AP Statistics Unit 1 Summary Statistics: Understanding Center, Spread, and Boxplots

Measure of center

A single value that summarizes a quantitative distribution as a “middle” or “typical” observation.

Mean (arithmetic average)

The balancing point of a distribution; computed by summing all observations and dividing by the sample size.

Sample mean (x̄)

The mean of a sample; notation x̄ = (1/n)∑(i=1 to n) xᵢ.

Summation symbol (∑)

A symbol meaning “add up” a sequence of values (for example, adding all xᵢ values).

Sample size (n)

The number of observations in a sample.

Balance point interpretation of the mean

Viewing each data value as a weight on a number line, the mean is where the line would balance; extreme values pull the mean toward them.

Mean minimizes sum of squared deviations

Among all possible centers, the mean makes the total of squared distances (xᵢ − center)² as small as possible.

Resistance (in statistics)

A property describing how much a statistic changes when extreme values (outliers) are present.

Mean is not resistant

A single extreme high or low value can noticeably change the mean.

Median

The middle value of an ordered data set; about half the observations are at or below it and about half are at or above it.

Finding the median (odd n)

After sorting, the median is the single middle observation when the sample size n is odd.

Finding the median (even n)

After sorting, the median is the average of the two middle observations when the sample size n is even.

Median is resistant

The median changes little (or not at all) when extreme values become more extreme, as long as they remain on the same side of the middle.

Right-skewed distribution (skewed right)

A distribution with a long right tail; typically the mean is greater than the median because large high values pull the mean upward.

Left-skewed distribution (skewed left)

A distribution with a long left tail; typically the mean is less than the median because small low values pull the mean downward.

Appropriate center for skewness/outliers

For skewed distributions or those with outliers, the median is often a better “typical” value than the mean.

Measure of variability (spread)

A statistic that describes how dispersed data values are (how much they vary around the center).

Range

A spread measure computed as max − min; gives overall width of the data.

Range sensitivity to outliers

Range is extremely sensitive to outliers because it depends only on the minimum and maximum values.

Quartiles (Q1 and Q3)

Cut points that split ordered data into quarters: Q1 is about the 25th percentile and Q3 is about the 75th percentile.

Interquartile range (IQR)

A resistant measure of spread for the middle 50% of the data; IQR = Q3 − Q1.

Standard deviation (sample, s)

A measure of spread describing a typical distance of data values from the sample mean; s = sqrt[(1/(n−1))∑(xᵢ − x̄)²].

Population parameters (μ and σ)

μ is the population mean and σ is the population standard deviation, used when describing an entire population rather than a sample.

Five-number summary

The set of five values: minimum, Q1, median, Q3, maximum; used to describe center and spread and to build boxplots.

Modified boxplot (1.5 IQR rule)

A boxplot that flags outliers using fences at Q1 − 1.5(IQR) and Q3 + 1.5(IQR); whiskers extend to the most extreme non-outlier values and outliers are plotted individually.