CHAPTER 5 PERCENTILES AND PERCENTILE RANKS

Transcription

1 CHAPTER 5 PERCENTILES AND PERCENTILE RANKS Percentiles and percentile ranks are frequently used as indicators of performance in bo e academic and corporate worlds. Percentiles and percentile ranks provide information about how a person or ing relates to a larger group. Relative measures of is type are often extremely valuable to researchers employing statistical techniques. For example, most nationally standardized tests scores are reported as percentile ranks. Many graduate schools admit only candidates who score in e upper fifty percent of ose taking certain standardized tests. Honor students are frequently determined by selecting ose at fall in e top ten percent of class. Researchers may want to know which Congresspersons fall in e bottom ten percent of e 1 ratings by ADA. These and oer applications for percentiles and percentile ranks emphasize e need for students to be knowledgeable in is important area of statistics. This chapter provides an explanation of e meaning of ese statistics and instructions for calculating em using data provided in frequency and grouped frequency distributions. Percentiles and percentile ranks are highly similar statistics. Percentiles are calculated as a means of dividing a distribution of values into 2 or more groups. They are used to determine where to draw e line between observed values wiin e distribution. For example: if a teacher wishes to determine e exam score at divides his class in half, wi 50% scoring above and 2 50% scoring below, he determines e point at marks e 50 percentile. In statistics, ere are percentile scales which have special names for various percentiles. Deciles are percentiles which 1 Receiving a low rating by Americans for Democratic Action would probably mean at a Congressperson would be very conservative. 2 The 50 percentile is identical to one of e measures of central tendency The Median 43

2 divide a distribution into ten equal sections. There are nine deciles in each distribution of values which correspond to e 10, 20, 30,...90 percentiles. Quartiles are points in a distribution which divide at distribution into quarters. The first quartile (Q 1) is e 25 percentile. The second quartile (Q 2) is e 50 percentile or e median. The 75 percentile is e ird quartile (Q 3). These quartiles are shown in Figure 5:1. FIGURE 5:1 LOWER QUARTILE, MEDIAN, UPPER QUARTILE A percentile rank is used to determine where a particular score or value fits wiin a broader distribution. For example: A student receives a score of 75 out of 100 on an exam and wishes to determine how her score compares to e rest of e class. She calculates a percentile rank for a score of 75 based on e reported scores of e entire class. Her percentile rank in is example 44

3 would be 80, meaning at 80 percent of scores on e exam were at or below 75. Calculating e percentile rank of a score of 75. This point is illustrated in Figure 5:2 Figure 5:2 PERCENTILE RANK FOR SCORE OF 75 If e data are properly organized and e appropriate formulas carefully followed, e sequential statistical steps for finding percentiles and percentile ranks are not difficult. Researchers working wi percentiles begin wi e desired percentage which is used to calculate e specific value at represents e appropriate dividing point wiin e distribution. Researchers working wi percentile ranks begin wi a specific score or value and calculate e percentage of cases falling at or below it. Even ough it will be unclear at is point as to how e formula for finding percentiles is actually used, e formula is given here and it will be explained momentarily exactly how it will be used. The formula for percentiles is as follows: 45

4 FORMULA FOR FINDING PERCENTILES (Simple Frequency Distribution) Definitions of e Symbols in e Formula k = The percentile one wishes to calculate. The answer will be a value. P= Represents e position wiin e distribution at marks e percentile one wishes to calculate. For example: if one wishes to find e 50 percentile and calculated value of p=5, e 50 percentile is equal to e 5 value in e distribution. Always begin wi e lowest value when counting and round e value of P to e nearest whole number. n= The total number of values in e distribution This is a simple formula at yields e precise location of each percentile line wiin a distribution. Unfortunately, researchers are often unable to obtain data sufficient for e use of is formula. Raer an report each specific observed value, data is often presented in e form of a grouped frequency distribution. The use of such distributions reduces precision of measurements, but it is still possible to calculate a close approximation of any given percentile wi a second formula. FORMULA FOR FINDING PERCENTILES (Grouped Frequency Distribution) Definitions of e Symbols in e Formula k = The percentile one wishes to calculate. The answer will be a value. 46

