Measures of Center and Variation

Sample mean Example: The birth weights in pounds: 9.2, 6.4, 10.5, 8.1, 7.8. The mean is (9.2 + 6.4 + 10.5 + 8.1 + 7.8) / 5 = 8.4 Figure 2.9 (p. 42) Dot diagram and the sample mean for the birth-weight data.

Sample median Example(cont.): The birth weights in pounds: 9.2, 6.4, 10.5, 8.1, 7.8. The median is 8.1 . Note: The median is not affected by a few very small or very large observations, whereas the presence of such extremes can have a considerable effect on the mean.

The sample 100 p-th percentile is a value such that after the data are ordered from smallest to largest, at least 100p % of the observations are at or below this value and at least 100(1-p) % are at or above this value.

An example of percentiles To determine the first quartile take p=.25 38 * .25 = 9.5 round 9.5 up to 10 10th ordered observation is 4.4 so the first quartile is Q1 = 4.4

Figure 2.10 (p. 49) Dot diagrams with similar center values but different amounts of variation.

Box on Page 50 Sample variance of n observations

Table 2.10 (p. 50) Calculation of Variance

Box on Page 51 Sample Standard Deviation To obtain a measure of variability in the same unit as the data, take the positive square root of the variance: Box on Page 51 Sample Standard Deviation

Other measures of variation Example of phone calls (table 8): Sample range = 53.3 – 1.6 = 51.7 Note: sample range is very sensitive to the existence of a very large or very small observation in the data set. Example of phone calls: Sample interquartile range = 17.5 – 4.4 = 13.1

Boxplots The center half of the data, from the first to the third quartile, is represented by a rectangle (box) with the median indicated by a bar. A line extends from Q3 to the maximum value and another from Q1 to the minimum. Figure 2.12 (p. 55) Boxplot of the length of phone call data in Table 8