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Psychology 10

Psychology 10. Analysis of Psychological Data January 27, 2014. The Plan for Today. Frequency distributions Grouping data Histograms and bar plots Percentiles Stem-and-leaf plots. Frequency Distributions. Purpose: to simplify the data and help us see structure. Definition:

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Psychology 10

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  1. Psychology 10 Analysis of Psychological Data January 27, 2014

  2. The Plan for Today • Frequency distributions • Grouping data • Histograms and bar plots • Percentiles • Stem-and-leaf plots

  3. Frequency Distributions • Purpose: to simplify the data and help us see structure. • Definition: • “A frequency distribution is an organized tabulation of the number of individuals located in each category on the scale of measurement.” • Yuck!

  4. Frequency Distributions • Let’s simplify that. A frequency distribution is the values that a variable takes on, along with the frequencies of those values. • This idea can be applied to grouped or ungrouped data. • Example where grouping would not be necessary: ages of undergraduate students in this room.

  5. Frequency Distributions • Often, though, there will be so many values of a variable that we need to group them in order to simplify. • Example: the exam scores we discussed last time. • Grouping: try to have “about 10” intervals. • Alternative: shoot for somewhere between 7 and 15 intervals.

  6. Final Exam Scores 53 38 38 38 35 52 33 51 49 39 43 52 47 52 50 41 51 17 47 33 51 52 39 37 48 52 36 43 20 35 50 45 55 43 42 41 46 30 35 43 49 45 45 50 36 36 49 45 45 48 52 35 45 41 25 29 36 47 42 48 27 28 41 48 42 48 38 49 48 46 36 37 44 16 45 42 37 47 43 40 38 43 40 49 37 40 40 33 47 48 20 40 40 31 36 49 42 32 40 42 42 48 42 32 31 30 29 42 46 44 39 48 48 42 54 49 45 54 43 52 45 54 49 41 50 23 41 45 37 39 55 43 28 52 49 39 50 54 46 37 41 48 42 28 31 34 51 31 46 50 49 34 22 46 45 52

  7. Final Exam Scores 16 17 20 20 22 23 25 27 28 28 28 29 29 30 30 31 31 31 31 32 32 33 33 33 34 34 35 35 35 35 36 36 36 36 36 36 37 37 37 37 37 37 38 38 38 38 38 39 39 39 39 39 40 40 40 40 40 40 40 41 41 41 41 41 41 41 42 42 42 42 42 42 42 42 42 42 42 43 43 43 43 43 43 43 43 44 44 45 45 45 45 45 45 45 45 45 45 45 46 46 46 46 46 46 47 47 47 47 47 48 48 48 48 48 48 48 48 48 48 48 49 49 49 49 49 49 49 49 49 49 50 50 50 50 50 50 51 51 51 51 52 52 52 52 52 52 52 52 52 53 54 54 54 54 55 55

  8. Grouping Data • Illustration using exam scores. • 10s, 20s, 30s, 40s, 50s seems natural, but only five groups. • High 10s, low 20s, high 20s… produces nine groups. • That meets the guideline of “about 10” and the guideline of “7 to 15” intervals.

  9. Frequency Distribution of ExamScores ScoresFrequency 15-19 2 20-24 4 25-29 7 30-34 13 35-39 26 40-44 35 45-49 43 50-54 24 55-59 2

  10. Some technical vocabulary • When we are dealing with continuous data, the intervals have real limits. • For example, the first two intervals in our exam distribution were 15-19 and 20-24. • The real limits of the first interval are 14.5 and 19.5. • The real limits of the second interval are 19.5 and 24.5.

  11. Some technical vocabulary • So the upper real limit of one interval is the lower real limit of the next higher interval. • The midpoint of an interval is exactly what it sounds like: it is the point that is midway between the real limits. • If we know that a datum falls in a particular interval, the midpoint represents our best guess of its value.

  12. Grouping and Frequency Distributions • Importance of having intervals of equal width: • The Divers’ Alert Network example (p 14 and 64).

  13. Histograms • The frequency distribution helps us understand the structure of the variable. • However, a picture could help even more. • Histograms are graphical representations of the distribution.

  14. Creating a histogram • Algorithm: • Identify real limits of all intervals; • Represent them on a number line; • Label the midpoints of the intervals; • Draw histobars with edges at the real limits and heights determined by frequency. • Illustration using exam scores.