5 P = (k 100) (n) where k is e percentile and n is e number of values in e distribution. For example, if one wanted to find e 50 percentile, and ere were 400 values (n) in e distribution, P would be e 200 value or (50 100) (400) = 200. L = lower limit of e critical interval. The critical interval is designated "critical" because it is e interval wiin which e percentile will occur. The percentile will be a value at or between e lowest and highest values of at interval. The critical interval is where P occurs. The lower limit of e critical interval is e lowest possible value of e critical interval. For example, e real lower limit for a critical interval $ would be $ because all values above $ would be rounded up to $200 and included in e interval. cf b = f = U = cumulative frequency of all e intervals below (but not including) e critical interval. frequency in e critical interval. Upper Limit of critical interval. This is e highest value at would be included in e critical interval. For example: e interval on an exam would typically have an upper limit of The next step in explaining percentiles will be to apply ese concepts to some real data. Suppose a researcher had data on consulting fees per day paid by e Environmental Protection Agency (EPA) for a sample of 400 consultants and wanted to know what e top ten percent earned per day. First e data are organized in a data matrix such as e one shown in Figure 5:3. Based on ese data, e above question could be answered by finding e appropriate percentile. 47

6 FIGURE 5:3 EPA CONSULTING FEES Interval Y f cf 1 $250-$ (Critical Interval) $200-$ P-or 360 Value occurs here 3 $160-$ $130-$ $100-$ $80-$ $60-$ $40-$ Solution: 3 (1) Question: What is e 90 percentile? Research Conclusion: The top ten percent make $ to $ per day. Ninety percent make at or below $ per day. 3 Finding e 90 percentile will enable e researcher to know what e top 10% earned per day. 48

7 Once e data are organized, calculating a percentile is a very logical process. It is obvious at e 360 (P) value is in interval 2 ($ ) because values 342nd to 368 are in is critical interval, and its lower limit is $ (L). The 90 Percentile is e 360 value and is between $200 and $249. There are 341 (cf) values below e critical interval and 27 (f) values in e critical interval. The upper limit of e critical interval is $ In order to reach e 360 value, one must have 19 of e 27 values in e critical interval because 19 plus e 341 values below e critical interval is equal to 360. The formula en produces an estimate of how far e 19 value will fall from e lower limit of e critical interval 70% (19 27) of e $50 (w) is $35. The $35 is added to e real lower limit of $199.50, and e value which is equivalent to e 90 percentile is $ Ten percent of e values in e distribution are above $234.50, and ninety percent are at or below $ Graphically is conclusion is shown in Figure 5:4. FIGURE 5:4 90TH PERCENTILE FOR EPA FEES 49

8 The preceding example demonstrates at calculation of a percentile begins wi determination of e desired percentage which is en used to find a value. Calculating a percentile rank involves e opposite procedure. One begins wi a value and calculates e percentage of cases falling below it. The formula for finding percentile ranks using a simple frequency distribution is as follows: Formula for Calculating Percentile Rank (Simple Frequency Distributions) PR= X p= Percentile Rank. The answer will be a percentage The position of e score wiin e distribution. Begin wi e lowest value and count e number of cases until reaching e score under consideration. Be sure to include e score under consideration and all ose of equal value when determining X p n= The total number of cases in e distribution This formula provides a simple and accurate value for e percentile rank of any given value wiin a distribution. As e discussion of percentiles demonstrated, however, researchers often lack e data necessary for e use of is formula. When data is presented in a grouped frequency distribution, e formula for calculating e percentile rank is as follows: 50

