AP Stats Test 1

Investigative question

A question that can be answered using data and accounts for variability in the answers.

Population

The entire group about which a statistical study wants information.

Sample

A subset of the population from which data are actually collected.

Observational unit

The individual person, object, or thing described by the data.

Variable

A characteristic recorded for each observational unit.

Categorical variable

A variable that places observations into groups or categories.

Quantitative variable

A variable with numerical values for which arithmetic makes sense.

Frequency

The number of observations in a category.

Relative frequency

The proportion or percentage of observations in a category; frequency divided by total.

Bar chart

A graph for categorical data with separated bars.

Pie chart

A graph for categorical data in which the entire circle represents 100% of the observations.

Misleading graph

A graph that distorts or misrepresents data through its scale, intervals, labels, proportions, or design.

Dotplot

A graph of quantitative data in which each dot represents an observation.

Symmetric distribution

A distribution whose left and right sides are approximately mirror images.

Right-skewed distribution

A distribution with a long tail toward larger values.

Left-skewed distribution

A distribution with a long tail toward smaller values.

Bimodal distribution

A distribution with two noticeable peaks.

SOCS

A method for describing a quantitative distribution: Shape, Outliers, Center, Spread.

Stem-and-leaf plot

A display of quantitative data that shows the distribution while preserving individual values.

Histogram

A graph of quantitative data grouped into intervals; the bars touch.

Median

The middle value of an ordered data set.

Mean

The sum of all observations divided by the number of observations.

Range

The maximum value minus the minimum value.

Standard deviation

A measure of the typical distance of observations from the mean.

IQR

The interquartile range; Q3 minus Q1 and the spread of the middle 50% of the data.

Q1

The first quartile, or 25th percentile.

Q3

The third quartile, or 75th percentile.

Outlier

An observation that is unusually far from the rest of the distribution.

1.5 × IQR rule

An observation is an outlier if it is below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR).

Five-number summary

Minimum, Q1, median, Q3, and maximum.

Boxplot

A graph based on the five-number summary; the box represents the middle 50% of the data.

Percentile

A measure of the relative position of an observation within a distribution.

z-score

A standardized score showing how many standard deviations an observation is from the mean.

Positive z-score

A value above the mean.

Negative z-score

A value below the mean.

z = 0

An observation exactly at the mean.

Mean and standard deviation

Usually appropriate for roughly symmetric distributions without strong outliers.

Median and IQR

Usually appropriate for skewed distributions or distributions with outliers.

Adding a constant to every value

Changes the center by that constant but does not change the shape or spread.

Subtracting a constant from every value

Changes the center by that amount but does not change the shape or spread.

Multiplying every value by a positive constant

Changes the center and spread by that factor but does not change the shape.

Dividing every value by a positive constant

Changes the center and spread by that factor but does not change the shape.

Bar chart vs histogram

Bar charts display categorical data with separated bars; histograms display quantitative data with touching bars.

Mean vs median

The mean is sensitive to outliers; the median is resistant to outliers.

Range vs IQR

Range uses the maximum and minimum; IQR measures the middle 50% and is resistant to outliers.

Percentile vs z-score

Percentile gives relative position; z-score gives distance from the mean in standard deviation units.

Right skew and center

For a right-skewed distribution, the mean is usually greater than the median.

Left skew and center

For a left-skewed distribution, the mean is usually less than the median.

Symmetry and center

For a roughly symmetric distribution, the mean and median are usually similar.

z-score formula

z = (x − mean) / standard deviation.

IQR formula

IQR = Q3 − Q1.

Range formula

Range = maximum − minimum.

Mean formula

Mean = sum of all values divided by the number of values.

Upper outlier fence

Q3 + 1.5(IQR).

Lower outlier fence

Q1 − 1.5(IQR).