1. Distributions (Chapter 1) Sonja Swanson
AP Statistics
Distributions
(Chapter 1)
Sonja Swanson

2. Variables
Variable ~ characteristic of a person or thing that can be expressed as a number
Value ~ actual number describing the person or thing
Quantitative Variable ~ takes numeric values for which arithmetic operations (e.g. differences and averages) make sense
Categorical Variable ~ records into which category a person or thing falls into
Distribution ~ pattern of variation of a variable

3. Types of Distributions  Stem Plots
AKA stem-and-leaf plot
Separates first digit(s) into a stem and last digit(s) onto the leaves
May also be displayed comparing two distributions back-to-back
Works best with small numbers of observations all greater than zero

4. Types of Distributions  Histograms
Breaks range of values into intervals
Displays the count/percent of observations falling within each interval

5. Types of Distributions  Time Plots
Plots measurement of values against time
Displays different patterns
Trend ~ long-term change in level of variable
Seasonal ~ regular rises and falls with seasons
Irregular Fluctuation ~ fluctuation due to unusual event (e.g. price shocks after crop failures)
Cycles ~ distinct up and down movements (similar to seasons)

6. Important Aspects of Distributions
Clusters ~ natural subgroups of values
Gaps ~ holes where no values fall
Outliers ~ values deviating far from the other observations
Skewedness ~ spread is unevenly weighed toward one side

7. Measuring Center
Mean ~ average, balance point, found by dividing the sum of the observations by the number of observations
Median ~ midpoint, 50th percentile, found by finding the middle value of the observations
Mode ~ most frequent value

8. Measuring Spread
Inter-Quartile Range ~ middle 50% of observations, Q3 – Q1
1.5 x IQR added to Q3 or subtracted from Q1 determines outlier boundaries
Variance ~ measures spread based on mean
Standard Deviation ~ square root of variance, most common measure of spread

9. Five Number Summary Used for box-plots
Minimum ~ lowest value of the observations
Q1 ~ 25th percentile, midpoint between median and minimum
Median ~ midpoint of observations
Q3 ~ 75th percentile, midpoint between median and maximum
Maximum ~ highest value of the observations

10. Normal Distribution
Symmetric
Single-peaked
Bell-shaped

11. Rule
68% of observations in a normal distribution fall within one standard deviation of the mean
95% fall within two standard deviations of the mean
99.7% fall within three standard deviations of the mean

12. Standard Normal Distributions
Standardized normal distributions have a mean of zero and a standard deviation of one
Z value
Equals the variable minus mean all divided by the standard deviation
Use Table A to find percent of observations falling below that value

13. Work Cited
McCabe, G.P., & Moore, D.S. (1993). Introduction to the Practice of Statistics. W.H. Freeman: New York.