MATH 1107 Elementary Statistics Lecture 5 The Standard Deviation as a Ruler

Following from the example in your book (page 100)…is it a more impressive performance to win the 800 meters by 8 seconds or the long jump by 60 cm? Was Babe Ruth a better hitter than Barry Bonds? MATH 1107 – Using the STD DEV for Relative Standing

MATH 1107 – Using the STD DEV for Relative Standing The only way to compare values in different units is to standardize the deviations from the means. In other words, we first have to convert all of the values into similar units – standard deviations from the respective means. THEN, we can compare them directly. This is done through the application of a Z-score: (y – y) z = s Value of interest Mean of data Std dev of data

MATH 1107 – Using the STD DEV for Relative Standing Continuing with the example in the book (page 101): The winning 800m time of 129 second was 8 seconds better than the mean of 137 seconds. The std dev of all times was 5 seconds. So: ( )/5 = -1.6 In other words, the winning time was 1.6 standard deviations below the mean. The winning long jump was 60cm longer than the average 6m jump. The std dev of all jumps was 30cm. So: 60/30 = 2 In other words, the winning jump was 2 standard deviations above the mean.

MATH 1107 – Using the STD DEV for Relative Standing Example from page 105: The SAT scores have a mean of 1000 and a std dev of 200. The ACT scores have a mean of 20.8 and a std dev of 4.8. What ACT score would be equivalent to an SAT score of 1220? First, lets convert the SAT score of 1220 to a Z-score: ( )/200 = 1.10 Next, lets solve for the ACT score that has a Z-score of 1.10: (x-20.8)/4.8 = 1.10 X=26.08

MATH 1107 – Using the STD DEV for Relative Standing Spot the Jack Russell weighs 19 pounds. The mean weight for a Jack Russell Terrier is 16 pounds with a std dev of 1.5 pounds. Desdi the Maine Coon cat weighs 18 pounds and frequently kicks Spot’s but around the house. The mean weight for a Maine Coon is 17 pounds with a std dev of.75 pounds. Which animal is most in need of a diet?

MATH 1107 – Using the STD DEV for Relative Standing The Rule (page 107): Many distributions in life are “normal” or bell shaped. When data follows a normal curve, the following is true:

MATH 1107 – Using the STD DEV for Relative Standing This is an important concept, because it allows us to determine the probabilities associated with certain outcomes. This is true, because the Z- scores can be converted into probabilities of occurrence.

MATH 1107 – Using the STD DEV for Relative Standing Example from page 107: Suppose it takes you 20 minutes to drive to campus, with a standard deviation of 2 minutes. 1. How often will you arrive on campus in less than 22 minutes? 2. How often will it take you more than 24 minutes? 3. 75% of the time you will arrive in X minutes or less. Solve for x.

MATH 1107 – Using the STD DEV for Relative Standing Fun EXCEL exercises!