Z Scores and Percentiles Lesson 3 1 Part

Z Scores and Percentiles Lesson 3. 1 Part 2

Histograms ¢ What do we remember about histograms? ¢ For a data set, What does the mean tell us? l What does the standard deviation tell us? l

Normal Distribution ¢ A family of distributions (histograms) with a symmetrical bell shape

Standard Normal Distribution A special case of the normal distribution with a mean of 0 and a standard deviation of 1.

Normal vs. Standard Normal ¢ Most normal curves are not standard normal curves l l They may be translated along the x axis (different means) They might be wider or thinner (different standard deviations) standard σ>1 σ<1

Why is this a problem? ¢ Imagine there are two MDM 4 U classes. Mr. X teaches one section, while Ms. Y teaches the other one. l They both have a quiz l Andy scored 60% on Mr. X’s test, while Mason scored a 70% on Ms. Y’s test. Who did better? l

Working between distributions l ¢ Andy scored 60% on Mr. X’s test, while Mason scored a 70% on Ms. Y’s test. Who did better? It is hard to compare these two different quizzes… maybe Mr. X’s was tougher, and a 60% on his quiz is better than a 70% on Ms. Y’s quiz How many standard deviations away from the mean are these scores? This would tell us how we should compare them.

Z-scores ¢ The z-score for a given piece of data is how far away it is from the mean – it counts the number of standard deviations ¢ For example: a z-score of 2 means the data is two standard deviations above the mean A positive z-score indicates that the value lies above the mean. A negative z-score indicates that the value lies below the mean. ¢ ¢

Calculating z-scores The data The mean number of standard deviations x is away from the mean The deviation The z-score is the deviation divided by the standard deviation The standard deviation

Ex. # 1: Calculating Z-Scores For the distribution with a mean of 20 and a standard deviation of 4, determine the number of standard deviations each piece of data lies above or below the mean. a) x = 11 b) x = 25

Why z-scores? convert to standard

Why z-scores? Z-Scores allow us to convert any normal distribution to a standard normal distribution ¢ This lets us compare distributions that have different means and standard deviations. ¢

Ex. #2 Cayley: got an 84, mean was 74, standard dev was 8 Lauren: got an 83, mean was 70, standard dev was 9. 8 Cayley’s z-score: Lauren’s z score was higher – she did “better” when compared with the rest of her class

Percentiles ¢ ¢ ¢ Percentiles are another way to talk about z-scores the kth percentile is the data value that is greater than k% of the population The table on page 606 and 607 relates zscores to percentiles. Lauren: her z-score was 1. 32 Looking up 1. 32 in the z-score table, you find 0. 9066 Lauren’s mark is at the 90. 66 percentile, or the 90 th percentile (always round down) She did better than 90% of the students

Ex. #3: Calculating Percentiles Fish in a lake have a mean length of 20 cm and a standard deviation of 5 cm. Find the percent of the population that is less than or equal to the following lengths (the percentile). a) 23 cm b) 11 cm c) 30 cm

Ex. #4: See the textbook question on page 145 (Example 5). a) b) c)

Practice Questions Page 148 #4, 5, 9, 14