9 Formula for Calculating Percentile Rank (Grouped Frequency Distributions) PR = Percentile rank. The answer will be a percentage cf b= cumulative frequency of all e values below e critical interval. 4 X = L = U = raw score or value for which one wants to find a percentile rank. lower limit of e critical interval. upper limit of critical interval. f= frequency of e values in e critical interval. In order to explain how percentile ranks are calculated, e same data used for calculating percentiles will be used. The data are organized in a solution matrix shown in Figure 5:5. Assume at a researcher wanted to know e percentage of consultants at made $ or more per day. The calculations would be as follows: 4 The critical interval for percentile ranks is e interval where X is located. 51

10 FIGURE 5:5 EPA CONSULTING FEES Interval Y f cf 1 $250-$ % 2 $200-$ % 3 $160-$ % 4 (critical interval) $130-$ % 5 $100-$ % 6 $80-$ % 7 $60-$ % 8 $40-$ % Steps for solution: Question: What is e percentile rank for $135.00? Conclusion: Round e resulting value to e nearest whole number. Therefore 58% make $ or less per day. 42% make $ or more per day. 52

11 As shown above, when calculating a percentile rank, an additional column is added to e solution matrix. This is called e cumulative percentage ( ) column. This addition is logical because it must be remembered at e final result will be a percentage. Finding e critical interval for calculating percentile ranks is a simple process because one knows e value (X). In is example, e value for X is $135. $ clearly falls in interval 4 ($ ). The real lower limit of at interval is $ After e cumulative percentage ( ) column has been calculated, it is clear at 55.25% ( ) of e values are located in intervals 5, 6, 7, and 8 which are all below e critical interval (4). One now knows at $ is a percentile rank somewhere between 55.25% and 73.25% or e highest points of intervals 4 and 5. The wid of e critical interval is $ The logic employed in step 2 is at $ is $5.50 above e real lower limit of e critical interval ($ minus $ is equal to $5.50). This measures e distance of X from e bottom of e interval. The calculation (f n 100) in step 1 ( ) is necessary to determine e percentage of e values in e critical interval. In is example, 18% of e values are in interval 4. Since $ is not at e top of e interval, one does not add all 18% in e interval to e 55.25% of e values below e interval. In is case, only.18 (5.5 30) of e 18% or 3.34% in e critical interval is added to e 55.25% below. The final result is 58%. One en concludes at 58% make $ or less, and 42% make $ and above per day. Graphically is is shown in Figure 5:6. 53

12 FIGURE 5:6 PERCENTILE RANKS FOR $135 FOR EPA FEES Percentiles and percentile ranks are useful statistical tools. This is especially true of e median or 50 percentile which is e measure of central tendency at is most useful in evaluating severely skewed distributions. There are many applications in statistical research which make practical use of percentiles and percentile ranks. A review of e sequential steps for calculating percentiles and percentile ranks appear at e end of is chapter. A major idea: The Median of a distribution of values is e 50 percentile. A percentile is a value and a percentile rank is a percent. 54

13 Step 1 Organize Data Matrix SEQUENTIAL STATISTICAL STEPS Finding Percentiles What is e first operation at must be performed in an effort to find percentiles? Construct a solution matrix. Step 2 k What percentile is to be obtained? It can be e median (50 ) or any oer percentile selected. Step 3 P What is e position of e percentile you wish to calculate? Solve for P [P=k 100(n)] to identify e critical interval. Step 4 L What is e real lower limit (L) of e critical interval? This is e lowest value at can be included in e interval. Step 5 cf What is e cumulative frequency (cf) of all e intervals below e critical interval? This should be part of your solution matrix Step 6 f What is e frequency of values wiin e critical interval? This e number of values at fall wiin at interval. Step 7 i What is e size or wid of e critical interval? The size is determined by subtracting e lower limit of e interval from e upper limit (U-L). Step 8 Maematical Computation Substitute e appropriate values wiin e formula and solve. Step 9 Conclusions Draw conclusions based on e final result of your data analysis. 55

