1. Lesson 3 - 4 Measures of Position

2. Objectives Determine and interpret z-scores Determine and interpret percentiles Determine and interpret quartiles Check a set of data for outliers

3. Vocabulary Z-Score – the distance that a data value is from the mean in terms of the number of standard deviations K Percentile – (P k ) divides the lower k th percentile of a set of data from the rest Quartiles – (Q i ) divides the whole data into four (25%) sets of data Outliers – extreme observations IQR (Interquartile range) – difference between third and first quartiles (IQR = Q 3 – Q 1 ) Lower fence – Q 1 – 1.5(IQR) Upper fence – Q 3 – 1.5(IQR)

4. Population z-Score Sample z-Score x – μ x – x z = ------------ σ s Mean of z is 0 and the standard deviation of z is 1. Allows comparisons of different distributions. Z-Scores

5. Smallest Data Value Largest Data Value Median Q1Q1 Q2Q2 Q3Q3 25% of the data Quartiles The index, i (position in sorted list), for the y%-tile will be i = (y/100)(n + 1) Where y is the percent and n is the number in the data set

6. Interquartile Range (IQR) IQR = Q3 – Q1 It is a measure of the spread of the data. It is used to help determine outlying data (data beyond the upper or lower fences). Upper Fence = Q3 + 1.5 IQR Lower Fence = Q1 – 1.5 IQR

7. Example 1 Which player had a better year in 1967? Carl Yastrzemski AL Batting Champ 0.326 Roberto Clemente NL Batting Champ 0.357 AL average 0.236 NL average 0.249 AL stdev 0.01072 NL stdev 0.01257 Roberto did, barely. His z-score was 8.60 and Yaz’s was 8.14

8. Example 2 Given the following set of data: 70, 56, 48, 48, 53, 52, 66, 48, 36, 49, 28, 35, 58, 62, 45, 60, 38, 73, 45, 51, 56, 51, 46, 39, 56, 32, 44, 60, 51, 44, 63, 50, 46, 69, 53, 70, 33, 54, 55, 52 What is the median? What is the Q1? What is the Q3? What is the IQR? 51 45 57 57- 45 = 12

9. Example 2 continued What is the upper fence? What is the lower fence? Are there any outliers? Q3 + 1.5(IQR) = 57 + 1.5(12) = 75 Q1 + 1.5(IQR) = 45 - 1.5(12) = 27 No! UF > max and LF > min

10. Summary and Homework Summary –Data sets should be checked for outliers as the mean and standard deviation are not resistant statistics and any conclusions drawn from a set of data that contains outliers can be flawed –Fences serve as cutoff points for determining outliers (data values less than lower or greater than upper fence are considered outliers) Homework: pg 172 - 174: 9-12, 14, 19