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Psychology 10

Psychology 10. Analysis of Psychological Data January 29, 2014. The Plan for Today. More on interpolation Measures of central tendency Mean, median, and mode The mean From raw data From grouped data The median The median and interpolation. More examples of interpolation.

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Psychology 10

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  1. Psychology 10 Analysis of Psychological Data January 29, 2014

  2. The Plan for Today • More on interpolation • Measures of central tendency • Mean, median, and mode • The mean • From raw data • From grouped data • The median • The median and interpolation

  3. More examples of interpolation Cumulative Score Percent 50-59 100 40-49 85 30-39 66 20-29 32 10-19 10 0-9 4

  4. Aspects of Shape • Central tendency • Variability • Symmetry • Modality

  5. Measures of central tendency • Central tendency helps us think about the typical value of a distribution. • Most commonly used measures of central tendency are the mean, the median, and the mode.

  6. The mode • The mode is rarely very useful for continuous data. • It can easily change depending on how data are grouped. • The mode isuseful for discrete data with only a few possible values. • Most often such variables will be measured at the nominal level.

  7. What is the mode here? 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 Answer: 42, 45 and 48.

  8. What is the mode here?

  9. What is the mode here?

  10. Example where the mode is useful • What is the most typical eye color? • Sometimes, though, the variables that have a few discrete values can be fully ratio level. • Example: number of persons in your family. • The mode can be useful for such data.

  11. Problems with the mode • Can vary widely depending on how we define it. • Can vary depending on how we group the variable.

  12. The mean • Notation: • Population: m. • Sample: M or • The second symbol there is read “xbar.” • Definition: the mean is the arithmetic average. • In mathematical notation,

  13. What is the mean final exam score? • The sum of the scores is 6526. • There are 156 of them. • 6526 / 156 = 41.83333 ≈ 41.8.

  14. The mean from grouped data • Our best guess of the value of an observation from an interval is the midpoint of that interval. • If there are six observations in an interval, our best guess is that the data set contained six observations at the midpoint of the interval. • Hence, the mean can be estimated by summing over intervals the number in the interval times the midpoint of the interval, and then dividing by N.

  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. 2×17 + 4×22 + 7×27 + 13×32 + 26×37 + 35×42 + 43×47 + 24×52 + 2×57 = 6542 6542 / 156 = 41.9359 ≈ 41.9 (Recall that the mean calculated from raw data was 41.8.)

  17. The mean from relative frequency tables • Similarly, the mean can be estimated by summing over intervals the midpoint of each interval times the proportion of the total sample in the interval.

  18. Relative Frequency Distribution of Exam Scores 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

  19. 17×.013 + 22×.026 + 27×.045 + 32×.083 + 37×.167 + 42×.224 + 47×.276 + 52×.154 + 57×.013 = 41.972 ≈ 42.0 (Recall that the mean calculated from raw data was 41.8, and from grouped data with raw frequencies was 41.9.)

  20. The median • The median is most easily defined as the centermost ordered observation, or, if there are an even number of observations, the mean of the two centermost ordered observations.

  21. Algorithm for calculatingthe median • Calculate (N +1 ) / 2. • If the result is a whole number, count that many observations up from the bottom of the ordered data set. • If the result ends in .5, round down; count that far up from the bottom of the ordered data set, and take the average of that observation and the next one.

  22. What is the median here? 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 N = 156. (156 + 1) / 2 = 78.5, so the median is the average of the 78th and 79th scores.

  23. What is the median here? 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 N = 156. (156 + 1) / 2 = 78.5, so the median is the average of the 78th and 79th scores.

  24. The median and interpolation • When the data are grouped, it is important to recognize the fact that several observations fall in the same interval as the median. • Another name for the median is 50th percentile. • Just another interpolation problem.

  25. Cumulative Relative Frequency Distribution of Exam Scores 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

  26. The median and interpolation • The interval containing the 50th percentile spans the values 33.3% to 55.8%. • That’s 55.8 - 33.3 = 22.5 units wide. • 50 is 5.8 down from the top, a proportion equal to 5.8 / 22.5 = .25778 of the width. • The score interval has an upper real limit of 44.5 and is 5 units wide. • Median = 44.5 - .25778×5 = 43.211 ≈ 43.

  27. Activity • Gather some data. • Frequency distribution. • Stem-and-leaf plot. • Mean and median.

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