Chapter 3 Section 4

Chapter 3Section 4 Measures of Position

1 2 3 4 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

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 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

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 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 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 20 would have a z-score of 0.0

Chapter 3 – Section 4 • The population z-score is calculated using the population mean and population standard deviation • 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

Chapter 3 – Section 4 • z-scores can be used to compare the relative positions of data values in different samples • 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

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

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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • Take the average of the 4th and 5th values, orP28 = (7 + 8) / 2 = 7.5, as the 28th percentile

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 • 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 • The topquarter are the values between Q3 and the maximum • Quartiles divide the data set into four equal parts • The topquarter are the values between Q3 and the maximum • The bottomquarter are the values between the minimum and Q1

Chapter 3 – Section 4 • Quartiles divide the data set into four equal parts • The interquartilerange (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 • 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

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

Chapter 3 Section 4