1. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 1 of 23 Chapter 3 Section 4 Measures of Position

2. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 2 of 23 Chapter 3 – Section 4 ●Learning objectives  Determine and interpret z-scores  Determine and interpret percentiles  Determine and interpret quartiles  Check a set of data for outliers 1 2 3 4

3. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 3 of 23 Chapter 3 – Section 4 ●Mean / median describe the “center” of the data ●Variance / standard deviation describe the “spread” of the data ●This section discusses more precise ways to describe the relative position of a data value within the entire set of data

4. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 4 of 23 Chapter 3 – Section 4 ●Learning objectives  Determine and interpret z-scores  Determine and interpret percentiles  Determine and interpret quartiles  Check a set of data for outliers 1 2 3 4

5. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 5 of 23 Chapter 3 – Section 4 ●The standard deviation is a measure of dispersion that uses the same dimensions as the data (remember the empirical rule) ●The distance of a data value from the mean, calculated as the number of standard deviations, would be a useful measurement ●The standard deviation is a measure of dispersion that uses the same dimensions as the data (remember the empirical rule) ●The distance of a data value from the mean, calculated as the number of standard deviations, would be a useful measurement ●This distance is called the z-score

6. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 6 of 23 Chapter 3 – Section 4 ●If the mean was 20 and the standard deviation was 6  The value 26 would have a z-score of 1.0 (1.0 standard deviation higher than the mean) ●If the mean was 20 and the standard deviation was 6  The value 26 would have a z-score of 1.0 (1.0 standard deviation higher than the mean)  The value 14 would have a z-score of –1.0 (1.0 standard deviation lower than the mean) ●If the mean was 20 and the standard deviation was 6  The value 26 would have a z-score of 1.0 (1.0 standard deviation higher than the mean)  The value 14 would have a z-score of –1.0 (1.0 standard deviation lower than the mean)  The value 17 would have a z-score of –0.5 (0.5 standard deviations lower than the mean) ●If the mean was 20 and the standard deviation was 6  The value 26 would have a z-score of 1.0 (1.0 standard deviation higher than the mean)  The value 14 would have a z-score of –1.0 (1.0 standard deviation lower than the mean)  The value 17 would have a z-score of –0.5 (0.5 standard deviations lower than the mean)  The value 20 would have a z-score of 0.0

7. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 7 of 23 Chapter 3 – Section 4 ●The population z-score is calculated using the population mean and population standard deviation ●The sample z-score is calculated using the sample mean and sample standard deviation

8. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 8 of 23 Chapter 3 – Section 4 ●z-scores can be used to compare the relative positions of data values in different samples  Pat received a grade of 82 on her statistics exam where the mean grade was 74 and the standard deviation was 12 ●z-scores can be used to compare the relative positions of data values in different samples  Pat received a grade of 82 on her statistics exam where the mean grade was 74 and the standard deviation was 12  Pat received a grade of 72 on her biology exam where the mean grade was 65 and the standard deviation was 10 ●z-scores can be used to compare the relative positions of data values in different samples  Pat received a grade of 82 on her statistics exam where the mean grade was 74 and the standard deviation was 12  Pat received a grade of 72 on her biology exam where the mean grade was 65 and the standard deviation was 10  Pat received a grade of 91 on her kayaking exam where the mean grade was 88 and the standard deviation was 6

9. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 9 of 23 Chapter 3 – Section 4 ●Statistics  Grade of 82  z-score of (82 – 74) / 12 =.67 ●Statistics  Grade of 82  z-score of (82 – 74) / 12 =.67 ●Biology  Grade of 72  z-score of (72 – 65) / 10 =.70 ●Statistics  Grade of 82  z-score of (82 – 74) / 12 =.67 ●Biology  Grade of 72  z-score of (72 – 65) / 10 =.70 ●Kayaking  Grade of 81  z-score of (91 – 88) / 6 =.50 ●Statistics  Grade of 82  z-score of (82 – 74) / 12 =.67 ●Biology  Grade of 72  z-score of (72 – 65) / 10 =.70 ●Kayaking  Grade of 81  z-score of (91 – 88) / 6 =.50 ●Biology was the highest relative grade

10. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 10 of 23 Chapter 3 – Section 4 ●Learning objectives  Determine and interpret z-scores  Determine and interpret percentiles  Determine and interpret quartiles  Check a set of data for outliers 1 2 3 4

11. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 11 of 23 Chapter 3 – Section 4 ●The median divides the lower 50% of the data from the upper 50% ●The median is the 50 th percentile ●If a number divides the lower 34% of the data from the upper 66%, that number is the 34 th percentile

12. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 12 of 23 Chapter 3 – Section 4 ●The computation is similar to the one for the median ●Calculation  Arrange the data in ascending order  Compute the index i using the formula ●If i is an integer, take the i th data value ●If i is not an integer, take the mean of the two values on either side of i

13. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 13 of 23 Chapter 3 – Section 4 ●Compute the 60 th percentile of 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 34 ●Compute the 60 th percentile of 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 34 ●Calculations  There are 14 numbers (n = 14)  The 60 th percentile (k = 60)  The index ●Compute the 60 th percentile of 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 34 ●Calculations  There are 14 numbers (n = 14)  The 60 th percentile (k = 60)  The index ●Take the 9 th value, or P 60 = 23, as the 60 th percentile

14. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 14 of 23 Chapter 3 – Section 4 ●Compute the 28 th percentile of 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Compute the 28 th percentile of 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Calculations  There are 14 numbers (n = 14)  The 28 th percentile (k = 28)  The index ●Compute the 28 th percentile of 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Calculations  There are 14 numbers (n = 14)  The 28 th percentile (k = 28)  The index ●Take the average of the 4 th and 5 th values, or P 28 = (7 + 8) / 2 = 7.5, as the 28 th percentile

15. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 15 of 23 Chapter 3 – Section 4 ●Learning objectives  Determine and interpret z-scores  Determine and interpret percentiles  Determine and interpret quartiles  Check a set of data for outliers 1 2 3 4

16. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 16 of 23 Chapter 3 – Section 4 ●The quartiles are the 25 th, 50 th, and 75 th percentiles  Q 1 = 25 th percentile  Q 2 = 50 th percentile = median  Q 3 = 75 th percentile ●Quartiles are the most commonly used percentiles ●The 50 th percentile and the second quartile Q 2 are both other ways of defining the median

17. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 17 of 23 Chapter 3 – Section 4 ●Quartiles divide the data set into four equal parts ●The top quarter are the values between Q 3 and the maximum ●Quartiles divide the data set into four equal parts ●The top quarter are the values between Q 3 and the maximum ●The bottom quarter are the values between the minimum and Q 1

18. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 18 of 23 Chapter 3 – Section 4 ●Quartiles divide the data set into four equal parts ●The interquartile range (IQR) is the difference between the third and first quartiles IQR = Q 3 – Q 1 ●The IQR is a resistant measurement of dispersion

19. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 19 of 23 Chapter 3 – Section 4 ●Learning objectives  Determine and interpret z-scores  Determine and interpret percentiles  Determine and interpret quartiles  Check a set of data for outliers 1 2 3 4

20. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 20 of 23 Chapter 3 – Section 4 ●Extreme observations in the data are referred to as outliers ●Outliers should be investigated ●Extreme observations in the data are referred to as outliers ●Outliers should be investigated ●Outliers could be  Chance occurrences  Measurement errors  Data entry errors  Sampling errors ●Extreme observations in the data are referred to as outliers ●Outliers should be investigated ●Outliers could be  Chance occurrences  Measurement errors  Data entry errors  Sampling errors ●Outliers are not necessarily invalid data

21. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 21 of 23 Chapter 3 – Section 4 ●One way to check for outliers uses the quartiles ●Outliers can be detected as values that are significantly too high or too low, based on the known spread ●One way to check for outliers uses the quartiles ●Outliers can be detected as values that are significantly too high or too low, based on the known spread ●The fences used to identify outliers are  Lower fence = LF = Q 1 – 1.5  IQR  Upper fence = UF = Q 3 + 1.5  IQR ●One way to check for outliers uses the quartiles ●Outliers can be detected as values that are significantly too high or too low, based on the known spread ●The fences used to identify outliers are  Lower fence = LF = Q 1 – 1.5  IQR  Upper fence = UF = Q 3 + 1.5  IQR ●Values less than the lower fence or more than the upper fence could be considered outliers

22. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 22 of 23 Chapter 3 – Section 4 ●Is the value 54 an outlier? 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Is the value 54 an outlier? 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Calculations  Q 1 = (4 + 7) / 2 = 5.5  Q 3 = (27 + 31) / 2 = 29 ●Is the value 54 an outlier? 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Calculations  Q 1 = (4 + 7) / 2 = 5.5  Q 3 = (27 + 31) / 2 = 29  IQR = 29 – 5.5 = 23.5  UF = Q 3 + 1.5  IQR = 29 + 1.5  23.5 = 64 ●Is the value 54 an outlier? 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Calculations  Q 1 = (4 + 7) / 2 = 5.5  Q 3 = (27 + 31) / 2 = 29  IQR = 29 – 5.5 = 23.5  UF = Q 3 + 1.5  IQR = 29 + 1.5  23.5 = 64 ●Using the fence rule, the value 54 is probably not an outlier

23. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 23 of 23 Summary: Chapter 3 – Section 4 ●z-scores  Measures the distance from the mean in units of standard deviations  Can compare relative positions in different samples ●Percentiles and quartiles  Divides the data so that a certain percent is lower and a certain percent is higher ●Outliers  Extreme values of the variable  Can be identified using the upper and lower fences