Data Presentation

1. Data Presentation

2. Descriptive Statistics • Descriptive statistics provide procedures to organize data we have collected from studies, summarize sample findings, and present these summaries in ways that can be easily communicated to others.

3. Descriptive Statistics • The goal of descriptive statistics is to summarize a collection of data in a clear and understandable way. • What is the pattern of scores over the range of possible values? • Where, on the scale of possible scores, is a point that best represents the set of scores? • Do the scores cluster about their central point or do they spread out around it?

4. Display • Graphs often make it easier to see certain characteristics and trends in a set of data. • Graphs for quantitative data. • Histogram • Frequency Polygon • Stem and Leaf Display • Graphs for qualitative data. • Bar Chart • Pie Chart

5. Data • Classifications or Scales • Nominal - groups subjects into mutually exclusive categories; numerals represent category labels only (sex, nationality, blood type, clinical diagnosis) • Ordinal - gives a quantitative order to the variable; numbers indicate rank order of observations (manual muscle test, functional status, pain) • Interval - equal units of measurement between each division, but no true zero thus can not represent absolute quantity (calendar years, IQ, temperature) • Ratio - interval scale with an absolute zero (distance, age, time, weight strength, blood pressure)

6. Scales of Measurement • Discrete variable: • Consists of separate, indivisible categories; no values between neighboring categories. • e.g., students in a class; psychiatric disorders • Continuous variable: • Divisible into an infinite number of fractional parts. • e.g., height, weight, time. Scores on continuous variables are actually intervals – therefore, they may have boundaries called real limits.

7. Scales of Measurement 1. nominal: Set of categories, but no quantitative distinctions between categories. example: professions

8. Scales of Measurement 2. ordinal: Categories ranked in terms of magnitude. example: ranking participants in a race

9. Scales of Measurement 3. interval: Ordered categories with equal intervals between them; however, ratios of magnitudes are not meaningful. example: IQ scores Is a person with IQ 200, twice as intelligent as the person with IQ 100?

10. Scales of Measurement 4. ratio: An interval scale with the additional feature of an absolute zero (ratios are meaningful). example: time measurements Is two hours twice as long as one hour?

11. Graduate students’ anxiety scores 51 50 50 50 51 48 46 48 46 47 50 48 49 46 50 47 47 47 49 49 45 46 46 47 46 46 47 47 44 45 46 46 48 47 46 45 48 48 48 47 47 44 49 47 48 47 49 47 45 48 49 48 48 49 45 49 47 45 47 44 48 46 46 48 48 48 47 47 46 47 44 45 44 46 49 46 47 46 45 47 47 49 43 47 46 45 45 47 48 48 47 48 43 48 46 46 48 45 46 47

12. First you must list your scores in order. • Next, record the number of times each score occurs.

13. Anxiety Scores 51 50 49 48 47 46 45 44 43 Relative Freq. Freq (f) 2 5 10 20 25 20 11 5 2 51 50 49 48 47 46 45 44 43 II IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII I IIII II .02 .05 .10 .20 .25 .20 .11 .05 .02 Check each time these numbers occur Total 100 1.00

14. Cumulative Frequency Distribution Records all subjects who obtained a particular score or lower. X freq Cumulative Freq 51 2 100 50 5 98 49 10 93 48 20 83 47 25 63 46 20 38 45 11 18 44 5 7 43 2 2

15. The Percentile Rank • This is a measure of relative standing, i.e., it tells us where a particular score falls in relation to the rest of the data set. • In fact, it tells us what percentage of scores in a data set fall at or below a particular score. • For a score at the pth percentile, p% of the scores fall at or below that score. • E.g. On his first stats test, a student scored at the 70 percentile. This means 70% of the class scored the same or lower than that student.

16. You need a cumulative frequency distribution when calculating percentile rank. Eg., The following scores are received on a stats exam marked out of 20. X freq Cumulative freq 20 1 15 19 2 14 16 2 12 14 1 10 12 4 9 11 2 5 10 3 3 What is the percentile rank of the score 14?

17. Step 1 - Count the number of scores at and below the score you’re looking at. • In this case, 10 scores fall at or below 14. • Step 2 - Divide this number by N and multiply by 100 to get the percentile rank. • In this case…10/15 X 100 = 67%. • The score falls at the 67th percentile.

18. When the score in question is obtained more than once, a couple of steps must be added. Eg. Graduate Students Anxiety Scores X freq Cumulative Freq 51 2 100 50 5 98 49 10 93 48 20 83 47 25 63 46 20 38 45 11 18 44 5 7 43 2 2 What is the percentile rank of the score of 46?

