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Numerical Measures of Position Standardization With the x & s a clear picture of the distribution of a set of values is possible. Often, an individual value is of interest rather than the entire set of values. 2 Standardization Using the x & s, a standard score can be calculated. Standardization allows for • the comparison of individual scores, • even those with different scales Ex. The ability to compare the results of 3 different writing measures Ex. Weights of an individual calculated in lbs and kg 3 Standardization and z Score Standardization is simply a transformation of a value to a linear standard score most frequently a z score. A z score tells us how far a raw value is above or below (±) the mean of a distribution (the group of values). The distance is expressed using standard deviation units. Standard Score z = (x - x ) / s 4 z Score z = (x - x ) / s Ex. Ashley has two test scores on listening comprehension; Test A = 49 and Test B = 99. Test A = 49 Test A has a M = 44 and a s = 4. What would her standard score be for Test A? zA = (49 - 44 ) / 4 = 5/4 = 1.25 Test B = 99 Test B has a M = 104 and a s = 3. What would her standard score be for Test B? zB = (99 - 104 ) / 3 = -5/3 = -1.67 5 z Scores and Normal Curve z scores can be plotted on the Normal Curve. There is a direct 1:1 ratio between standard deviation and z scores. ± 1s = ± 1z ± 2s = ± 2z ± 3s = ± 3z 6 z Scores and Normal Curve z scores can be plotted on the normal curve S -3 -2 -1 M 0 z -3 -2 -1 0 Test A = 1.25 z Test B = -1.67 z +1 +2 +3 +1 +2 +3 7 z Score Regardless of the scale, z scores remain the same Patients’ z scores remain the same even though the measurement scale changes Weight in kg Student x z A 48 B Weight in lbs. x z -1.52 105.2 -1.52 52 -0.17 114.4 -0.17 C 54 -0.51 118.8 -0.51 D 60 1.18 123.2 1.18 8 Percentile Rank Percentile ranks (%ile) are based on the percentage areas under the curve. A percentile rank (PRx) describes the percentage of scores in the distribution that fall above and below the given standard score. Ex. A value at the 65th percentile falls at or above 65% of the values in the distribution. A values at the 43rd percentile falls at or above 43% of the values in the distribution. 9 Percentile Rank The formula for Percentile Rank is as follows: PRx = B + 0.5 fx 100 = n where B = number of values below x where fx = number of values equal to x With this measure, rank order is a must!! 10 Percentile Rank Shoe Sizes for a sample of 12 male students 13 11 10 13 11 10 8 12 9 9 8 9 What is the percentile rank for a shoe size of 12? Rank Order 8 8 9 9 9 10 10 11 11 12 13 13 PR12 = 9 + 0.5(1) 12 X 100 PR12 = 79.17 A value of 12 corresponds approx. to the 79th percentile. 79% of the sampled male students have a shoe size of 12 or smaller. 11 Distributions Percent of values under portions of the normal curve 34.13% 34.13% 13.59% 13.59% 2.14% 2.14% 0.13% S 0.13% -3 -2 -1 0 1 2 3 M 12 Percentile Rank and Normal Curve Percentile Rank can be plotted on the normal curve Size 12 = 79th Size 10 = 50th Size 11 = 67th S -3 -2 -1 M 0 z -3 -2 -1 0 +1 +2 +3 %ile 0.1 2 16 50 84 98 99.9 +1 +2 +3 13