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Measures of Center
& Variability

Learn to summarize data distributions using mean, median, IQR, and MAD: the essential tools for describing what's typical and how spread out data really is.

AP Statistics

What Are Measures of Center?

A measure of center is a single number that represents the "typical" value in a data set. It tells you where the middle of the data is.

Mean
The arithmetic average: add all values, then divide by how many there are.
Median
The middle value when data is ordered from least to greatest.

What Are Measures of Variability?

A measure of variability (or spread) describes how spread out the data values are from the center. Two data sets can have the same mean but very different spreads.

IQR
Interquartile Range: the spread of the middle 50% of the data. Paired with the median.
MAD
Mean Absolute Deviation: the average distance each data point is from the mean.

Same Center, Different Spread

2 4 6 8 10 2 4 6 8 10 Data Set A: Low Spread
\( \text{mean} = 6 \)
 Data Set B: High Spread
\( \text{mean} = 6 \)

Computing the Mean

The mean is the sum of all data values divided by the number of values:

The mean uses every data point, which makes it sensitive to outliers. A single extreme value can pull the mean far from where most data sits.

Finding the Median

1
Order all values from least to greatest.
2
If there is an odd number of values, the median is the middle value.
3
If there is an even number of values, the median is the mean of the two middle values.

Mean vs. Median

Mean Median
What it measures Arithmetic average Middle position
Uses every value? Yes No (position only)
Affected by outliers? Yes No (resistant)
Best for Symmetric data Skewed data
Paired with MAD IQR

Interquartile Range: IQR

The IQR measures the spread of the middle 50% of data. It ignores the lowest 25% and highest 25%, making it resistant to outliers.

Min Q1 Median Q3 Max IQR

Mean Absolute Deviation: MAD

The MAD tells you, on average, how far each data point is from the mean.

1
Find the mean of the data set.
2
Subtract the mean from each value and take the absolute value.
3
Find the mean of those absolute deviations.

Summarizing Data: Step by Step

1
Order the data from least to greatest.
2
Check for outliers or skew. Look at a dot plot or the ordered list.
3
Choose your pair: symmetric → mean \(+\) MAD; skewed/outliers → median \(+\) IQR.
4
Compute center (mean or median).
5
Compute spread (MAD or IQR).
6
Interpret in context: "The typical student scored about ___, and scores varied by about ___."

Example 1: Mean & MAD

Five students scored the following on a quiz: 70, 75, 80, 85, 90. Find the mean and MAD.

70 75 80 85 90
\( \text{mean} = 80 \)

Example 2: Median & IQR

Daily temperatures (\(^\circ\text{F}\)) for one week: 58, 60, 62, 63, 65, 68, 91. The 91 is an outlier, so use median + IQR.

58 60 62 63 65 68 91
\( \text{median} = 63 \)
 outlier

Example 3: Choosing the Right Pair

A class recorded how many books they read last month: 1, 2, 2, 3, 3, 3, 4, 4, 5, 22. Which measures best summarize this data?

Mean
\(\frac{1+2+2+3+3+3+4+4+5+22}{10} = \frac{49}{10} = 4.9\)
Median
10 values → average of 5th and 6th: \(\frac{3+3}{2} = 3\)

Common Mistakes

✕ Forgetting to Order
Finding the median without sorting the data first leads to the wrong middle value.
✕ Wrong Pair
Using mean \(+\) IQR or median \(+\) MAD. Always pair mean with MAD, median with IQR.
✕ Ignoring Outliers
Reporting the mean for skewed data without noting that outliers distort it. Check for outliers first.

Summary: Choosing Your Measures

Measure of Center Measure of Spread Best When
Pair 1 Mean MAD Data is symmetric, no outliers
Pair 2 Median IQR Data is skewed or has outliers

Question 1 of 5

The data set \(\{4, 6, 7, 8, 10\}\) has a mean of:

Question 2 of 5

Find the median of \(\{12, 5, 8, 3, 15, 9\}\).

Question 3 of 5

A data set has values \(\{2, 4, 6, 8, 10, 12, 14\}\). What is the IQR?

Question 4 of 5

A distribution is strongly skewed right with one very large outlier. Which pair of measures should you use to summarize the data?

Question 5 of 5

The data set \(\{10, 10, 10, 10, 10\}\) has a mean of \(10\). What is the MAD?