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

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, which you treat as a guess for people in general. This is a(n) A.Descriptive statistic B.Estimator C.Inferential statistic D.Parameter

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

Distributions 9/4

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

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

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 – 1 st column for value, 2 nd column for frequency x f(x) {5,7,3,7,2,5,5,3,7,5,3,11,7,5,3,5}

Histogram UnitsVariable Label Values Frequency of this value Graphical representation of a distribution, showing frequency of each value

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) = ? F(3) = ? 3 4 x f(x) F(x) f(3) f(4)

Quantile Quantile - the value of X that's greater than a certain fraction of the data Percentile - quantile defined by a certain percentage 90 th %ile 25 th %ile Interpolation {1,2,2,4,5,5,7,8,8,8} {8,2,5,5,7,1,8,2,4,8} 50 th percentile = 590 th percentile = 8

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

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

Density 100% 2% 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 Household Income Density

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

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

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

Review Data: {2, 5, 6, 8, 5, 6, 4, 3, 2, 1, 4, 9} What is f(4)? A.2 B.4 C.6 D.8

Review Data: {2, 5, 6, 8, 5, 6, 4, 3, 2, 1, 4, 9} What is the 75 th percentile? A.2 B.6 C.8 D.9

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