Unit 2: Density Curves and the Normal Distribution Standardization (z-scores) 2.2.1

SAT versus ACT A pair of twin bothers, Larry and Barry, are seniors in high school and applying to colleges. They are fairly competitive and want to know who has the better college entrance exam score. Larry took the SAT and scored 1900. Barry took the ACT and scored 29. Who did better? Eli- SAT/ACT…Larry and Barry connected to each one…Larry/SAT and Barry/ACT

Thoughts… “1900 is WAY higher than 29…so Larry did WAY better than Barry!” We don’t have enough information What information would you need? Eli- Larry wins

More Information The highest possible scaled score for the SAT is 2400 The highest possible scaled score for the ACT is 36 http://professionals.collegeboard.com/testing/sat-reasoning/scores/averages https://secure-media.collegeboard.org/digitalServices/pdf/sat/sat-percentile-ranks-composite-crit-reading-math-writing-2014.pdf http://www.act.org/newsroom/data/2013/states.html http://www.act.org/newsroom/data/2013/pdf/profile/National2013.pdf

More Thoughts… “1900 is WAY higher than 29…so Larry did WAY better than Barry!” Larry scored 1900/2400 which is about 79%. Barry scored 20.9/29 which is about 72%...so Larry did better. We don’t have enough information What information would you need? Eli-Larry still wins

More Information According to College Board, the mean national composite (total) score for the SAT in 2013 was 1497. According to ACT.org, the mean national composite score for the ACT in 2013 was 20.9 http://professionals.collegeboard.com/testing/sat-reasoning/scores/averages https://secure-media.collegeboard.org/digitalServices/pdf/sat/sat-percentile-ranks-composite-crit-reading-math-writing-2014.pdf http://www.act.org/newsroom/data/2013/states.html http://www.act.org/newsroom/data/2013/pdf/profile/National2013.pdf

More Thoughts… “1900 is WAY higher than 29…so Larry did WAY better than Barry!” Larry scored 1900/2400 which is about 79%. Barry scored 20.9/29 which is about 72%...so Larry did better. Larry scored 1900…which is 403 points better than the mean of 1497. Barry scored 29 which is only 8.1 points higher than the mean. So Larry did better. We don’t have enough information What information would you need? Eli-Larry still wins

More Information According to College Board, the standard deviation for the national composite (total) score for the SAT in 2013 is 322. According to ACT.org, the standard deviation for the national composite score for the ACT in 2013 is 5.4. http://professionals.collegeboard.com/testing/sat-reasoning/scores/averages https://secure-media.collegeboard.org/digitalServices/pdf/sat/sat-percentile-ranks-composite-crit-reading-math-writing-2014.pdf http://www.act.org/newsroom/data/2013/states.html http://www.act.org/newsroom/data/2013/pdf/profile/National2013.pdf

More Thoughts… “1900 is WAY higher than 29…so Larry did WAY better than Barry!” Larry scored 1900/2400 which is about 79%. Barry scored 20.9/29 which is about 72%...so Larry did better. Larry scored 1900…which is 403 points better than the mean of 1497. Barry scored 29 which is only 8.1 points higher than the mean. So Larry did better. Larry scored 403/322 = 1.25 STANDARD DEVIATIONS higher than the mean. Barry scored 8.1/5.4 = 1.5 STANDARD DEVIATIONS better than the mean

Larry or Barry? Barry did better! Larry scored 403/322 = 1.25 STANDARD DEVIATIONS higher than the mean. Barry scored 8.1/5.4 = 1.5 STANDARD DEVIATIONS better than the mean Relative to the rest of the people that took the SAT, Larry did 0.72 standard deviations BETTER than the average. Relative to the rest of the people that took the ACT, Barry did 1.5 standard deviation BETTER than the average Since Barry scored a high number of standard deviations above the mean than Larry… Eli-Barry wins! Barry did better!

z-score According to Stattrek, A z-score (standardized score) indicates how many standard deviations an element is from the mean http://stattrek.com/statistics/dictionary.aspx?definition=z%20score

z-score A positive z-score represents a data point HIGHER than the mean A negative z-score represents a data point LOWER than the mean A z-score of 0 represents a data point equal to the mean http://stattrek.com/statistics/dictionary.aspx?definition=z%20score

Formula for z-score value of interest mean Standardized score Later… standard deviation

Example #1 Pepper is a student in AP Biology and AP English. She had a test in both classes last Friday. She got an 85 on her Biology test and an 80 on her English test. The mean for both tests was 75. The standard deviation for the Biology test was 10 and the standard deviation for the English test was 5. Relative to the rest of the class, which test did she score “better”? Eli-Bio versus English test

Example #1 Solution Biology Test English Test Score = 85 Mean = 75 Pepper is a student in AP Biology and AP English. She had a test in both classes last Friday. She got an 85 on her Biology test and an 80 on her English test. The mean for both tests was 75. The standard deviation for the Biology test was 10 and the standard deviation for the English test was 5. Relative to the rest of the class, which test did she score “better”? Biology Test English Test Score = 85 Mean = 75 Standard deviation = 10 Score = 80 Standard deviation = 5 Since z = 1 for both the Biology and English tests, Pepper scored 1 standard deviation higher than the mean for BOTH exams. She did the same on both tests relative to the rest of her class.

Example #2 Diego ran the 400 meter race in 49 seconds and the 200 meter race in 22.8 seconds . Suppose that the mean time for the 400 meter race for a varsity high school athlete is 51 seconds with a standard deviation of 1.5 seconds. Suppose the mean time for the 200 meter race for a varsity high school athlete is 23.5 seconds with a standard deviation of 0.85 seconds. Which time should he be most proud of? Explain.

Example #2 Solution 400 meter race 200 meter race Diego ran the 400 meter race in 49 seconds and the 200 meter race in 22.8 seconds . Suppose that the mean time for the 400 meter race for a varsity high school athlete is 51 seconds with a standard deviation of 1.5 seconds. Suppose the mean time for the 200 meter race for a varsity high school athlete is 23.5 seconds with a standard deviation of 0.85 seconds. Which time should he be most proud of? Explain. 400 meter race 200 meter race Diego’s time = 49 seconds Mean = 51 Standard deviation = 1.5 Diego’s time = 22.8 seconds Mean = 23.5 Standard deviation = 0.85 Since the z-score for the 400 meter race (z = -1.33) is LOWER than the z-score for the 200 meter race (z = -0.82), Diego ran faster in the 400 meter race relative to the rest of the field. He should be most proud of the 400 meter time.

In Video Quiz In the 2015 Women’s World Cup in Canada, the United States scored 5 goals in the final game against Japan to win the World Cup. Their z-score for this game was 2.75. Assuming that the standard deviation for goals scored in the 2015 world cup was 0.8, what was the mean number of goals scored in the 2105 World Cup? Eli- World Cup Soccer!

In Video Quiz Solution -5 -5 (0.8) (0.8) In the 2015 Women’s World Cup in Canada, the United States scored 5 goals in the final game against Japan to win the World Cup. Their z-score for this game was 2.75. Assuming that the standard deviation for goals scored in the 2015 world cup was 0.8, what was the mean number of goals scored in the 2105 World Cup? (0.8) (0.8) -5 -5

Wrap Up After this lesson you should be able to… Define z-score/standardized score Use the formula for z-score to standardize data for comparison purposes Use the formula for z-score to find the mean or standard deviation