Measuring Relative Achievements: Percentile rank and Percentile point

Transcription

1 Measuring Relative Achievements: Percentile rank and Percentile point

2 Consider an example where you receive the same grade on a test in two different classes. In which class did you do better?

3 Why do we need to transform scores? 1. It gives meaning to the scores and allows some kind of interpretation of the scores, 2. It allows direct comparison of two scores For example, a score of 86 on the first test might not mean the same thing as a score of 86 on the second test.

4 Percentile Rank: The percentage of cases in a sample or population that are equal to or lower than a particular score Percentile point: The score associated with a percentile rank, sometimes referred to as percentile

5 Example: if a student's height of 185 cm is the 90th percentile, then 90% of students have heights less than 185 cm, and 10% have heights greater than 185 cm. if a baby's weight of 3000 g is the 40th percentile, then 40% of babies have weights less than 3000 g; 60% have weights greater than 3000 g.

6 Example: if a raw score of 86 on the test might transformed into a percentile rank 95 and interpreted as You did better than 95% of the students who did the test. In this case the student would feel pretty good about the test. On the other hand, a percentile of 30 was obtained, the student might wonder what he or she was doing wrong

7 Percentile rank vs percentile point the percentile rank which is a number between 0 and 100 indicating the percent of cases in a norm group falling at or below that score. The percentile is a point on a scale of scores at or below which a given percent of the cases falls.

8 Question: Emily scored 586 on her police procedures exam. Only 15 % of those taking the exam did better than Emily. What is Emily s percentile rank? For this particular test, what is the 85 th percentile point?

9 Answer: Eighty-five percent of the group writing test did as well as or more poorly than Emily, and 15 % did better. Therefore PR 586 = 85. P 85 = 586

10 Procedure for finding the percentile rank First, rank order the scores from lowest to highest. Next, for each different score, add the percentage of scores that fall below the score to one-half the percentage of scores that fall at the score. The result is the percentile rank for that score.

11 Example: Suppose the obtained scores from 11 students were: Find the percentile rank for the score 31

12 Example: suppose the obtained scores from 11 students were: The first step would be to rank order the scores from lowest to highest

13 Example: Computing the percentage falling below a score of 31, for example, gives the value of 4/11 =.364 or 36.4 %. The 4 in the numerator reflects that of four scores (25, 28, 29, and 29) were less than 31. The 11 in the denominator is N, or the number of scores.

14 Example: The percentage falling at a score of 31 would be 1/11 =.0909 or 9.09%. The numerator being the number of scores with a value of 31 and the denominator again being the number of scores. One-half of 9.09 would be Adding the percentage below to one-half the percentage within would yield a percentile rank of or 40.95%. The percentile rank for the score 31 is 40.95%

15 Exercise: Suppose the obtained scores from 11 students were: Find the percentile rank for the score 33

16 Solution: suppose the obtained scores from 11 students were: The first step would be to rank order the scores from lowest to highest

17 Solution: Computing the percentage falling below a score of 33, for example, gives the value of 6/11 =.5454 or %. The 6 in the numerator reflects that of six scores (25, 28, 29, 30, 31and 32) were less than 33. The 11 in the denominator is N, or the number of scores.

18 Solution: The percentage falling at a score of 33 would be 3/11 =.2727 or 27.27%. The numerator being the number of scores with a value of 33 and the denominator again being the number of scores. One-half of would be Adding the percentage below to one-half the percentage within would yield a percentile rank of or 68.18%. The percentile rank for the score 33 is 68.18%

19 Percentile Rank Formula The percentile rank corresponding to a given value (X) is computed by using the following formula:

20 Application of this algebraic procedure to the score values of 31 and 33 would give the following results: Note that the results are within the rounding error of the percentile rank computed earlier using the procedure described in words.

21 Exercise: Calculate the percentile rank for all the scores:

22 Solution:

23 Percentile rank grouped frequency PR x = [f w (x- L)/i + f b ]100 N Percentile rank X = Score value whose rank you wish to determine L = lower exact limit of the interval in which the score falls N = total number of cases in the distribution f b = cumulative frequency below that interval f w = number of scores in that interval i = interval width

24 Interval f cf

25 Find PR 135 PR x = [f w (x- L)/i + f b ]100 N PR 135 = 3 ( ) /5 + 17] = 69.2 Find PR 146.

26 Percentile Locator Formula For large data sets, one can calculate the locator L to help find a requested percentile. It is computed as follows. Percentile Locator Formula k is the percentile being sought n is the number of elements in our data set.

27 Percentile Locator Formula Once L is obtained, it must be checked to see if it is a whole number. If it is a whole number, the value of Pk is the mean of the Lth data element and the next higher data element. If it is not a whole number, L must be rounded up to the next larger whole number. The value of Pk is then the Lth data element, counting from the lowest. There is an essential difference between rounding up and rounding off. If we round off we get 3. Whereas, if we round up we get 4.

28 Example: For the following data, find the value corresponding to the 25 th percentile 2, 3, 5, 6, 8, 10, 12, 15, 18, 20 Solution: Compute: : (25)/100 = 2.5

29 Example: 2, 3, 5, 6, 8, 10, 12, 15, 18, 20 Round up 2.5 will give you 3 Start at the lowest value and count over to third value, which is 5. Hence, the value 5 corresponds to the 25 th percentile

30 Example: For the following data, find the value corresponding to the 60 th percentile 2, 3, 5, 6, 8, 10, 12, 15, 18, 20 Solution: Compute: : (60)/100 = 6

31 Example: 2, 3, 5, 6, 8, 10, 12, 15, 18, 20 If it is a whole number, the value of Pk is the mean of the Lth data element and the next higher data element. 6 is a whole number, thus, Pk mean of 6 th and 7 th score which is ( )/2 = correspond to the 60 th percentile. Anyone scoring 11 would have done better than 60% of the class

32 Exercise: Suppose the obtained scores from 7 students were: 12, 18, 35, 42, 47, 49, 50 Find the percentile point for 60 th percentile

33 Solution : The first step would be to rank order the scores from lowest to highest. 12, 18, 35, 42, 47, 49, 50 Using the formula L = (60. 7)/100 = 4.2

34 Solution (cont.) Round it up will give you 5 12, 18, 35, 42, 47, 49, 50 Start at the lowest value and count over to 5 th score, which is 47. Hence, the value 47 corresponds to the 60 th percentile

35 Percentiles Grouped Frequency P PR = L + [ N (PR/100) f b ]i f w Percentile: L = lower exact limit of the score or interval containing the percentage of cases of interest N = total number of cases in the distribution f b = cumulative frequency below the score or interval containing the percentile f w = number of cases within the interval containing the percentile i = interval width

36 Interval f cf

37 Determine P 90 P PR = L + [ N (PR/100) f b ]i f w

38 Determine P 90 P PR = L + [ N (PR/100) f b ]i f w P 90 = [ 25 (90/100) 22 ]5 2 =

39 For the data below what is PR 5? What is P 70 Interval f

40 PR 5 PR x = [f w (x- L)/i + f b ]100 N PR 5 = [6 (5-3.5)/3 + 4 ] = 20

41 P 70 P PR = L + [ N (PR/100) f b ]i f w P 70 = L + [ N (PR/100) f b ]i f w P 70 = [ 35 (70/100) 10 ]3 =

42 Thank You