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Review. Which of these is a parameter? The average height of all people The time it takes rat #3 to learn the maze The number of subjects in your experiment The average memory score of subjects in your sample. Review.

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Review

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  1. Review Which of these is a parameter? • The average height of all people • The time it takes rat #3 to learn the maze • The number of subjects in your experiment • The average memory score of subjects in your sample

  2. Review You want to know the average time people can hold their breath. You measure 10 people and find that their average is 104 seconds. This is a(n) • Descriptive statistic • Estimator • Inferential statistic • Parameter

  3. Review Curious how heavy your dishes are, you weigh each one and then calculate the average. This is a(n) • Descriptive statistic • Estimator • Inferential statistic • Parameter

  4. Distributions 9/5

  5. Outline • Distributions • Frequency • Histograms • Cumulative frequency • Quantiles • Continuous variables • Shape of a distribution

  6. Distribution • The set of values present in a sample or population • Which values occur • How often • Starting point for statistics • Every statistic is computed from sample distribution • Every parameter is a property of population distribution • Need ways of representing or talking about distributions

  7. Frequency • Easiest way to characterize distribution • How often each value occurs f(x) = frequency of value x Sample: {1, 6, 3, 8, 6, 4}. f(6) = ? • Frequency table • Shows frequencies of all values • 1st column for value, 2nd column for frequency xf(x) 2 1 3 4 {5,7,3,7,2,5,5,3,7,5,3,11,7,5,3,5} 5 6 7 4 11 1

  8. Frequencyof this value Units Values Variable Label Histogram Graphical representation of a distribution, showing frequency of each value

  9. Cumulative Frequency • Number of scores below or equal to a given value F(x) = cumulative frequency for value x {4,3,4,5,3,4,2,4,3,4} f(3) = ? 3 xf(x) F(x) F(3) = ? 4 2 1 1 3 3 4 4 5 9 f(3) f(4) 5 1 10

  10. 25th %ile Quantile • Quantile - the value of X that's greater than a certain fraction of the data • Percentile - quantile defined by a certain percentage {8,2,5,5,7,1,8,2,4,8} {1,2,2,4,5,5,7,8,8,8} 50thpercentile = 5 90thpercentile = 8 90th %ile Interpolation

  11. Continuous vs. Discrete Variables • Discrete variable • Can only take certain values (usually integers) • Counts: people, test score, stories, … • Continuous variable • Infinite set of values, in principle • Height, weight, temp, IQ, … • For any two scores, there are other possible scores in between

  12. Histograms of Continuous Variables • Plotting unique scores isn’t useful • Bins or intervals • Ranges for grouping continuous variables • Best width depends on number of data 73.5 71.5 72.5

  13. 100% Density • Frequency only well-defined for discrete variables • f(x): scores exactly equal to x • 0 almost everywhere for continuous variables • Density function • Describes theoretical distribution of continuous variable • Allows determination of number of scores in any range, by integration • Usually shown as proportion of total population (probability), not frequency 2% Density Household Income

  14. Density Household Income

  15. Shape of a Distribution • Information beyond average score & variability • Broad, often qualitative property • Need "nice" shape to do statistics • Normal distribution • Gold standard for good shape • Symmetric, unimodal, thin tails

  16. Bad Shape • Skew: Asymmetric distribution • Extreme scores in one direction bias results • Positive skew vs negative skew - which tail is bigger • Solutions • Only consider order of scores (“ordinal data”) • Transform: Do statistics on new variable

  17. Bad Shape • Multimodal: More than one peak • Suggests there are multiple constituent populations • Learners vs. non-learners • Solution: discretize • Do statistics on proportion of learners

  18. Review Data: {2, 5, 6, 8, 5, 6, 4, 3, 2, 1, 4, 9} What is f(4)? • 2 • 4 • 6 • 8

  19. Review Data: {2, 5, 6, 8, 5, 6, 4, 3, 2, 1, 4, 9} What is the 75th percentile? • 2 • 6 • 8 • 9

  20. Review Find the bimodal distribution A. C. B. D. Density Density Density Score Score Score Density Score

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