.5+20.5)/2 = 1820.5 – 25.5 5 (20.5+25.5)/2 = 2325.5 – 30.5 4 (25.5+30.5)/2 = 2830.5 – 35.5 3 (30.5+35.5)/2 = 3335.5 – 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) =
Example 15: Miles RunClass Boundaries Frequency x, Midpoint
5.5 – 10.5 1 (5.5+10.5)/2 = 810.5 – 15.5 2 (10.5+15.5)/2 = 1315.5 – 20.5 3 (15.5+20.5)/2 = 1820.5 – 25.5 5 (20.5+25.5)/2 = 2325.5 – 30.5 4 (25.5+30.5)/2 = 2830.5 – 35.5 3 (30.5+35.5)/2 = 3335.5 – 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)
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 810.5 – 15.5 2 (10.5+15.5)/2 = 1315.5 – 20.5 3 (15.5+20.5)/2 = 1820.5 – 25.5 5 (20.5+25.5)/2 = 2325.5 – 30.5 4 (25.5+30.5)/2 = 2830.5 – 35.5 3 (30.5+35.5)/2 = 3335.5 – 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)
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 810.5 – 15.5 2 (10.5+15.5)/2 = 1315.5 – 20.5 3 (15.5+20.5)/2 = 1820.5 – 25.5 5 (20.5+25.5)/2 = 2325.5 – 30.5 4 (25.5+30.5)/2 = 2830.5 – 35.5 3 (30.5+35.5)/2 = 3335.5 – 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)
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 810.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 2615.5 – 20.5 3 (15.5+20.5)/2 = 1820.5 – 25.5 5 (20.5+25.5)/2 = 2325.5 – 30.5 4 (25.5+30.5)/2 = 2830.5 – 35.5 3 (30.5+35.5)/2 = 3335.5 – 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)
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 810.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 2615.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 5420.5 – 25.5 5 (20.5+25.5)/2 = 2325.5 – 30.5 4 (25.5+30.5)/2 = 2830.5 – 35.5 3 (30.5+35.5)/2 = 3335.5 – 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)
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 810.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 2615.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 5420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 11525.5 – 30.5 4 (25.5+30.5)/2 = 2830.5 – 35.5 3 (30.5+35.5)/2 = 3335.5 – 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)
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 810.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 2615.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 5420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 11525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 11230.5 – 35.5 3 (30.5+35.5)/2 = 3335.5 – 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)
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 810.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 2615.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 5420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 11525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 11230.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 9935.5 – 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)
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 810.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 2615.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 5420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 11525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 11230.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 9935.5 – 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) ( · )
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 810.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 2615.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 5420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 11525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 11230.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 9935.5 – 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) ( · )
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 810.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 2615.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 5420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 11525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 11230.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 9935.5 – 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) ,
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 810.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 2615.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 5420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 11525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 11230.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 9935.5 – 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) ,
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 2615.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 5420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 11525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 11230.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 9935.5 – 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) ,
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 16915.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 5420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 11525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 11230.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 9935.5 – 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) ,
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 16915.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 32420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 11525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 11230.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 9935.5 – 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) ,
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 16915.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 32420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 52925.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 78430.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 9935.5 – 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) ,
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 16915.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 32420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 52925.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 78430.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 108935.5 – 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) ,
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 16915.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 32420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 52925.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 78430.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 108935.5 – 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) ,
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 16915.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 32420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 52925.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 78430.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 108935.5 – 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) , ( · )
Example 15: Miles RunClass Boundaries Frequency x, Midpoint f∙x x2 f∙x2
5.5 – 10.5 1 (5.5+10.5)/2 = 8 1∙8 = 8 82 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 16915.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 32420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 52925.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 78430.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 108935.5 – 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) , ( · )
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 16915.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 32420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 52925.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 78430.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 108935.5 – 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) , ( · )
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 32420.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 52925.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 78430.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 108935.5 – 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) , ( · )
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 97220.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 52925.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 78430.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 108935.5 – 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) , ( · )
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 97220.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 264525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 78430.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 108935.5 – 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) , ( · )
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 97220.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 264525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 313630.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 108935.5 – 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) , ( · )
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 97220.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 264525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 313630.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 326735.5 – 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) , ( · )
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 97220.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 264525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 313630.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 326735.5 – 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) , ( · )
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 97220.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 264525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 313630.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 326735.5 – 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) ( · )
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 97220.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 264525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 313630.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 326735.5 – 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) ( · )
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 97220.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 264525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 313630.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 326735.5 – 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 − 240100380 = 26100
380 ≈ 68.7
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 97220.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 264525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 313630.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 326735.5 – 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
: = 26100380 ≈ 68.7
Example 15: Miles RunClass 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 = 6410.5 – 15.5 2 (10.5+15.5)/2 = 13 2∙13 = 26 132 = 169 2∙169 = 33815.5 – 20.5 3 (15.5+20.5)/2 = 18 3∙18 = 54 182 = 324 3∙324 = 97220.5 – 25.5 5 (20.5+25.5)/2 = 23 5∙23 = 115 232 = 529 5∙529 = 264525.5 – 30.5 4 (25.5+30.5)/2 = 28 4∙28 = 112 282 = 784 4∙784 = 313630.5 – 35.5 3 (30.5+35.5)/2 = 33 3∙33 = 99 332 = 1089 3∙1089 = 326735.5 – 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
: = 26100380 ≈ 68.7 : = = 26100
380 ≈ 8.3
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