  15. Frequency Distribution of Exam Scores ScoresFrequency 15-19 2 20-24 4 25-29 7 30-34 13 35-39 26 40-44 35 45-49 43 50-54 24 55-59 2

  16. Bar Plots • Histograms are appropriate for interval or ratio level data. • If the level of measurement is nominal or ordinal, a bar plot is appropriate. • A bar plot differs from a histogram in that the bars don’t touch. • Illustration using exam grades.

  17. Frequency Distribution of ExamScores Letter ScoresGradeFrequency > 50 A 20 43 – 50 B 59 38 – 42 C 35 32 – 37 D 23 < 32 F 19

  18. Other forms of frequency distributions • Cumulative frequency distributions • Shows the count of data up to and including this value or interval • Relative frequency distribution • Shows the proportion of observations with this value or in this interval

  19. Cumulative Frequency Distribution of Exam Scores ScoresFrequency 15-19 2 20-24 6 25-29 13 30-34 26 35-39 52 40-44 87 45-49 130 50-54 154 55-59 156

  20. Relative Frequency Distribution of ExamScores Relative ScoresFrequencyFrequency 15-19 2 .013 20-24 4 .026 25-29 7 .045 30-34 13 .083 35-39 26 .167 40-44 35 .224 45-49 43 .276 50-54 24 .154 55-59 2 .013

  21. Cumulative Relative Frequency Distribution of ExamScores Cumulative Cumulative Relative ScoresFrequencyFrequency 15-19 2 .026 20-24 6 .038 25-29 13 .083 30-34 26 .167 35-39 52 .333 40-44 87 .558 45-49 130 .833 50-54 154 .987 55-59 156 1.000

  22. Percentiles • The percentile rank of a score is the proportion of the distribution falling at or below that score. • The percentile is the score that has a particular proportion falling below it. • For example, if the percentile rank of the value “25” is 55 for some distribution, then 25 is the 55th percentile of the distribution.

  23. Calculating percentiles • If we are lucky, it is easy to calculate percentiles and percentile ranks. • For example, we can see from the cumulative relative frequency table that exactly 55.8% of the exam distribution falls at or below 44. • Hence, the percentile rank of 44 is 55.8, and 44 is the 55.8thpercentile.

  24. Calculating percentiles • Actually, though, things are more complicated because the percentile falls within an interval of grouped data. • For the example in the previous slide, what’s really true is that 55.8% of the distribution is below 44.5. • Under such circumstances, we need to interpolate.

  25. How to interpolate • Figure out what interval contains the value • Calculate the width of the interval • Calculate how far into the interval the value is (as a proportion) • Figure out how wide the interval is in relative frequency units • Move that same proportion up through the relative frequency range of the interval

  26. Interpolation • Example: Find the percentile rank of a score of 44 in the exam data. • 44 is the last value in the sixth interval. The real limits of the interval are 39.5 to 44.5, so it is 5 units wide. • The cumulative relative frequency for that interval is .558, and for the next lower interval, it’s .333. That’s a width of .225.

  27. Interpolation • 44 is (44 – 39.5) / 5 = .9 (or 90%) of the way up from the bottom of the interval. • .90 times the width of the interval’s proportion of the data is .90*.225 = .2025 • So the percentile rank of 44 is .2025 plus the percentile rank of the lower real limit. • .2025 + .333 = .5355, the 53.6th percentile.

  28. Interpolation in the other direction • Identify the interval containing the percentile. • Figure out how far into that interval the percentile is. • Move that same distance through the range of the interval to get the corresponding value.

  29. Interpolation • Example: find the 25th percentile of the exam score distribution. • We can see from the table that this is in the interval 35 to 39 (or, really, 34.5 to 39.5). • The interval is .166 relative frequency units wide. • .25 - .167 = .083, so we want to find .083 / .166 = .5 of the way through the interval. • .5 * the width of the score interval (5) is 2.5, so the 25th percentile is 34.5 + 2.5 = 37.

  30. Check by working backwards • What is the percentile rank of 37? • It’s in the interval 34.5 – 39.5, which is 5 units wide. • 37 is exactly half way through that interval. • The percentile range of the interval is .167 to .333, which is .166 units wide. • .5 * .166 + .167 = .25, so 37 is the 25th percentile of the distribution.

  31. Activity • Stem-and-leaf plots.

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