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 1 3 2 5 3 4 4 2 5 3 6 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 = 1 and f = 3 , the product x ⋅ f = 1 ⋅ 3 = 3. For x = 2 and f = 5 , the product x ⋅ f = 2 ⋅ 5 = 10. For x = 3 and f = 4 , the product x ⋅ f = 3 ⋅ 4 = 12. For x = 4 and f = 2 , the product x ⋅ f = 4 ⋅ 2 = 8. For x = 5 and f = 3 , the product x ⋅ f = 5 ⋅ 3 = 15. For x = 6 and f = 2 , the product x ⋅ f = 6 ⋅ 2 = 12. The sum of these products is 3 + 10 + 12 + 8 + 15 + 12 = 60. Therefore, ∑ ( x ⋅ f ) = 60. The sum of the frequencies is 3 + 5 + 4 + 2 + 3 + 2 = 19. Therefore, ∑ f = 19. The mean is then ∑ ( x ⋅ f ) ∑ f = 60 19 = 3

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Author: Pirnot

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Chapter 14.3, Problem 24E

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

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

Solution Summary: The author explains how to compute the mean and standard deviation of a frequency distribution.

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To find:

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