Chapter Three

1. Chapter Three Frequency Distributions and Percentiles

2. Raw Scores (Data) • Dr. Peabody gave a statistics exam to students in his Introduction to Statistics course: • 55, 57.5, 65, 67.5, 67.5, 72.5, 75, 75, 75, 77.5, 77.5, 77.5, 77.5, 82.5, 82.5, 82.5, 82.5, 82.5, 82.5, 85, 85, 85, 85, 85, 85, 85, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 92.5, 92.5, 95, 95, 95, 97.5

3. New Statistical Notation The number of times a score occurs is the score’s frequency, which is symbolized by f A distribution is the general name for any organized set of data N is the total number of scores in the data Chapter 3 - 3

4. SimpleFrequency Distributions Chapter 3 - 4

5. Simple Frequency Distribution A simple frequency distribution shows the number of times each score occurs in a set of data The symbol for a score’s simple frequency is simply f Chapter 3 - 5

6. Raw Scores Following is a data set of raw scores. We will use these raw scores to construct a simple frequency distribution table. Chapter 3 - 6

7. Simple FrequencyDistribution Table Chapter 3 - 7

8. Graphing a SimpleFrequency Distribution A frequency distribution graph shows the scores on the X axis and their frequency on the Y axis The type of measurement scale (nominal, ordinal, interval, or ratio) determines whether we use A bar graph A histogram A polygon Chapter 3 - 8

9. A Simple Frequency Bar Graph Is Used for Nominal and Ordinal Data Chapter 3 - 9

10. A Histogram Is Used for a Small Range of Different Interval or Ratio Scores Chapter 3 - 10

11. A Frequency Polygon Is Used for a Large Number of Different Scores Chapter 3 - 11

12. Types of Simple Frequency Distributions Chapter 3 - 12

13. The Normal Distribution A bell-shaped curve Called the normal curve or a normal distribution It is symmetrical The far left and right portions containing the low-frequency extreme scores are called the tails of the distribution Chapter 3 - 13

14. An Ideal Normal Distribution Chapter 3 - 14

15. Skewed Distributions A skewed distribution is not symmetrical as it has only one pronounced tail A distribution may be either negatively skewed or positively skewed Whether a skewed distribution is negative or positive corresponds to whether the distinct tail slopes toward or away from zero Chapter 3 - 15

16. Negatively Skewed Distribution A negatively skewed distribution contains extreme low scores that have a low frequency, but does not contain low frequency extreme high scores Chapter 3 - 16

17. Positively Skewed Distribution A positively skewed distribution contains extreme high scores that have a low frequency, but does not contain low frequency extreme low scores Chapter 3 - 17

18. Bimodal Distribution A bimodal distributionis a symmetrical distribution containing two distinct humps Chapter 3 - 18

19. Rectangular Distribution A rectangular distribution is a symmetrical distribution shaped like a rectangle Chapter 3 - 19

20. Relative Frequency and the Normal Curve Chapter 3 - 20

21. Relative Frequency Relative frequency is the proportion of time the score occurs The symbol for relative frequency is rel. f The formula for a score’s relative frequency is Chapter 3 - 21

22. A RelativeFrequency Distribution Chapter 3 - 22

23. Finding Relative Frequency Using the Normal Curve The proportion of the total area under the normal curve that is occupied by a group of scores corresponds to the combined relative frequency of those scores. Chapter 3 - 23

24. Computing CumulativeFrequency and Percentile Chapter 3 - 24

25. Cumulative Frequency Cumulative frequency is the frequency of all scores at or below a particular score The symbol for cumulative frequency is cf To compute a score’s cumulative frequency, we add the simple frequencies for all scores below the score with the frequency for the score Chapter 3 - 25

26. A Cumulative Frequency Distribution Chapter 3 - 26

27. Percentile A percentile is the percent of all scores in the data that are at or below the score If the scores cumulative frequency is known, the formula for finding the percentile is Score’s Percentile = Chapter 3 - 27

28. Finding Percentiles The percentile for a given score corresponds to the percent of the total area under the curve that is to the left of the score. Chapter 3 - 28

29. Percentiles Normal distribution showing the area under the curve to the left of selected scores. Chapter 3 - 29

30. Grouped FrequencyDistributions Chapter 3 - 30

31. Grouped Distributions In a grouped distribution, scores are combined to form small groups, and then we report the total f, rel. f, or cf of each group Chapter 3 - 31

32. A Grouped Distribution Chapter 3 - 32

33. Example 1 Using the following data set, find the relative frequency of the score 12 Chapter 3 - 33

34. Example 1 The simple frequency table for this set of data is as follows. Chapter 3 - 34

35. Example 1 The frequency for the score of 12 is 1. N = 18. Therefore, the rel. f of 12 is Chapter 3 - 35

36. Example 2 What is the cumulative frequency for the score of 14? Answer: The cumulative frequency of 14 is the frequency of all scores at or below 14 in this data set cf = 1 + 1 + 1 + 4 + 6 = 13 The cf for the score of 14 in this data set is 13 Chapter 3 - 36

37. Example 3 What is the percentile for the score of 14? Answer: The percentile of 14 is the percentage of all scores at or below 14 in this data set 0.056 + 0.056 + 0.056 + 0.222 + 0.333 = 0.72 Another way to calculate this percentile is Chapter 3 - 37

38. Key Terms • negatively skewed distribution • normal curve • normal distribution • percentile • positively skewed distribution • proportion of the total area under the curve • bar graph • bimodal distribution • cumulative frequency • distribution • frequency • frequency polygon • grouped distribution • histogram Chapter 3 - 38

39. Key Terms (Cont’d) • simple frequency distribution • tail • ungrouped distribution rectangular distribution relative frequency relative frequency distribution simple frequency Chapter 3 - 39