Statistics 1: Introduction to Probability and Statistics Section 3-4 Measures of Position or Relative Standing Where is this data value
with respect to the other values in the population or in the sample? Measures of position Z-scores Percentiles
Measures of position Z-scores position with respect to mean scale is in sigmas; the number of standard deviations away from the mean z-score with sample
statistics x x z s z-score with population parameters
x z z-score practice Given : mean = 38 and st. dev. = 6
If x = 28, the z-score = ? If x = 42, the z-score = ? If x = 46, the z-score = ? z-score practice Given : mean = 38 and st. dev. = 6
If x = 28, the z-score = - 1.67 If x = 42, the z-score = 0.67 If x = 46, the z-score = 1.33 What makes a z-score unusual ? A z-score will be considered
unusual if its absolute value is greater than 2. -3.44 is unusual 1.91 is not unusual 2.08 is unusual Which z-score is the most unusual ?
For the following z-scores, -1.67, 0.67, and 1.33, -1.67 is the most unusual, because |-1.67| is biggest, or farthest away from the mean Measures of position Percentiles
position with respect to order in the sorted data set scale is percent 0% to 100%. The k Percentile; Pk th
Pk is the value that divides the lowest k% of the data from the highest (100-k)% of the data Easier said than done The k Percentile; Pk th
Examples P30 is the value that divides the lowest 30% of the data from the highest 70% of the data P70 divides the lowest 70% of the data from the highest 30% of the data
Percentiles: problem #1 For a specified x value, determine what percentile it represents, that is, the percent (k) of the data that are less than x. X = Pk
Problem #1 Given x, what is k in Pk? N values < X k = [-------------------]*100% N values total The k Percentile; Pk
th Data in sorted order : 8, 12, 15, 16, 27 30, 36, 37, 44, 56 (n 10) The k Percentile; Pk
th Data in sorted order : 8, 12, 15, 16, 27 30, 36, 37, 44, 56 P70 37 because 7 out of 10 values are 37
But why not do this? N values > X k = [-------------------]*100% N values total Problem #2 Given k, what value = Pk? L location of Pk in the data
k L *n 100 Problem #2 Given k, what value = Pk? If L is not a whole number
then round it UP! Now, the value at location L in the sorted data Pk Problem #2 Given k, what value = Pk? If L is a whole number, then P average of two
values : the value at location L the value at location L 1 The 70 Percentile; P70 th 8, 12, 15, 16, 27
30, 36, 37, 44, 56 70 L * 10 7 100 th th Average 7 and 8 values
P70 36.5 The 63 Percentile; P63 rd 8, 12, 15, 16, 27 30, 36, 37, 44, 56 63
L * 10 6.3 100 Round 6.3 up to 7 P63 36 Percentile Aliases Deciles :
D1 , D2 , , D9 P10 , P20 , , P90 Quartiles : Q1 , Q2 , Q3 P25 , P50 , P75 Percentile Aliases
Median, D5 , Q2 : all aliases for the 50th percentile, P50