1. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   1

2. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   2  A class of 40 students has just returned the Perceptual Speed test score. Aside from the primary question about your grade, you’d like to know more about how you stand in the class  How does your score compare with other in the class? What was the range of performance  What more can you learn by studying the scores?

3. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   3 Score of PERCEPTUAL SPEED Test 2947454048 4945 4049483748465567 3653255833 4342 3251485447403844 4650284452495648 Taken from Guilford p.55

4. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   4 OVERVIEW  When a researcher finished the data collect phase of an experiment, the result usually consist pages of numbers  The immediate problem for the researcher is to organize the scores into some comprehensible form so that any trend in the data can be seen easily and communicated to others  This is the jobs of descriptive statistics; to simplify the organization and presentation of data  One of the most common procedures for organizing a set of data is to place the scores in a FREQUENCY DISTRIBUTION

5. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   5 GROUPED SCORES  After we obtain a set of measurement (data), a common next step is to put them in a systematic order by grouping them in classes  With numerical data, combining individual scores often makes it easier to display the data and to grasp their meaning. This is especially true when there is a wide range of values.

6. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   6 TWO GENERAL CUSTOMS IN THE SIZE OF CLASS INTERVAL 1.We should prefer not fewer than 10 and more than 20 class interval. ○ More commonly, the number class interval used is 10 to 15. ○ An advantage of a small number class interval is that we have fewer frequencies which to deal with ○ An advantage of larger number class interval is higher accuracy of computation

7. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   7 TWO GENERAL CUSTOMS IN THE SIZE OF CLASS INTERVAL 2. Determining the choice of class interval is that certain ranges of units (scores) are preferred. Those ranges are 2, 3, 5, 10, and 20. These five interval sizes will take care of almost all sets of data

8. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   8 Score of PERCEPTUAL SPEED Test 2947454048 4945 4049483748465567 3653255833 4342 3251485447403844 4650284452495648 Taken from Guilford p.55

9. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   9 HOW TO CONSTRUCT A GROUPED FREQUENCY DISTRIBUTION Step 1: find the lowest score and the highest score Step 2: find the range by subtracting the lowest score from the highest Step 3: divide the range by 10 and by 20 to determine the largest and the smallest acceptable interval widths. Choose a convenient width (i) within these limits

10. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   10 Score of PERCEPTUAL SPEED Test 2947454048 4945 4049483748465567 3653255833 4342 3251485447403844 4650284452495648 Range = 42  42 : 10 = 4,2 and 42 : 20 = 2,1

11. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   11 WHERE TO START CLASS INTERVAL  It’s natural to start the interval with their lowest scores at multiples of the size of the interval.  When the interval is 3, to start with 24, 27, 30, 33, etc.; when the interval is 4, to start with 24, 28, 32, 36, etc.

12. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   12 HOW TO CONSTRUCT A GROUPED FREQUENCY DISTRIBUTION Step 4: determine the score at which the lowest interval should begin. It should ordinarily be a multiple of the interval. Step 5: record the limits of all class interval, placing the interval containing the highest score value at the top. Make the intervals continuous and of the same width Step 6: using the tally system, enter the raw scores in the appropriate class intervals Step 7: convert each tally to a frequency

13. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   13 Score of PERCEPTUAL SPEED Test 2947454048 4945 4049483748465567 3653255833 4342 3251485447403844 4650284452495648 Taken from Guilford p.55

14. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   14 FREQUENCY DISTRIBUTION TABLE SCORE 66 - 68 63 - 65 60 - 62 57 -59 54 - 56 51 - 53 48 - 50 45 - 47 42 - 44 39 - 41 36 - 38 33 - 35 30 - 32 27 - 29 24 - 26 SCORE 64 - 67 60 - 63 56 - 59 52 - 55 48 - 51 44 - 47 40 - 43 36 - 39 32 - 35 28 - 31 24 - 27 X max = 67 X min = 25 RANGE = 42 Interval = 3 C.i = 15 Interval = 4 C.i = 11

15. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   15 TALLYING THE FREQUENCIES  Having adopted a set of class intervals, we locate it within its proper interval and write a tally mark in the row for that interval.  Having completed the tallying, we count up the number case within each group to find the frequency (f), or the total number of case within the interval.

16. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   16 FREQUENCY DISTRIBUTION TABLE SCORETALLIESf 66 - 68 63 - 65 60 - 62 57 -59 54 - 56 51 - 53 48 - 50 45 - 47 42 - 44 39 - 41 36 - 38 33 - 35 30 - 32 27 - 29 24 - 26 SCORETALLIESf 64 -67 60 - 63 56 - 59 52 - 55 48 - 51 44 - 47 40 - 43 36 - 39 32 - 35 28 - 31 24 - 27

17. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   17 FREQUENCY DISTRIBUTION TABLE SCORETALLIESf 66 - 681 63 - 650 60 - 620 57 -591 54 - 563 51 - 533 48 - 5010 45 - 476 42 - 444 39 - 413 36 - 383 33 - 352 30 - 321 27 - 292 24 - 261 SCORETALLIESf 64 -671 60 - 630 56 - 592 52 - 554 48 - 5111 44 - 478 40 - 435 36 - 393 32 - 353 28 - 312 24 - 271

18. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   18 SCORE LIMITS OF CLASS INTERVAL  The intervals are therefore labeled 24 to 27, 28 to 31, 32 to 35 and so on.  The top and bottom for each interval are called the score limit.  They are useful for labeling the intervals and in tallying score within the intervals.

19. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   19 EXACT LIMITS OF CLASS INTERVAL  In computations, however, it’s often necessary to work with exact limits.  A score of 40 actually means from 39.5 to 40.5 and that a score of 55 means from 54.5 to 55.5  Thus the interval containing scores 39 through 41 actually covers a range from 38.5 to 41.5

20. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   20 PERCEPTUALSPEEDPERCEPTUALSPEED SCOREfXcLower Exact LimitUpper Exact Limit 64 -67165.5 60 - 63061.5 56 - 59257.5 52 - 55453.5 48 - 511149.5 44 - 47845.5 40 - 43541.5 36 - 39337.5 32 - 35333.5 28 - 31229.5 24 - 27125.5

21. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   21 PERCEPTUALSPEEDPERCEPTUALSPEED SCOREf XcLower Exact LimitUpper Exact Limit 64 -671 65.563.567.5 60 - 630 61.559.563.5 56 - 592 57.555.559.5 52 - 554 53.551.555.5 48 - 5111 49.547.551.5 44 - 478 45.543.547.5 40 - 435 41.539.543.5 36 - 393 37.535.539.5 32 - 353 33.531.535.5 28 - 312 29.527.531.5 24 - 271 25.523.527.5

22. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   22 WARNING!!  Although grouped frequency distribution can make easier to interpret data, some information is lost.  In the table, we can see that more people scored in the interval 48 – 51 than in any other interval  However, unless we have all the original scores to look at, we would not know whether the 11 scores in this interval were all 48s, all, 49s, all 50s, or all 51 or were spread throughout the interval in some way  This problem is referred to as GROUPING ERROR  The wider the class interval width, the greater the potential for grouping error

23. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   23 STEM and LEAF DISPLAY  In 1977, J.W. Tukey presented a technique for organizing data that provides a simple alternative to a frequency distribution table or graph  This technique called a stem and leaf display, requires that each score be separated into two parts.  The first digit (or digits) is called the stem, and the last digit (or digits) is called the leaf.

24. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   24 DataStem & Leaf Display 83 82 63 62 93 78 71 68 33 76 52 97 85 42 46 32 57 59 56 73 74 74 81 76 34567893456789 3 2 1 6 5 2 3 2 6 6 2 7 9 4 3 8 4 6 2 8 3 2 1 3 7

25. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   25 GROUPED FREQUENCY DISTRIBUTION HISTOGRAM AND A STEM AND LEAF DISPLAY 34567893456789 2 3 2 6 6 2 7 9 2 8 3 1 6 4 3 8 4 6 3 5 2 1 3 7 76543217654321 30 40 50 60 70 80 90 0

26. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   26  Place the following scores in a frequency distribution table 2, 3, 1, 2, 5, 4, 5, 5, 1, 4, 2, 2 XfXf 5432154321 3214232142 LEARNING CHECK

27. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   27  A set of scores ranges from a high of X=142 to a low X=65 ○ Explain why it would not be reasonable to display these scores in a regular frequency distribution table ○ Determine what interval width is most appropriate for a grouped frequency distribution for this set of scores ○ What range of values would form the bottom interval for the grouped table? LEARNING CHECK

28. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   28 Initstereng!! Aoccdrnig to a rscheearch at an Elingsh uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoatnt tihng is that frist and lsat ltteer is at the rghit pclae. The rset can be a toatl mses and you can sitll raed it wouthit porbelm. Tihs is bcuseae we do not raed ervey lteter by it slef but the wrod as a wlohe.

29. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   29 MAKING GRAPH POLIGON and HISTOGRAM

30. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   30 MAKING GRAPH POLIGON

31. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   31 PERCEPTUALSPEEDPERCEPTUALSPEED SCOREfXcLower Exact Limit 64 -67165.563.567.5 60 - 63061.559.563.5 56 - 59257.555.559.5 52 - 55453.551.555.5 48 - 511149.547.551.5 44 - 47845.543.547.5 40 - 43541.539.543.5 36 - 39337.535.539.5 32 - 35333.531.535.5 28 - 31229.527.531.5 24 - 27125.523.527.5

32. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   32 POLIGON X f 0 29.5 37.5 45.5 53.5 61.5 25.5 33.5 41.5 49.5 57.5 65.5 12 10 8 6 4 2 21.569.5 Class Interval’s MIDPOINT

33. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   33 PERCEPTUAL SPEED X f 0 29.5 37.5 45.5 53.5 61.5 25.5 33.5 41.5 49.5 57.5 65.5 12 10 8 6 4 2 21.569.5

34. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   34 MAKING GRAPH HISTOGRAM

35. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   35 PERCEPTUALSPEEDPERCEPTUALSPEED SCOREfXcLower Exact Limit 64 -67165.563.567.5 60 - 63061.559.563.5 56 - 59257.555.559.5 52 - 55453.551.555.5 48 - 511149.547.551.5 44 - 47845.543.547.5 40 - 43541.539.543.5 36 - 39337.535.539.5 32 - 35333.531.535.5 28 - 31229.527.531.5 24 - 27125.523.527.5

36. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   36 HISTOGRAM X f 0 27.5 35.5 43.5 51.5 59.5 67.5 23.5 31.5 39.5 47.5 55.5 63.5 12 10 8 6 4 2 Class Interval’s EXACT LIMIT

37. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   37 POLIGON and HISTOGRAM X f 0 27.5 35.5 43.5 51.5 59.5 67.5 23.5 31.5 39.5 47.5 55.5 63.5 12 10 8 6 4 2

38. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   38 THE SHAPE OF A FREQUENCY DISTRIBUTION Symmetrical It is possible to draw a vertical line through the middle so that one side of the distribution is a mirror image of the other Skewed The scores tend to pile up toward one end of the scale and taper off gradually at the other end positive negative

39. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   39  Describe the shape of distribution for the data in the following table XfXf 5432154321 4631146311 LEARNING CHECK The distribution is negatively skewed

40. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   40 PERCENTILES and PERCENTILE RANKS  The percentile system is widely used in educational measurement to report the standing of an individual relative performance of known group. It is based on cumulative percentage distribution.  A percentile is a point on the measurement scale below which specified percentage of the cases in the distribution falls  The rank or percentile rank of a particular score is defined as the percentage of individuals in the distribution with scores at or below the particular value  When a score is identified by its percentile rank, the score called percentile

41. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   41  Suppose, for example that A have a score of X=78 on an exam and we know exactly 60% of the class had score of 78 or lower….…  Then A score X=78 has a percentile of 60%, and A score would be called the 60 th percentile Percentile Rank refers to a percentage Percentile refers to a score

42. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   42 Xfcfc% 5432154321 1584215842 CUMMULATIVE FREQUENCY and CUMULATIVE PERCENTAGE 20 19 14 6 2 100% 95% 70% 30% 10% 1.What is the 95 th percentile? Answer: X = 4.5 2.What is the percentile rank for X = 3.5 Answer: 70%

43. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   43 INTERPOLATION  It is possible to determine some percentiles and percentile ranks directly from a frequency distribution table  However, there are many values that do not appear directly in the table, and it is impossible to determine these values precisely

44. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   44 INTERPOLATION Using the following distribution of scores we will find the percentile rank corresponding to X=7 Xfcfc% 10 9 8 7 6 5 284641284641 25 23 15 11 5 1 100 92 60 44 20 4 Notice that X=7 is located in the interval bounded by the real limits of 6.5 and 7.5 The cumulative percentage corresponding to these real limits are 20% and 44% respectively

45. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   45 Scores (X) – percentage 7.544% 7.0…….. ?? 6.520% STEP 1 For the scores, the width of the interval is 1 point. For the percentage, the width is 24 points STEP 2 Our particular score is located 0.5 point from the top of the interval. This is exactly halfway down the interval STEP 3 Halfway down on the percentage scale would be ½ (24 points) = 12 points STEP 4 For the percentage, the top of the interval is 44%, so 12 points down would be 32%

46. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   46 Using the following distribution of scores we will use interpolation to find the 50 th percentile X f cf c% 20 - 24 15 - 19 10 - 14 5 - 9 0 - 4 2 3 10 2 20 18 15 12 2 100 90 75 60 10 A percentage value of 50% is not given in the table; however, it is located between 10% and 60%, which are given. These two percentage values are associated with the upper real limits of 4.5 and 9.5

47. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   47 Scores (X) – percentage 9.560% ??…….. 50% 4.510% STEP 1 For the scores, the width of the interval is 5 point. For the percentage, the width is 50 points STEP 2 The value of 50% is located 10 points from the top of the percentage interval. As a fraction of the whole interval this is 1/5 of the total interval STEP 3 Using this fraction, we obtain 1/5 (5 points) = 1 point The location we want is 1 point down fom the top of the score interval STEP 4 Because the top of the interval is 9.5, the position we want is 9.5 – 1 = 8.5  the 50 th percentile = 8.5

48. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   48  On a statistics exam, would you rather score at the 80 th percentile or at the 40 th percentile?  For the distribution of scores presented in the following table, LEARNING CHECK X f cf c% 40 - 49 30 - 39 20 - 29 10 - 19 0 - 9 4 6 10 3 2 25 21 15 5 2 100 84 60 20 8 a.Find the 60 th percentile b.Find the percentile rank for X=39.5 c.Find the 40 th percentile d.Find the percentile rank for X=32

49. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   49 SCOREFrequency 57 -591 54 – 563 51 – 534 48 – 508 45 – 479 42 – 447 39 – 416 36 – 385 33 – 353 30 – 322 27 – 291 24 – 261 H O M E W O R K a.Make a polygon or histogram graph for the distribution of scores presented in the following table b.Describe the shape of distribution c.Find the 25 th, 50 th, and 75 th percentile d.Find the percentile rank for X=25, X=50, and X=75

50. © aSup-2007 FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE   50 SCOREFrequencyXc Exact Limit LowerUpper 57 - 5915856.559.5 54 – 5635553.556.5 51 – 5345250.553.5 48 – 5084947.550.5 45 – 4794644.547.5 42 – 4474341.544.5 39 – 4164038.541.5 36 – 3853735.538.5 33 – 3533432.535.5 30 – 3223129.532.5 27 – 2912826.529.5 24 – 2612523.526.5 e.Make a polygon or histogram graph for the distribution Histogram Polygon Xc E.L.