FREQUENCY DISTRIBUTIONS AND PERCENTILES

Transcription

1 FREQUENCY DISTRIBUTIONS AND PERCENTILES

2 New Statistical Notation Frequency (f): the number of times a score occurs N: sample size

3 Simple Frequency Distributions

4 Raw Scores The scores that we have directly measured. number of correct answers on a test people s height temperature measured during the day

5 Raw Scores Here is a data set of some raw scores:

6 How to construct a simple frequency table: Score f N=18

7 Example 1 How to construct a simple frequency table: Score f N=10

8 Example How to construct a simple frequency table: Score f N=9

9 Graphing a Simple Frequency Distribution Scores on the X axis Frequencies 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

10 Bar Graph Used for: Nominal Data Gender, marital stat. Ordinal Data rank in class

11 Bar Graph Adjacent bars do not touch Why?

12 Bar Graph Party Com. Soc. Dem. Rep. f N=18

13 Histogram Used for: Interval scores IQ, temperature Ratio scores Weight, height, time

14 Histogram Adjacent bars touch Why?

15 Histogram Score f N=36

16 Drawing a Polygon

17 Polygon Used when we have large number of interval or ratio scores When a histogram is hard to create

18 Histogram vs. Polygon It is easier to draw a polygon when you have a lot of scores Bars get thinner and thinner

19 Types of Simple Frequency Distributions

20 Normal Distribution

21 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 lowfrequency extreme scores are called the tails of the distribution

22 Normal Distribution

23 The Normal Distribution Most variables are normally distributed in the population IQ Height of females Height of males

24 Non-normal Distributions Skewed Bimodal Rectangular

25 Skewed Distributions Not symmetrical A distribution may be either negatively skewed or positively skewed

26 Negatively Skewed Distribution Has extreme low scores that have a low frequency Does not have extreme high scores that have low frequency

27 Positively Skewed Distribution Has extreme high scores that have a low frequency Does not have extreme low scores that have low frequency

28 Bimodal Distribution A symmetrical distribution containing two distinct humps

29 Rectangular Distribution A symmetrical distribution shaped like a rectangle

30 Examples Negative Skew

31 Examples Bimodal Distribution

32 Examples Negative Skew

33 Examples Bimodal Distribution

34 Examples Positive Skew

35 Examples Positive Skew

36 Relative Frequency and the Normal Curve

37 Relative Frequency Relative frequency (rel. f) : the proportion of time the score occurs 0 rel. f 1 The formula is: rel. f = f N

38 Relative Frequency Gives us a frame of reference. It is easier to interpret For ex: In an exam,7 students got 100. Is the class successful? Depends on how many students there are (N) If N is 14, then rel. f = 7/14=0.5

39 Relative Frequency Table Score f Rel. f N=18

40 A Relative Frequency Distribution

41 N=36 Relative Frequency Tables Example 1 Score f Rel. f

42 Relative Frequency and the Normal Curve combined relative frequency: the proportion of the total area under the normal curve that is occupied by a group of scores.

43 Cumulative Frequency and Percentile

44 Cumulative Frequency Cumulative frequency (cf ): the frequency of all scores at or below a particular score To compute a score s cumulative frequency, we add the simple frequencies for all scores below the score with the frequency for the score

45 Cumulative Frequency Table Score f cf N=18

46 A Cumulative Frequency Distribution

47 Percentile Percentile: the percent of all scores in the data that are at or below the score Formula: Percentile = (cf/n)*100

48 Percentile Score f cf percentile N=18

49 Finding Percentiles in Graphs 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.

50 Percentiles Normal distribution showing the area under the curve to the left of selected scores.

51 Grouped Frequency Distributions

52 Grouped Distributions Grouped distribution: scores are combined to form small groups we report the total f, rel. f, or cf of each group

53 A Grouped Distribution

54 An Example: Grouped Frequency Distribution Record the limits of all class intervals, placing the interval containing the highest score value at the top. Count up the number of scores in each interval. Hotel Rates Frequency Las Vegas Hotel Rates

55 Frequency Table Guidelines Intervals should not overlap, so no score can belong to more than one interval. Make all intervals the same width. Make the intervals continuous throughout the distribution (even if an interval is empty). Place the interval with the highest score at the top. Choose a convenient interval width. Hotel Rates Frequency

56 An Example: Grouped Frequency Distribution Proportion (Relative Frequency) Divide frequency of each class by total frequency. Hotel Rates Frequency Proportion N = 35

57 An Example: Grouped Frequency Percentage Proportion *100 Distribution Hotel Rates Frequency Proportion Percent

58 An Example: Grouped Frequency Cumulative Frequency Distribution Shows total number of observations in each class and all lower classes. Hotel Rates Frequency Proportion Percent Cumulative Frequency

59 An Example: Grouped Frequency Distribution Cumulative Proportion (Cumulative Relative Frequency): Divide Cumulative Frequency by Total Frequency Percentile Rank Cumulative Proportion * 100 Hotel Rates Frequency Proportion Percent Cumulative Frequency Cumulative Proportion Percentile

60 Example 1 Using the following data set, find the relative frequency of the score

61 The simple frequency table for this set of data is as follows. Example 1

62 Example 1 The frequency for the score of 12 is 1. N = 18. Therefore, the rel. f of 12 is Rel.f = f/n =1/18 =.06

63 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 = = 13 The cf for the score of 14 in this data set is 13

64 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 = 0.72 Another way to calculate this percentile is cf N

65 Example 4 Data set: 1,4,5,3,2,5,7,3,4,5. Find the mistakes below. Score f cf N=6 The N should be 10; the scores should be listed in descending order; a row containing the score of 6 should be included; the cf for the score of 1 should be 1; and the cf for all other scores should be corrected so that at the score of 7 the cf is 10.

66 Example 5 Organize the scores below in a table showing Simple frequency Relative frequency Cumulative frequency Draw a relative frequency histogram Draw a simple frequency polygon

67 Example 5 Score f rel.f cf

68

69