12.4 – Measures of Position In some cases, the analysis of certain individual items in the data set is of more interest rather than the entire set. It is necessary at times, to be able to measure how an item fits into the data, how it compares to other items of the data, or even how it compares to another item in another data set. Measures of position are several common ways of creating such comparisons.

12.4 – Measures of Position The z-Score The z-score measures how many standard deviations a single data item is from the mean.  

Example: Comparing with z-Scores 12.4 – Measures of Position Example: Comparing with z-Scores Two students, who take different history classes, had exams on the same day. Jen’s score was 83 while Joy’s score was 78. Which student did relatively better, given the class data shown below? Jen Joy Class mean 78 70 Class standard deviation 4 5

Example: Comparing with z-Scores 12.4 – Measures of Position Example: Comparing with z-Scores Jen 83 Joy 78 Class mean 78 70 Class standard deviation 4 5 Jen’s z-score: Joy’s z-score: 83 – 78 78 – 70 = 1.25 = 1.6 4 5 Joy’s z-score is higher as she was positioned relatively higher within her class than Jen was within her class.

12.4 – Measures of Position Percentiles A percentile measure the position of a single data item based on the percentage of data items below that single data item. Standardized tests taken by larger numbers of students, convert raw scores to a percentile score. If approximately n percent of the items in a distribution are less than the number x, then x is the nth percentile of the distribution, denoted Pn.

12.4 – Measures of Position Percentiles Example: The following are test scores (out of 100) for a particular math class. 44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100 Find the fortieth percentile. 40% = 0.4 The average of the 12th and 13th items represents the 40th percentile (P40). 0.4(30) 12 40% of the scores were below 74.5.

Other Percentiles: Deciles and Quartiles 12.4 – Measures of Position Other Percentiles: Deciles and Quartiles Deciles are the nine values (denoted D1, D2,…, D9) along the scale that divide a data set into ten (approximately) equal parts. 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90% Quartiles are the three values (Q1, Q2, Q3) that divide the data set into four (approximately) equal parts. 25%, 50%, and 75%

12.4 – Measures of Position Other Percentiles: Deciles and Quartiles Example: Deciles The following are test scores (out of 100) for a particular math class. 44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100 Find the sixth decile. Sixth decile = 60% The average of the 18th and 19th items represents the 6th decile (D6). 60% = 0.6 0.6(30) 60% of the scores were at or below 82. 18

Other Percentiles: Deciles and Quartiles 12.4 – Measures of Position Other Percentiles: Deciles and Quartiles Quartiles For any set of data (ranked in order from least to greatest): The second quartile, Q2 (50%) is the median. The first quartile, Q1 (25%) is the median of items below Q2. The third quartile, Q3 (75%) is the median of items above Q2.

12.4 – Measures of Position Other Percentiles: Deciles and Quartiles Example: Quartiles The following are test scores (out of 100) for a particular math class. 44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100 Find the three quartiles. Q1= 25% The 8th item represents the 1st quartile (Q1) 25% = 0.25 0.25(30) 25% of the scores were below 72. 7.5

12.4 – Measures of Position Other Percentiles: Deciles and Quartiles Example: Quartiles The following are test scores (out of 100) for a particular math class. 44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100 Find the three quartiles. Q2= 50% = median The average of the 15th and 16th items represents the 2nd quartile (Q2) or the median 50% = 0.5 0.5(30) 50% of the scores were below 78.5. 15

12.4 – Measures of Position Other Percentiles: Deciles and Quartiles Example: Quartiles The following are test scores (out of 100) for a particular math class. 44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100 Find the three quartiles. Q3= 75% The 23rd item represents the 3rd quartile (Q3) 75% = 0.75 0.75(30) 75% of the scores were below 88. 22.5

the first quartile, the median, the third quartile, 12.4 – Measures of Position Box Plots A box plot or a box and whisker plot is a visual display of five statistical measures. The five statistical measures are: the lowest value, the first quartile, the median, the third quartile, the largest value. the lowest value the largest value        

12.4 – Measures of Position Box Plots Example: The following are test scores (out of 100) for a particular math class. 44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100 Q1= 25% = 72 Q2= 50% = median= 78.5 Q3= 75%= 88 Lowest = 44 Largest = 100