Describing Center: Mean and Median Section 5.4

Transcription

1 Describing Center: Mean and Median Section 5.4

2 Look at table 5.2 at the right. We are going to make the dotplot of the city gas mileages of midsize cars.

3 How to describe the center of a distribution: x Mean (average value): This is the average of a set of observations, or the average value. To find mean of a set of observations, add their values (x 1, x 2, x 3,, x n ) and divide by the number of observations, n. x = (x 1 + x 2 + +x n )/n

4 Example 1 Find the mean city mileage for the 13 midsize cars in Table 5.2 x x1 x 2... xn n miles per gallon mpg

5 Suppose we take the outlier (Toyota Prius) out of the data. What would the mean city mileage be? x miles per gallon mpg 13

6 The single outlier adds more than 2 mpg to the mean city mileage. This illustrates an important weakness of the mean as a measure of center: The mean is sensitive to the influence of a few extreme observations (non-resistant). These may be outliers, but a skewed distribution that has no outliers will also pull the mean toward its long tail.

7 Median middle value The midpoint or center of an ordered list; middle value of a set of observations; half fall below the median and half fall above. Arrange observations in increasing order (smallest to largest). If the number of observations is odd, the median M is the center observation in the ordered list. If the number of observations is even, the median M is the average of the two center observations in the ordered list. The position of the median is (n + 1)/2.

8 Example 2 Find the median of city mileage of the 13 midsize cars. 1. Arrange the data in increasing order Since the number of observations is odd, the median position is (13 + 1)/2 = 7 th observation.

9 What is the median if we drop the Toyota Prius? Since the number of observations is even, the median position is (12 + 1)/2 = 6.5 which means it is the mean of the 6 th and 7 th observations. (20 +21)/2 = 20.5

10 You see that the median resists the influence of extreme observations better than the mean does. A very high value like the Toyota Prius is simply one observation to the right of center and removing it hardly changed the median at all. In fact, removing an extreme outlier can leave the median completely unchanged while significantly changing the mean

11 Another common numerical summary of a distribution is the: Mode the most frequently occurring value. Like the mean and median, it is a measure of location for a distribution. However, the mode is not necessarily a good measure of center.

12 Example 3 A. Find the mode of the city mileage in table 5.2. The mode is 21 since it occurs 3 times. B. Find the mode of the highway mileage in table 5.2. The modes are 27, 29, 30, 33.

13 We can also identify the mode(s) of a histogram. The mode of this histogram is the class $34,000 to $38,000 since it is the tallest.

14 Measures of Center and Distributions For a symmetrical distribution, the mean, median and the mode are the same.

15 Measure of Center and Distribution For skewed distributions, the mean is pulled to the side of the skew.

16 Worksheet Homework