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14.3 Measures of Dispersion

14.3 Measures of Dispersion. Compute the range of a data set. Calculate the standard deviation of a distribution. Understand how the standard deviation measures the spread of a distribution. Use the coefficient of a variation to compare the standard deviations of different distributions.

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14.3 Measures of Dispersion

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  1. 14.3 Measures of Dispersion • Compute the range of a data set. • Calculate the standard deviation of a distribution. • Understand how the standard deviation measures the spread of a distribution. • Use the coefficient of a variation to compare the standard deviations of different distributions.

  2. The range is a crude measure of the spread of a data set. • The range of a data set is the difference between the largest and smallest data values in the set. • Because of the problem of outliers, the range is not a very effective measure of the spread of a data set.

  3. The standard deviation is a reliable measure of dispersion. • Standard deviation – a measure based on calculating the distance of each data value from the mean. • If x is a data value in a set whose mean is x, then x – x is called x’s deviation from the mean.

  4. We denote the standard deviation of a sample of n data values by s, which is defined as follows: s = Σ(x – x)2 n – 1 • To compute the standard deviation for a sample consisting of n data values: • Compute the mean of the data set; call it x. • Find (x – x)2 for each score x in the data set. • Add the squares found in step 2 and divide this sum by n – 1 which is called the variance. • Compute the square root of the number found in step 3.

  5. Formula for computing the sample standard deviation for a frequency. • We calculate the standard deviation, s, of a sample that is given as a frequency distribution as follows: s = Σ(x – x)2 • f n – 1 x is the mean of the distribution, f is the frequency of data value x, and n = Σf (the number of data values in the distribution).

  6. Example Assume that the professor in a class assigns grades based on the following: • A: scores which are 1.5 or more standard deviations above the mean. • B: scores which are between 0.5 and 1.5 standard deviations above the mean. • C: scores which are between 0.5 standard deviations below the mean and 0.5 standard deviations above the mean. • D: scores which are between 0.5 and 1.5 standard deviations below the mean. • F: scores which are 1.5 or more standard deviations below the mean. • The scores in the class are 85, 78, 95, 81, 70, 88, 84, 90, 68, and 74. • What grade does a person earning a 78 get? • What grade does a person earning an 88 get?

  7. We use the coefficient of variation to compare the standard deviations of different data sets. • For a set of data with mean x and standard deviation s, we define the coefficient of variation, denoted by CV as: CV = s • 100% x

  8. Classwork/Homework • Classwork – Page 815 (5 – 13 odd, 23, 33) • Homework – Page 815 (6 – 14 even, 24, 34)

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