Chapter 3 Section 4 Measures of Position

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 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 The standard deviation is a measure of dispersion that uses the same dimensions as the data (remember the empirical rule) 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

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) 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 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) If the mean was 20 and the standard deviation was 6 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)

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 The population z-score is calculated using the population mean and population standard deviation

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 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 z-scores can be used to compare the relative positions of data values in different samples

Chapter 3 – Section 4 Statistics Biology Kayaking 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 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 Statistics Grade of 82 z-score of (82 – 74) / 12 = .67

Chapter 3 – Section 4 The quartiles are the 25th, 50th, and 75th percentiles Q1 = 25th percentile Q2 = 50th percentile = median Q3 = 75th percentile 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 Quartiles divide the data set into four equal parts The top quarter are the values between Q3 and the maximum Quartiles divide the data set into four equal parts Quartiles divide the data set into four equal parts Quartiles divide the data set into four equal parts

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 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 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

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 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 = Q1 – 1.5  IQR Upper fence = UF = Q3 + 1.5  IQR

Chapter 3 – Section 4 Is the value 54 an outlier? Calculations 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 Calculations Q1 = (4 + 7) / 2 = 5.5 Q3 = (27 + 31) / 2 = 29 IQR = 29 – 5.5 = 23.5 UF = Q3 + 1.5  IQR = 29 + 1.5  23.5 = 64 Using the fence rule, the value 54 is not an outlier 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 Q3 = (27 + 31) / 2 = 29 IQR = 29 – 5.5 = 23.5 UF = Q3 + 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 Q1 = (4 + 7) / 2 = 5.5 Q3 = (27 + 31) / 2 = 29 Is the value 54 an outlier? 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54

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