Summary Statistics: Center & Spread

You’ll be able to

Compute and interpret the mean, median, standard deviation, and IQR
Explain which measures are resistant to outliers and choose accordingly
Predict how the mean and median compare under different skew

Two measures of center

The mean (x-bar) is the arithmetic average — add the values and divide by n. The median is the middle value when the data are ordered (the average of the two middle values if n is even). The mean is the "balance point" of the distribution; the median is the "50th percentile." They agree when data are symmetric but diverge when data are skewed.

Resistance: who gets dragged by outliers

A statistic is resistant if extreme values barely move it. The median and IQR are resistant — adding one billionaire to a room shifts the middle value hardly at all. The mean and standard deviation are not resistant — a single huge value drags them upward. This is why we report the median and IQR for skewed data and reserve the mean and standard deviation for roughly symmetric data.

Measuring spread

The range (max − min) uses only two values and is very sensitive to outliers. The interquartile range (IQR = Q3 − Q1) is the width of the middle 50% and is resistant. The standard deviation measures the typical distance of a value from the mean; it is 0 only when every value is identical and grows as data spread out. Standard deviation, like the mean, is pulled by outliers.

Sample standard deviation

s = sqrt( Σ(x_i − x-bar)² / (n − 1) )

Find each deviation from the mean, square it, average the squares using n − 1 (not n), then take the square root. Dividing by n − 1 corrects the tendency of a sample to underestimate the population spread.

Mean vs. median under skew

skewed right: mean > median · skewed left: mean < median · symmetric: mean ≈ median

The mean chases the tail. Whichever side the long tail is on, that is the side the mean sits relative to the median.

Worked example

Five houses on a street are worth $200k, $220k, $240k, $260k, and $1,080k. Find the mean and median, and state which better represents a typical home.

1.Order the data (already ordered): 200, 220, 240, 260, 1080 (in $k).
2.Median = middle value = $240k.
3.Mean = (200 + 220 + 240 + 260 + 1080) / 5 = 2000 / 5 = $400k.
4.The $1,080k home is an outlier; it drags the mean up to $400k, far above four of the five homes, while the median stays at $240k.

Answer: Median = $240k, mean = $400k. The median ($240k) better represents a typical home because it is resistant to the one very expensive house, whereas the non-resistant mean is inflated by that outlier.

Checkpoint

A data set of household incomes is strongly skewed right. Which pair of summary statistics best describes its center and spread?

On the exam

When a prompt says "skewed" or mentions outliers, default to median and IQR. When it says "roughly symmetric" or "bell-shaped," the mean and standard deviation are appropriate. Matching the summary to the shape is a routine AP decision point.

Checkpoint

For a distribution that is skewed to the left, how do the mean and median typically compare?

Answer the 2 checkpoints as you read.