Z Scores. Normal vs. Standard Normal Standard Normal Curve: Most normal curves are not standard normal curves They may be translated along the x axis.

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Z Scores

Normal vs. Standard Normal Standard Normal Curve: Most normal curves are not standard normal curves They may be translated along the x axis (different means) They might be wider or thinner (different standard deviations) 0 0

Why is this a problem? Imagine there are two MDM4U classes. Mr. X teaches one section, while Ms. Y teaches the other one. They both have a quiz Ali scored 60% on Mr. X’s test, while Sandy scored a 70% on Ms. Y’s test. Who did better?

Working between distributions Ali scored 60% on Mr. X’s test, while Sandy 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 means 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 example: a z-score of 2 means the data is two standard deviations above the mean

Understanding z-scores The number of standard deviations is x away from the mean The standard deviation The mean The data The deviation

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

Why z-scores? Z-Scores allow us to convert any normal distribution to a standard normal distribution This lets us compare distributions

Example: How do two students compare if one has a mark of 82% in a class with an average of 72% and a standard deviation of 6, and the other has a mark of 81% in a class with an average of 68% and a standard deviation of 7.6?

Steps for comparing Student 1:Student 2: 82 = 72 + z(6) Z = = 68 + z(7.6) Z = 1.71 Student 2 is doing better than student 1 since it is better to be 1.71 sd above the average than 1.67 above.

Percentiles and Z-Score Table What percent of students have a mark less than or equal to a student with a mark of 85% in a class with an average of 80% and a standard deviation of 11.5%? Looking up 0.43 in the z-score table, you find The student with 85% has a mark in the 66 th percentile. She did better than 66% of the students Find the z-score.

Student 2: got an 81, mean was 68, standard dev was 7.6 her z-score was 1.71 Looking up 1.71 in the z-score table, you find Student 1’s mark is at the percentile, or the 95 th percentile (always round down) She also did better than 95% of the students