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Chapter 3 Section 4 Measures of Position Chapter 3 – Section 4 ● Learning objectives 1 Determine and interpret z-scores 2 Determine and interpret percentiles 3 Determine and interpret quartiles 4 Check a set of data for outliers 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 Chapter 3 – Section 4 ● Learning objectives 1 Determine and interpret z-scores 2 Determine and interpret percentiles 3 Determine and interpret quartiles 4 Check a set of data for outliers 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 ● This distance is called the z-score 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) 26 20 6 1 If the mean was 20 and the standard deviation was 6 The value 14 would have a z-score of –1.0 (1.0 standard deviation lower than the mean) 14 20 6 1 If the mean was 20 and the standard deviation was 6 The value 17 would have a z-score of –0.5 (0.5 standard deviations lower than the mean) 17 20 6 05 . If the mean was 20 and the standard deviation was 6 The value 20 would have a z-score of 0.0 20 20 6 0 Chapter 3 – Section 4 ● The population z-score is calculated using the population mean and population standard deviation z x ● The sample z-score is calculated using the sample mean and sample standard deviation xx z s 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 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 Chapter 3 – Section 4 ● 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 ● Remember the Empirical Rule: 68-95-99.7 ● A manufacture of bolts has a quality-control policy that requires it to destroy any bolts that are more than 2 standard deviations from the mean. The mean of the bolts is 8 cm with a standard deviation of 0.05 cm. ● For what lengths will the bolts be destroyed? ● What percentage of the bolts will be destroyed? ● A manufacture of bolts has a quality-control policy that requires it to destroy any bolts that are more than 2 standard deviations from the mean. The mean of the bolts is 8 cm with a standard deviation of 0.05 cm. ● For what lengths will the bolts be destroyed? ● A manufacture of bolts has a quality-control policy that requires it to destroy any bolts that are more than 2 standard deviations from the mean. The mean of the bolts is 8 cm with a standard deviation of 0.05 cm. ● For what lengths will the bolts be destroyed? ● What percentage of the bolts will be destroyed? Chapter 3 – Section 4 ● Learning objectives 1 Determine and interpret z-scores 2 Determine and interpret percentiles 3 Determine and interpret quartiles 4 Check a set of data for outliers Chapter 3 – Section 4 ● The median divides the lower 50% of the data from the upper 50% ● The median is the 50th percentile ● If a number divides the lower 34% of the data from the upper 66%, that number is the 34th percentile 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 k i n 1 100 ● If i is an integer, take the ith data value ● If i is not an integer, take the mean of the two values on either side of i Chapter 3 – Section 4 ● Compute the 60th percentile of 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 34 ● Calculations There are 14 numbers (n = 14) The 60th percentile (k = 60) The index k 60 14 1 9 i n 1 100 100 ● Take the 9th value, or P60 = 23, as the 60th percentile Chapter 3 – Section 4 ● Compute the 28th percentile of 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ● Calculations There are 14 numbers (n = 14) The 28th percentile (k = 28) The index k 28 i n 1 14 1 4.2 100 100 ● Take the average of the 4th and 5th values, or P28 = (7 + 8) / 2 = 7.5, as the 28th percentile Chapter 3 – Section 4 ● Learning objectives 1 Determine and interpret z-scores 2 Determine and interpret percentiles 3 Determine and interpret quartiles 4 Check a set of data for outliers Chapter 3 – Section 4 ● The quartiles are the 25th, 50th, and 75th percentiles Q1 = 25th percentile / also median of the lower 50% Q2 = 50th percentile = median Q3 = 75th percentile / also median of the upper 50% ● Quartiles are the most commonly used percentiles ● The 50th percentile and the second quartile Q2 are both other ways of defining the median Chapter 3 – Section 4 ● Quartiles divide the data set into four equal parts ● The top quarter are the values between Q3 and the maximum ● The bottom quarter are the values between the minimum and Q1 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 = Q3 – Q1 ● The IQR is a resistant measurement of dispersion Chapter 3 – Section 4 ● Learning objectives 1 Determine and interpret z-scores 2 Determine and interpret percentiles 3 Determine and interpret quartiles 4 Check a set of data for outliers Chapter 3 – Section 4 ● 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 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 ● The fences used to identify outliers are Lower fence = LF = Q1 – 1.5 IQR Upper fence = UF = Q3 + 1.5 IQR ● Values less than the lower fence or more than the upper fence could be considered outliers Chapter 3 – Section 4 ● Is the value 54 an outlier? 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ● Calculations Q1 = (4 + 7) / 2 = 5.5 / or median of lower 50% is 7 Q2 = (16 + 19)/2 = 17.5 Q3 = (27 + 31) / 2 = 29 / or median of upper 50% is 27 IQR = 29 – 5.5 = 23.5 / or 27 – 7 = 20 UF = Q3 + 1.5 IQR = 29 + 1.5 23.5 = 64 Or UF = Q3 + 1.5 IQR = 29 + 1.5 20 = 59 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