Explanation Definition: 1. While computing the mean of a frequency distribution, write all products x ⋅ f of the values and their frequencies in a new column of the table. Represent the sum of these products by ∑ ( x ⋅ f ) . Denote the sum of the frequencies by ∑ f . The mean is then ∑ ( x ⋅ f ) ∑ f . 2. If x is a data value in a set whose mean is x ¯ , then x − x ¯ is called x’s deviation from the mean. 3. We denote the standard deviation of a sample of n data values by s = ∑ ( x − x ¯ ) 2 n − 1 . 4. We calculate the standard deviation, s, of a sample that is given as a frequency distribution as s = ∑ ( x − x ¯ ) 2 ⋅ f n − 1 . Here f is the frequency of data value x , and n = ∑ f is the number of data values in the distribution. Calculation: Observe the bar graph carefully. From the graph understand the frequency of each data value. x f 0 13 1 11 2 5 3 0 4 0 5 8 6 12 7 9 8 6 9 5 10 0 11 0 12 2 First find the mean of the given data set. To compute the mean, let us find all products x ⋅ f of the given values and their respective frequencies. For x = 0 and f = 13 , the product x ⋅ f = 0 ⋅ 13 = 0. For x = 1 and f = 11 , the product x ⋅ f = 1 ⋅ 11 = 11. For x = 2 and f = 5 , the product x ⋅ f = 2 ⋅ 5 = 10. For x = 3 and f = 0 , the product x ⋅ f = 3 ⋅ 0 = 0. For x = 4 and f = 0 , the product x ⋅ f = 4 ⋅ 0 = 0. For x = 5 and f = 8 , the product x ⋅ f = 5 ⋅ 8 = 40. For x = 6 and f = 12 , the product x ⋅ f = 6 ⋅ 12 = 72. For x = 7 and f = 9 , the product x ⋅ f = 7 ⋅ 9 = 63. For x = 8 and f = 6 , the product x ⋅ f = 8 ⋅ 6 = 48. For x = 9 and f = 5 , the product x ⋅ f = 9 ⋅ 5 = 45. For x = 10 and f = 0 , the product x ⋅ f = 10 ⋅ 0 = 0. For x = 11 and f = 0 , the product x ⋅ f = 11 ⋅ 0 = 0. For x = 12 and f = 2 , the product x ⋅ f = 12 ⋅ 2 = 24. The sum of these products is 0 + 11 + 10 + 0 + 0 + 40 + 72 + 63 + 48 + 45 + 0 + 0 + 24 = 313. Therefore, ∑ ( x ⋅ f ) = 313. The sum of the frequencies is 13 + 11 + 5 + 0 + 0 + 8 + 12 + 9 + 6 + 5 + 0 + 0 + 2 = 71. Therefore, ∑ f = 71. The mean is then ∑ ( x ⋅ f ) ∑ f = 313 71 = 4.40845070422535. Next find the standard deviation. To begin with, find the deviation from the mean for each value. For x = 0 the deviation from the mean is x ⋅ x ¯ = − 4.40845070422535. For x = 1 the deviation from the mean is x ⋅ x ¯ = − 3.40845070422535. For x = 2 the deviation from the mean is x ⋅ x ¯ = − 2.40845070422535. For x = 3 the deviation from the mean is x ⋅ x ¯ = − 1

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ISBN: 9780134462356

Author: Pirnot

Publisher: PEARSON

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Chapter 14.3, Problem 23E

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The mean and standard deviation of the distribution given by the following graph: