Comparing Sets of Data
Comparing two data sets takes two numbers each, not one. A measure of center tells you which set's values run higher; a measure of spread tells you which set is more consistent. Two classes can have the same average score while one class is steady and the other swings wildly — one number alone hides that.
Side-by-side box plots are the standard tool for the job, because both comparisons are visible at a glance: compare the median lines for center, and compare the box widths (the IQRs) for spread. A smaller spread means the data is more consistent.
Compare center and spread together
For center, compare medians (or means, if both distributions are symmetric). The set with the greater median typically has larger values overall.
For spread, compare the IQRs or the ranges. On box plots, the IQR is simply the length of the box from to — a shorter box means the middle half of the data is packed tighter, so that set is more consistent.
Always state both comparisons in context. "Class B scored higher on average (median vs. ), but Class A's scores were more consistent (IQR vs. )" is a complete comparison; either sentence alone is only half of one.
In the side-by-side plots below, Team 2's median line sits farther right (higher center), but its box is also wider (larger IQR, less consistent) — both comparisons read straight off the picture.
Reading side-by-side box plots
The line inside each box is that set's median — compare their positions on the shared number line. The box edges are and , and the whiskers reach the minimum and maximum.
Be careful what a box plot cannot tell you: the mean, the number of data values, or any individual value between the five summary numbers. Questions that ask you to compare means from box plots alone cannot be answered.
Adding a constant to every value
Adding the same number to every value slides the whole data set over: the mean, median, minimum, and maximum all shift by that number, but the spread stays the same. The gaps between values did not change, so the range, IQR, and standard deviation are untouched.
Multiplying every value by a constant is different — that stretches the data, so both the center and the spread get multiplied by the constant.
Worked examples
Example 1: compare two classes with box plots
Class A's box plot shows minimum , , median , , maximum . Class B's shows minimum , , median , , maximum . Compare the classes.
Answer: Class B has the greater median ( vs. ), but Class A's scores are more consistent ( vs. )
Example 2: a 5-point curve
Every score in a class gets a 5-point curve. The old mean was and the old range was . Find the new mean and the new range.
Answer: New mean , new range
Example 3: same center, different consistency
Two runners both average seconds per lap. Runner A's times have an IQR of seconds; Runner B's have an IQR of seconds. Who is more consistent?
Answer: Runner A — same average lap, far more consistent times
Try one yourself
Common questions
Which measures should I use to compare two skewed data sets?
Median for center and IQR for spread. Skewed data drags the mean and the range toward its tail, so those two can make an unfair comparison. The median and IQR resist the tails.
Can I compare the means of two data sets using box plots?
No. A box plot shows the five-number summary, and the mean is not part of it. Box plots let you compare medians, quartiles, ranges, and IQRs — for means you need the actual data.
What does a longer box on a box plot mean?
A larger IQR — the middle half of that data set is more spread out, so the data is less consistent. It says nothing about which set has more values or a higher center.
If every value doubles, what happens to the mean and range?
Both double. Multiplying every value by a constant scales the center and the spread. Only adding a constant leaves the spread alone.
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