Describing, Exploring, and Comparing Data

Measures of Central Tendency

Measures of central tendency summarize a data set with a single value that represents the center of its distribution.

• Mean (Arithmetic Average): The sum of all data values divided by the number of values.

• Mean (Frequency Table): Used when data are presented with frequencies.

• Midrange: The value halfway between the lowest and highest values in the data set.

Formulas:

• Mean:

• Mean (Frequency Table):

• Midrange:

Measures of Variation

Measures of variation describe the spread or dispersion of data values.

• Standard Deviation (s): Measures the average distance of data values from the mean.

• Variance (s2): The square of the standard deviation.

• Standard Deviation (Frequency Table): Used when data are grouped by frequency.

Formulas:

• Standard Deviation:

• Standard Deviation (Frequency Table):

• Variance:

z-Score

The z-score indicates how many standard deviations a value is from the mean.

• Formula:

Percentiles

Percentiles indicate the relative standing of a value within a data set.

• Percentile of value x:

• Converting from kth percentile to data value:

Probability

Basic Probability Rules

Probability quantifies the likelihood of events occurring.

• Mutually Exclusive Events:

• Not Mutually Exclusive:

• Independent Events:

• Dependent Events:

• Rule of Complements:

• Conditional Probability:

Counting Principles

Counting principles are used to determine the number of ways events can occur.

• Permutations (no elements alike):

• Permutations (with alike elements):

• Combinations:

Discrete Probability Distributions

Mean and Standard Deviation

For a discrete random variable, the mean and standard deviation summarize the expected value and variability.

• Mean:

• Standard Deviation:

Binomial Probability Distribution

Key Properties and Formulas

The binomial distribution models the number of successes in a fixed number of independent trials with the same probability of success.

• Probability:

• Mean:

• Variance:

• Standard Deviation:

• Standard Score:

Sampling Distributions

Sample Mean and Proportion

Sampling distributions describe the distribution of sample statistics over repeated sampling.

• Mean of Sample Means:

• Standard Error of the Mean:

• Standard Score for Sample Mean:

Estimation and Confidence Intervals

Confidence Interval for Proportion

• Interval:

• Margin of Error:

Confidence Interval for Mean (σ known)

• Interval:

• Margin of Error:

Confidence Interval for Mean (σ unknown)

• Interval:

• Margin of Error:

Sample Size Determination

• For Proportion (unknown p):

• For Proportion (known and ):

• For Mean:

Hypothesis Testing

Test Statistics

• For Proportion:

• For Mean (σ known):

• For Mean (σ unknown):

• Degrees of Freedom:

• For Two Proportions: Additional info: Formula inferred for clarity.

Correlation and Regression

Equation of Regression Line

• Regression Line:

• Additional info: a = intercept, b = slope; formulas for a and b are not provided but are standard in statistics.

Goodness-of-Fit and Contingency Tables

Chi-Square Test Statistic

• Test Statistic:

• Expected Frequency:

• Degrees of Freedom:

Summary Table of Key Formulas