19. Step 1 - Count the number of scores below the score you’re looking at. • In this case, 18 scores fall below 46. • Step 2 - Divide the number of scores the same as the one you’re looking at by 2. • In this case, 20 people scored 46. • 20/2 = 10.

20. Step 3 - Add this number to the total from step 1. • 10 + 18 = 28. • Step 4 - Divide this number by N and multiply by 100 to get the percentile rank. • 28/100 x 100 = 28%. • The score 46 falls at the 28th percentile.

21. The Percentile Rank (backwards) • We know how to find the percentile rank that corresponds to a score, but what if we want to do the reverse? • What if we want to find the score that corresponds to a certain percentile rank.

22. E.g., The following scores are obtained on an exam marked out of 20. X freq Cumulative freq 20 1 15 19 2 14 16 2 12 14 1 10 12 4 9 11 2 5 10 3 3 What score is at the 75th percentile?

23. Step 1- Multiply the decimal form of the percentile rank by N. • 0.75 X 15 = 12 • This tells you are looking for the 12th score in the cumulative frequency distribution. • Step 2- Locate this score on the cumulative frequency distribution. • The score at the 75th percentile is 16.

25. Frequency Distributions • Simply a way of organizing and making sense of a data set. It’s difficult to get a sense of what the scores are really like when you just look at a data set. • E.g., A hundred graduate students are given a test to measure their anxiety. They receive the following scores. Scores can range from 40 (low) to 55 (high).

26. What is the pattern of scores? • Create a Frequency Distribution • Frequency distributions organize raw data or observations that have been collected. • Ungrouped Data • Listing all possible scores that occur in a distribution and then indicating how often each score occurs. • Grouped Data • Combining all possible scores into classes and then indicating how often each score occurs within each class. • Easier to see patterns in the data, but lose information about individual scores.

27. Las Vegas Hotel Rates An Example: GroupedFrequency Distribution • Find the lowest and highest score (order scores from lowest to highest). • 891 is highest score. • 52 is lowest score. • Find the range by subtracting the lowest score from the highest score. • 891-52 = 839 • Divide range by 10. • 839/10 = 83.9 • Round off to the nearest convenient width. • 100 • Determine the scores at which the lowest interval should begin (an interval of the class width). • 0

28. 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. Las Vegas Hotel Rates

29. 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. • For most work, use 10 class intervals. • Choose a convenient interval width. • When possible, make the lower score limit a multiple of the interval width.

30. An Example: Grouped Frequency Distribution • Proportion (Relative Frequency) • Divide frequency of each class by total frequency. • Used when you want to compare the frequencies of one distribution with another when the total number of data points is different.

31. An Example: Grouped Frequency Distribution • Percentage • Proportion *100

32. An Example: Grouped Frequency Distribution • Cumulative Frequency • Shows total number of observations in each class and all lower classes.

33. An Example: Grouped Frequency Distribution • Cumulative Proportion (Cumulative Relative Frequency): • Divide Cumulative Frequency by Total Frequency • Percentile Rank • Cumulative Proportion * 100

34. Table 1: Examination scores for 80 students

35. 8 9 1 8 8 2 9 9 9 6 7 3 5 0 9 2 9 8 0 9 6 2 8 6 4 2 2 8 9 9 2 3 7 8 6 8 8 0 3 0 5 0 9 2 1 7 1 8 5 0 5 3 2 3 9 1 3 7 5 5 0 4 6 2 0 9 2 2 1 1 1 3 9 8 3 3 0 1 Stem-and-leaf display 2 3 4 5 6 7 8 9 9 2

36. 2 3 4 5 6 7 8 9 8 9 1 2 6 7 8 8 9 9 9 0 0 2 3 5 6 8 9 9 9 9 2 2 2 2 3 4 6 6 7 8 8 8 8 9 9 0 0 0 0 1 1 2 2 3 3 5 5 5 7 8 8 9 0 0 1 2 2 2 2 3 3 4 5 5 6 7 9 9 1 1 1 3 3 8 9 0 1 3 Stem-and-leaf display

37. Stem-and-leaf display Break each number into its tens and units digits. Tally together values which share the tens digit. The ten digits will then be aligned vertically with the units digits displayed to the side.

38. Table 2: Responses of young boys to removal of toy Frequency distribution of categorical data Organizing data? Isn’t this table the original raw data?

39. Table 4: Response to removal of toy by gender of child Comparing distributions More girls withdraw?

40. Percentage distribution Comparing distributions

41. Comparing distributions • Making comparisons between distributions is a procedure often used. • If the total numbers of cases are equal, the frequency distributions can be used to make comparisons • In general, we use percentage distributions to make comparison.

42. Grouped distribution • Grouped frequency/percentage distributions present raw (unprocessed) data in a more readily usable form. • The price for this is the loss of some information. • Worthwhile.