1. Distributions

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

3. 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

4. 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
x f(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

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

6. Cumulative Frequency {4,3,4,5,3,4,2,4,3,4} f(3) = ? 3 F(3) = ? 4
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
f(4)
x f(x) F(x) 2 3 1 4 5
F(3) = ? 4 1 4
f(3) 9 10

7. 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}
50th percentile = 5
90th percentile = 8
{1,2,2,4,5,5,7,8,8,8}
25th %ile
Interpolation
90th %ile

8. Continuous vs. Discrete Variables
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

9. 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
71.5
72.5
73.5

10. Density 2% 100% 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
100%
Household Income
Density 2%

11. 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

12. 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

13. 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