Z scores and the empirical rule for mound shaped data

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Z-scores and the Empirical Rule For mound-shaped data, we can apply Z-scores to the Empirical Rule to help us create guidelines for identifying “unusual” observations. For a given data set, the mean Z-score is 0, while the standard deviation is 1. If the data is mound- shaped, then: 68% of our observations will have Z-scores between -1 and 1 95% of our observations will have Z-scores between -2 and 2 99% of our observations will have Z-scores between -3 and 3 EXAMPLE: Return to our GMAT scores ( μ = 663 points, σ = 37 points). For the student who had a GMAT score of 719 points, is this an exceptionally high score? What about a student who had a GMAT test score of 800 points? If a student had a test score of 600 points, would this be considered a really bad score? 23
Calculating a Z-score determines how many standard deviations any given observation is away from it’s mean. Since Z-scores are unitless, we can use them to compare observations from two data sets. EXAMPLE: Students who want to get into medical school are required to write the Medical College Admissions Test (MCAT). According to an October 2016 article in Maclean’s magazine, the average MCAT score for students entering University of Toronto’s medical program in fall 2015 was 11.03 points. Let’s assume MCAT scores have a mound shape distribution, with a standard deviation of 0.75 points. () Which student performed better on their respective test: the student who wrote the GMAT and ob- tained a score of 719, or a student who wrote the MCAT and obtained a score of 12.53? PERCENTILES: describe the location of an observation relative to the other observations in the data set, when the data is listed in ascending order . EXAMPLES: A student who scores 736 points on the GMAT is in the 95 th percentile of all students who wrote the GMAT. Quartiles and deciles are specific percentile values. Quartiles divide the data into 4 quarters, while deciles divide the data into 10 sections: 24
Interquartile Range: IQR = Q 3 - Q 1 . There is no single defined method for calculating a percentile. Di erent procedures may result in slightly di erent values for the percentiles. One way to calculate a percentile/quartile value is: To find the p th percentile: Take n p 100 If the number calculates to a decimal value, round up. The p th percentile is then this ordered value in the data set. If the number calculates to a whole number, the p th percentile is then the mean of this ordered value and the one above it from our data set.

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