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

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

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

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

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)

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

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

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

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

Normal Distribution Symmetric Single-peaked Bell-shaped

68-95-99.7 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

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

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