11-5 Analyzing Data

Quick Review

You can use measures of central tendency to analyze data. The mean, x ¯ , x bar , comma  equals the sum of the values divided by the number of values. The median is the middle value of a data set in numerical order. The mode is the most frequently occurring value. A data value substantially different from the rest of the data is an outlier. A box-and-whisker plot summarizes information about the range, the median, and the first and third quartiles of a data set.

Example

Find the mean, median, mode, and range of the following set of numbers, and identify any outliers.

3, 3, 4, 6, 19

The mean is 3 + 3 + 4 + 6 + 19 5 = 7 . fraction 3 plus 3 plus 4 plus 6 plus 19 , over 5 end fraction . equals 7 . .

The median is the middle data value, which is 4.

The mode is 3, which occurs twice.

The range is 19 − 3 = 16 . 19 minus 3 equals 16 .

The value 19 is very different from the others and is an outlier.

Exercises

29. Identify the outlier of this set of values.

    17, 15, 16, 15, 9, 18, 16

Find the mean, median, and mode for each set of values.

30. 1, 1, 3, 3, 5, 5, 6, 7, 9, 9, 9, 10, 10
31. 0, 3, 3, 7, 7, 8, 21, 22, 25
32. 8, 9, 11, 12, 13, 15, 16, 18, 18, 18, 27
33. 11, 6, 9, 4, 19, 10, 15, 2

Find the range, Q 1 , q sub 1 , comma  and Q 3 q sub 3  for each set of values.

34. 25, 25, 30, 35, 45, 45, 50, 55, 60, 60
35. 20, 23, 25, 36, 37, 38, 39, 50, 52, 55
36. 36, 36, 48, 65, 75, 82, 92, 101

11-6 Standard Deviation

Quick Review

The range of a data set is one measure of variation, used to describe the spread of data. Another measure of variation is the standard deviation, defined as

σ = ∑ ( x − x ¯ ) 2 n sigma equals . square root of fraction sum . open x minus , x bar , close squared , over n end fraction end root

where x ¯ x bar  is the mean of the data set and n is the number of values. The variance is σ 2 . sigma squared , .

Example

Find the mean, variance, and standard deviation for the following data values: 1, 3, 4, 6, 8, 11, 23.

The mean is x ¯ = 1 + 3 + 4 + 6 + 8 + 11 + 23 7 = 8 . x bar , equals . fraction 1 plus 3 plus 4 plus 6 plus 8 plus 11 plus 23 , over 7 end fraction . equals 8 . .

The sum of the squares of the differences is

( − 7 ) 2 + ( − 5 ) 2 + ( − 4 ) 2 + ( − 2 ) 2 + 0 2 + 3 2 + 15 2 = 328 . open negative 7 close squared . plus . open negative 5 close squared . plus . open negative 4 close squared . plus . open negative 2 close squared . plus , 0 squared , plus , 3 squared , plus , 15 squared , equals 328 .

So the variance is σ 2 = 328 7 ≈ 46.9 , sigma squared , equals , 328 over 7 , almost equal to , 46.9 . comma  and the standard deviation is σ ≈ 46.9 ≈ 6.8 . sigma almost equal to , square root of 46.9 , almost equal to 6.8 . .

Exercises

For each pair of data sets, which is likely to have the greater standard deviation?

37. heights of three people

    heights of twenty people

38. ages of thirty college students

    ages of thirty high school students

39. gas mileages of eighteen sport utility vehicles

    gas mileages of eighteen automobiles of various types

Find the mean and the standard deviation for each set of values.

40. 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 10, 10, 12, 20
41. 15, 17, 19, 20, 14, 23, 12
42. 3.1, 4.5, 7.8, 7.9, 8.0, 9.6, 11.6

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