Measures of Spread
Two data sets can have exactly the same mean and still look nothing alike — one tightly clustered, one scattered everywhere. A measure of spread describes how far apart the values in a data set are, which is the half of the story a measure of center leaves out.
Algebra 1 uses three measures of spread: the range, the interquartile range (IQR), and the standard deviation. The range is the fastest to compute, the IQR is the most resistant to outliers, and the standard deviation is the most complete.
The three measures of spread
Range: maximum minimum. One subtraction, but fragile — a single extreme value changes it completely.
Interquartile range: , the spread of the middle half of the data. Because it ignores the top and bottom quarters, outliers cannot touch it.
Standard deviation: the typical distance of a value from the mean. A small standard deviation means the values huddle near the mean; a large one means they scatter. In Algebra 1 you usually interpret it or let a calculator compute it rather than grinding it out by hand.
A box plot makes the difference between range and IQR visible: the whiskers reach the minimum and maximum (the range), while the box runs only from to (the IQR, the middle half).
Finding quartiles
Order the data and find the median. Then is the median of the lower half and is the median of the upper half. When the data set has an odd number of values, leave the overall median out of both halves.
For the median is . The lower half is , so ; the upper half is , so ; and .
Which measure to trust
Pair your measure of spread with your measure of center. Symmetric data with no outliers: mean with standard deviation. Skewed data or data with outliers: median with IQR, because the extreme values pull both the mean and the range but leave the median and IQR alone.
Worked examples
Example 1: find the range
Find the range of .
Answer: Range
Example 2: find the IQR
Find the interquartile range of .
Answer:
Example 3: same range, different spread
Set A is and Set B is . Compare their spreads.
Answer: Same range, but Set B's values are genuinely more spread out ( vs. )
Try one yourself
Common questions
Why is the IQR better than the range when there are outliers?
The range depends only on the two most extreme values, which is exactly where outliers live. The IQR is built from and , the edges of the middle half, so an outlier at either end never enters the calculation.
What does a standard deviation of 0 mean?
Every value in the data set is identical. No value sits any distance from the mean, so the typical distance is . Any variation at all makes the standard deviation positive.
Do I include the median when splitting the data into halves?
With an odd number of values, no — set the middle value aside and take the median of each remaining half. With an even number of values, the data splits cleanly into two halves and nothing is left out.
Can two data sets have the same center but different spreads?
Yes, easily. and both have a mean of , but the second set is far more spread out. That is exactly why you report a measure of spread alongside a measure of center.
Want the video version?
Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.