Measures of Position -techniques that divide a set of data into equal groups.

1. Ungrouped Data

1.1. Quartile - the three values of the variable that divide an ordered data set into four equal parts. Q1, Q2 and Q3 determine the values for 25%, 50% and 75% of the data. Q2 coincides with the median. Sample Problem : Find the 1st, 2nd and 3rd quartiles of the scores of 15 students in their 20 items math quiz. Their scores are: 8, 9, 10, 12, 12, 12, 13, 13, 14, 16, 16, 18, 19, 19, 20. Q1 = 1/4 (n+1) = 1/4 (15+1) = 16/4 = 4 Q1 = 4th score = 12 Q2 = 1/2 (n+1) = 1/2 (16) = 8 Q2 = 8th score = 13 Q3 = 3/4 (n+1) = 3/4 (16) = 12 Q3 = 12th score = 18

1.2. Decile - nine values of the variable that divide an ordered data set into ten equal parts. The deciles determine the values for 10%, 20%... and 90% of the data. D5 coincides with the median. Sample Problem Find the 1st, 4th and 6th deciles of the scores of 15 students in their 20 items math quiz. Their scores are: 8, 9, 10, 10, 14, 14, 14, 14, 14, 16, 16, 18, 19, 19, 20. D1 = 1/10 (n+1) = 1/10 (16) = 1.6 *note round off to the nearest integer. D1= 2nd score = 9 D4 = 4/10 (n+1) = 4/10 (16) = 6,4 D4 = 6th score = 14 D6 = 6/10 (n+1) = 6/10 (16) = 9.6 D6 = 10th score = 16

1.3. Percentile - the percentiles are the 99 values of the variable that divide an ordered data set into 100 equal parts. The percentiles determine the values for 1%, 2%... and 99% of the data. P50 coincides with the median. Sample Problem Find the 10th, 30th and 40th percentiles of the scores of 10 students in their 20 items math quiz. Their scores are: 8, 9, 10, 10, 14, 16, 18, 19, 19, 20. P10 = k (n+1)/ 100 = 10 (11) / 100 = 1 = 1.1 *note round off to the nearest integer. P10 = 1st score = 8 P30 = 30(11)/ 100 = 330/100 = 3.3 P30 = 3rd score = 10 P40 = 40(11) / 100 = 440/100 = 4.4 P40 = 4th score = 10

2. Grouped Data

2.1. Quartile Calculate the quartiles of the distribution for the following table: Ci fi Fi [50, 60) 8 8 [60, 70) 10 18 [70, 80) 16 34 [80, 90) 14 48 [90, 100) 10 58 [100, 110) 5 63 [110, 120) 2 65 65 Calculating the Second Quartile N/4 = 2(65)/4 = 32.5 Q2 = LB + (32.5 - cf (before Qk) / f of Qk) i Q2 = 70 + (32.5 - 18 / 16) 10 = 79.06

2.2. Decile Calculate the deciles of the distribution for the following table: Ci fi Fi [50, 60) 8 8 [60, 70) 10 18 [70, 80) 16 34 [80, 90) 14 48 [90, 100) 10 58 [100, 110) 5 63 [110, 120) 2 65 65 Calculating the Third Decile 3(65)/10 = 19.5 D3 = LB + (19.5 - cf (before Dk) / f of Dk) i D3 = 70 + (19.5 - 18 / 16) 10 = 70.94

2.3. Percentile Calculate the percentiles of the distribution for the following table: Ci fi Fi [50, 60) 8 8 [60, 70) 10 18 [70, 80) 16 34 [80, 90) 14 48 [90, 100) 10 58 [100, 110) 5 63 [110, 120) 2 65 65 Calculating the 35th percentile 35(65)/100 = 2275/100 = 22.75 P35 = LB + (22.75 - cf (before Pk) / f of Pk) i P35 = 70 + (22.75 - 18 / 16) 10 = 72.97