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Lesson 3 - 4

Lesson 3 - 4. Measures of Position. Objectives. Determine and interpret z-scores Determine and interpret percentiles Determine and interpret quartiles Check a set of data for outliers. Vocabulary.

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Lesson 3 - 4

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  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 – (Pk) divides the lower kth percentile of a set of data from the rest • Quartiles – (Qi) divides the whole data into four (25%) sets of data • Outliers – extreme observations • IQR (Interquartile range) – difference between third and first quartiles (IQR = Q3 – Q1) • Lower fence – Q1 – 1.5(IQR) • Upper fence – Q3 – 1.5(IQR)

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

  5. Quartiles Median SmallestData Value LargestData Value Q1 Q2 Q3 25% ofthe data 25% ofthe data 25% ofthe data 25% ofthe data 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 – Q1It 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, 52What 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 Q3 + 1.5(IQR) = 57 + 1.5(12) = 75 What is the upper fence? What is the lower fence? Are there any outliers? 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

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