Better AP Stats Master Review

Individual in statistics

The person, animal, or object described by a set of data.

Variable

A characteristic that changes from one individual to another.

Categorical variable

A variable that places individuals into groups or categories.

Quantitative variable

A variable that measures numerical values where arithmetic makes sense.

Discrete quantitative variable

A numerical variable with separated countable values, often integers with gaps.

Continuous quantitative variable

A numerical variable that can take any value in an interval.

Frequency table

A table showing the number of times each category or value occurs.

Relative frequency table

A table showing the proportion or percentage of observations in each category.

Cumulative frequency

The running total of counts up to a certain category or value.

Cumulative relative frequency

The running total of proportions or percentages up to a certain value.

Bar graph

A graph for categorical data using separated bars to compare counts or percentages.

Pie chart

A circular graph showing parts of a whole, but it can be misleading if area or angles are distorted.

Rule for all data displays

Include a title, labels, appropriate scales, and avoid distorting size or area.

Why distorted graphs are dangerous

They can make differences look bigger or smaller than they actually are.

Histogram

A graph for quantitative data that groups values into intervals and shows frequency.

Why histograms are useful

They show the overall shape of large quantitative data sets.

Limitation of histograms

Individual data values are not visible.

Dotplot

A graph that shows each data value as a dot above a number line.

Strength of dotplots

They show individual data points clearly.

Weakness of dotplots

They become messy with large data sets.

Stemplot

A display that separates data into stems and leaves while preserving individual values.

When stemplots work best

For small to moderate data sets with values that can be neatly split.

CUSS for describing a distribution

Center, Unusual features, Shape, Spread.

Center of a distribution

The typical or middle value, often measured by mean or median.

Mean

The arithmetic average of a data set.

Median

The middle value when data are ordered.

When mean is preferred

When the distribution is roughly symmetric and has no strong outliers.

When median is preferred

When the distribution is skewed or has outliers.

Resistant statistic

A statistic not strongly affected by outliers.

Median as resistant

The median is resistant because extreme values do not greatly change its position.

IQR as resistant

IQR is resistant because it uses the middle 50% of the data.

Nonresistant statistic

A statistic strongly affected by outliers.

Mean as nonresistant

The mean changes when extreme values are added.

Standard deviation as nonresistant

Standard deviation increases when outliers create larger distances from the mean.

Shape of a distribution

The overall pattern, including symmetry, skew, clusters, and modality.

Symmetric distribution

A distribution where the left and right sides are roughly mirror images.

Skewed right

A distribution with a long tail to the right.

Skewed left

A distribution with a long tail to the left.

Direction of skew

The direction of the tail, not the side with most of the data.

Unimodal

A distribution with one clear peak.

Bimodal

A distribution with two clear peaks.

Multimodal

A distribution with more than two peaks.

Uniform distribution

A distribution where values occur with roughly equal frequency.

Unusual features

Gaps, clusters, or outliers that stand out from the general pattern.

Gap

A region in the distribution with no observations.

Cluster

A group of observations concentrated near one another.

Outlier

A data value that falls far from the rest of the data.

IQR

The interquartile range, calculated as Q3 minus Q1.

Q1

The first quartile, or the median of the lower half of the data.

Q3

The third quartile, or the median of the upper half of the data.

Five-number summary

Minimum, Q1, median, Q3, maximum.

Boxplot

A graph showing the five-number summary and possible outliers.

1.5 IQR rule

Outliers are below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR).

Standard deviation

The typical distance of data values from the mean.

Sample standard deviation

Usually written as s, used when describing a sample.

Population standard deviation

Usually written as σ, used when describing a population.

Parameter

A number that describes a population.

Statistic

A number that describes a sample.

Why parameters are often unknown

We usually cannot measure every individual in the population.

Why statistics vary

Different random samples usually produce different values.

Z-score

A standardized value showing how many standard deviations a value is from the mean.

Z-score formula

z = (x − mean) / standard deviation.

Meaning of positive z-score

The value is above the mean.

Meaning of negative z-score

The value is below the mean.

Meaning of z = 0

The value equals the mean.

Why z-scores have no units

They measure distance in standard deviations, not original units.

Why z-scores are useful

They allow comparison of values from different distributions.

Effect of adding a constant to data

Measures of position shift by that constant, but spread does not change.

Effect of subtracting a constant from data

Measures of position decrease by that constant, but spread stays the same.

Effect of multiplying data by a positive constant

Measures of position and spread are multiplied by that constant.

Effect of dividing data by a positive constant

Measures of position and spread are divided by that constant.

Effect of changing units from inches to feet

Both position and spread are rescaled.

Association between categorical variables

A relationship where the distribution of one variable changes depending on the other variable.

No association in categorical data

Conditional distributions are approximately the same across groups.

Two-way table

A table showing counts for two categorical variables.

Marginal distribution

The distribution of one categorical variable using the row or column totals.

Conditional distribution

The distribution of one variable among only individuals in a specific category of another variable.

Joint relative frequency

A proportion involving one cell of a two-way table compared to the total.

Marginal relative frequency

A row or column total divided by the grand total.

Conditional relative frequency

A cell count divided by its row or column total, depending on the condition.

Segmented bar graph

A graph showing conditional distributions within categories.

Side-by-side bar graph

A graph comparing categorical distributions across groups.

Best graph for comparing conditional distributions

A segmented bar graph or side-by-side bar graph.

Explanatory variable

The variable that may explain or predict changes in another variable.

Response variable

The variable being measured or predicted.

Scatterplot

A graph showing the relationship between two quantitative variables.

What to describe in a scatterplot

Direction, form, strength, and unusual points.

Positive association

As x increases, y tends to increase.

Negative association

As x increases, y tends to decrease.

Linear form

The pattern in a scatterplot is roughly a straight line.

Nonlinear form

The pattern in a scatterplot curves or does not follow a straight line.

Strong association

Points fall close to a clear pattern.

Weak association

Points are widely scattered around the pattern.

Correlation r

A number measuring the direction and strength of a linear relationship.

Range of r

−1 ≤ r ≤ 1.

r close to 1

Strong positive linear association.

r close to −1

Strong negative linear association.

r close to 0

Weak or no linear association.

What r does not measure

Form, unusual features, or causation.

Correlation has no units

r is standardized, so it does not use the original units of x or y.