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: From the given table understand each data value. x f 49 1 50 1 51 3 52 3 First find the mean of the given data set. To compute the mean, let us find the sum of given data values. Begin by finding the sum of the given data. Sum of 49, 51, 52, 51, 51, 50, 52, 52 is 408. There are 8 values in the given data. Therefore, the mean is x ¯ = 408 8 = 51. Now find the standard deviation. Observe that there are some repeated values in the given data. Therefore, we will use a frequency table to compute the standard deviation. The frequency of 49 is 1. That is, 49 comes only once in the data set. Similarly, the frequency of 50 is 1. The frequency of 51 is 3. The frequency of 52 is 3. Now find the product x ⋅ f for each data value. Notice that 49 ⋅ 1 = 49 , 50 ⋅ 1 = 50 , 51 ⋅ 3 = 153 , and 52 ⋅ 3 = 156. The deviation is the difference between each product value and the mean. Therefore, the deviation of 49 from the mean 51 is 49 − 51 = − 2. The deviation of 50 from the mean 51 is 50 − 51 = − 1. The deviation of 51 from the mean 51 is 51 − 51 = 0. The deviation of 52 from the mean 51 is 52 − 51 = 1. Square each of the deviation value

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Author: Pirnot

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Chapter 14.3, Problem 26E

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To find:

The standard deviation of A’s income from the mean using the following table:

Family A B C D E F G H

Annual income ($ thousands) 49 51 52 51 51 50 52 52

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Chapter 14.3, Problem 25E

Chapter 14.3, Problem 27E

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The standard deviation of A’s income from the mean using the following table: Family A B C D E F G H Annual income ($ thousands) 49 51 52 51 51 50 52 52

To find: The standard deviation of A’s income from the mean using the following table: Family A B C D E F G H Annual income ($ thousands) 49 51 52 51 51 50 52 52

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Chapter 14 Solutions

To find:

The standard deviation of A’s income from the mean using the following table:

Family A B C D E F G H

Annual income ($ thousands) 49 51 52 51 51 50 52 52