14 Step 1 Organize Data Matrix SEQUENTIAL STATISTICAL STEPS Finding Percentile Ranks What is e first operation at must be performed in an effort to find percentile ranks? Construct a solution matrix. Step 2 Step 3 X c% What is e raw value or score for which you wish to calculate a percentile rank (PR)? The value may be any value which occurs wiin e distribution. It must be a whole number. What is e cumulative percentage of all e values below e critical interval? cf n. Add e frequencies and divide by e total number of values in e distribution. Step 4 Step 5 i L What is e real lower limit (L) of e critical interval? This is e lowest value at can be included in e interval in which X is located. What is e size or wid of e critical interval? The size is determined by subtracting e lower limit of e interval from e upper limit (U-L). Step 6 What is e frequency of values in e critical interval divided by e total number of values and multiplied by 100? The result is e percentage of e values in e critical interval Step 7 Maematical Computation Substitute e appropriate values wiin e formula and solve. Step 8 Conclusions Draw conclusions based on e final result of your data analysis. 56

15 Exercises Chapter 5 (1) Define e following terms: (A) Percentile (B) Percentile Rank (C) Quartile (D) Decile (E) Critical Interval (F) Cumulative Frequency (G) Cumulative Percentage (H) Wid (2) The average number of days a sample of Social Security recipients had to wait for approval of eir forms in 1996 were as follows: 70, 70, 90, 20, 20, 20, 21, 25, 26, 27, 28, 30, 30, 31, 31, 31, 32, 34, 35, 36, 36, 67, 74, 84, 71, 14, 36, 86, 40, 40, 41, 45, 46, 47, 48, 49, 50, 50, 52, 53, 55, 58, 59, 15, 91, 100, 101. Beginning wi e value 10, organize ese data in intervals (wids) of 15 and find e median and e percentile rank of 30 days waiting time. What conclusions can you draw from your answers about e waiting time of e recipients? Calculate e mean and mode for is distribution. (3) Find e 75 percentile or 3rd quartile and percentile rank of 130 and 145 for e following distribution. Identify e mean and median? X f

16 (4) A class of 15 students received e following scores on a quiz 0,3,3,6,6,6,9,12,12,12,12,15,15,15,15 a) Construct a simple frequency distribution b) Calculate e percentile rank for a student s score of 12 c) Find e 75 percentile (5) The following are final course grades: 101, 94, 89, 89, 89, 88, 85, 82, 81, 78, 77, 77, 77, 76, 76, 74, 73, 73, 73, 72, 71, 71, 71, 69, 67, 67, 63, 54, 46, 45, 34 Beginning wi e value 30, organize ese data in intervals of 9 and calculate e median. What is e mean? What is e 8 decile? What is e first quartile? What is e percentile rank of 68? (6) Charles and Mary scored 29 and 31 on e ACT test. The determining factor for a college scholarship is at a student's score be in e top 10% of eir graduating class. Charles and Mary's graduating class obtained e following ACT scores. ACT Scores f By making use of percentiles and percentile ranks, did Charles and Mary receive a scholarship? Joe obtained a score of 23. One of e criteria for being admitted to e 58

17 college to which he applied is to be in e upper half of his class. Did Joe get admitted? What is e mean ACT score for e class? Show all work and cross check. (7). Find e 3rd, 5, and 9 deciles for e following distribution of scores on an achievement test. What are e median and mean scores? Class interval f What are e percentile ranks for 41, 31, and 25? (8) The following values are gross average daily earnings for construction (X) and manufacturing (Y) workers in X = 48, 51, 57, 64, 70, 71, 83, 91, 72, 63, 64, 48, 70, 36, 30, 27, 76, 66, 78, 88, 89, 87, 45, 50, 93, 20, 21, 24 Y = 48, 52, 57, 65, 73, 74, 82, 84, 86, 87, 92, 99, 34, 35, 28, 49, 55, 53, 71, 75, 90, 62, 26, 66, 50, 20, 21, 22 Beginning wi 20, organize ese data in intervals of 5 and calculate e mean, median and mode for each distribution. What is e percentile rank for $50.00? The top 20% of which group makes less? 59