AP Statistics

1
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
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

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.