1. Chapter 12, Part 2 STA 291 Summer I 2011

2. Mean and Standard Deviation The five-number summary is not the most common way to describe a distribution numerically. The most common way is to use the mean to measure center and the standard deviation to measure spread.

3. Mean

4. Example The following are the number of jobs a sample of six graduating students applied for: 171523 7 913 Find the mean.

5. Standard Deviation If you are using the mean to describe the center of a distribution, you should use the standard deviation to describe spread. The standard deviation measures the average distance of the observations from the mean. Symbolically, standard deviation is denoted by s.

6. Standard Deviation (cont.)

7. Example (Job Applications) ObservationMeanObs. – Mean(Obs. – Mean) 2 17 15 23 7 9 13 Sum of (Obs. – Mean) 2 n – 1 Variance = [Sum of (Obs. – Mean) 2 ] / (n – 1) Standard Deviation = Square Root of Variance

8. Standard Deviation Properties

9. Relating the Two Methods What method is preferable? – five number summary or – mean and standard deviation The preferable method is largely determined by shape. If we have a symmetric distribution, reporting the mean and standard deviation is preferable. If we have a skewed distribution, reporting the five number summary is preferable.

10. Relating the Two Methods (cont.) Consider the following data sets: Data Set 1: 10 64 71 73 77 85 89 92 Data Set 2: 60 64 71 73 77 85 89 92 mean of Data Set 1 = 70.125 mean of Data Set 2 = 76.375

11. Relating the Two Methods (cont.) If a distribution is symmetric, the mean and median will be close to each other. If you have a symmetric distribution, it is preferable to use the mean, since the mean uses all the data points. If the distribution is skewed, the mean is pulled toward the long tail. In this situation, the median is a better measure of center, since it is less affected by skewness.

12. Relating the Two Methods (cont.) In the computation of the mean, all of the values are weighted equally. As a result, the mean is strongly influenced by extreme observations. On the other hand, the median is simply the middle value, so extreme observations have a substantially lesser effect on it. Thus, if a data set has outliers, the median is often a better measure of the center of the distribution.

13. Mean, Median, and Shape If the mean and median are close together, the distribution is roughly symmetric. If the mean is significantly larger than the median, the distribution is right-skewed. If the mean is significantly smaller than the median, the distribution is left-skewed.

14. Examples Suppose we are told that the mean score for an exam was 79.2 and the median score was 83.5. What is the shape of the distribution of exam scores? Now, suppose the mean score was 81.2 and the median score was 81.1. What is the shape of the distribution of exam scores?