**Unformatted text preview: **Chapter 3 Section 4
Measures of Position Overview 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
Determine and interpret z-scores 2 Determine and interpret percentiles 3 Determine and interpret quartiles 4 Check a set of data for outliers
1 Measures of Position 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 Measures of Position The population z-score is calculated using the population mean and population standard deviation
x- z= The sample z-score is calculated using the sample mean and sample standard deviation
x-x z= s Measures of Position 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 Measures of Position 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 Manual Solution 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 91 z-score of (91 88) / 6 = .50 Biology was the highest relative grade MINITAB Solution
1. 2. 3. Enable Commands from the Editor menu item Open the MINITAB file containing the data Type the following command to obtain the z-score: MTB > let k1=(x-m)/s MTB > print k1
where: let is a command to execute an algebraic operation k1 is a storage location for future reference containing the z-score x is a data value for conversion to a z-score m is the mean of the data s is the standard deviation of the data
Note: substitute the correct numbers from the problem at hand for the items in italics Measures of Position Learning objectives
Determine and interpret z-scores 2 Determine and interpret percentiles 3 Determine and interpret quartiles 4 Check a set of data for outliers
1 Measures of Position 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 Measures of Position 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 Measures of Position th k 60 i = ( n + 1) = ( 14 + 1) = 9 100 100 Measures of Position Compute the 28th percentile of 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54
th k ( n + 1) = 28 ( 14 + 1) = 4.2 i = 100 100 Measures of Position We can also obtain the percentile (or percentile rank) to which a specific data value corresponds. The kth percentile of a data value x is computed as follows: 1. Arrange the data in ascending order 2. Compute the percentile rank using less than x number of data values the formula
Percentile of x = n * 100 Round the answer to the nearest integer Measures of Position Find the percentile rank of 15 using the data below
1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 Calculations 5 Percentile of 15 = *100 = 35.7 14 The percentile rank of 15 is 36. Approximately 36% of the data values are less than 15. Measures of Position Learning objectives
Determine and interpret z-scores 2 Determine and interpret percentiles 3 Determine and interpret quartiles 4 Check a set of data for outliers
1 Measures of Position 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 other ways of designating the median Measures of Position Quartiles divide the data set into four equal parts 1 Measures of Position 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 Measures of Position 3 th k 75 i = ( n +1) = (14 +1) =11.25 100 100 MINITAB There is no specific MINITAB command to calculate percentiles or quartiles We will use the DESCRIBE command to obtain quartiles MTB > describe cx
where: cx is the column containing the data Note: substitute the correct numbers from the problem at hand for the items in italics Chapter 3 Section 4 Learning objectives
Determine and interpret z-scores 2 Determine and interpret percentiles 3 Determine and interpret quartiles 4 Check a set of data for outliers
1 Outliers 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 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 The fences used to identify outliers are Outliers Is the value 54 an outlier? 1, 3, 4, 8, 15, 16, 19, 23, 24, 27, 31, 54 1, 3, 4, 7, 7, 8, 15,16, 19, 23, 24, 27, 31, 33,33, 54 Calculations Example The following data represent the number of inches of rain in Chicago, Illinois, during the month of April for 20 randomly selected years.
0.97 1.14 1.85 2.34 2.47 2.78 3.41 3.48 3.94 3.97 4.00 4.02 4.11 4.77 5.22 5.50 5.79 6.14 6.28 7.69 Example (cont.)
1. Compute the z-score corresponding to the rainfall in 1971 of 0.97 inch. Interpret this result. ...

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