Say two classes both averaged exactly 70 on a math test. Does that mean their score distributions look similar? Not necessarily. In one class, everyone might be clustered between 65 and 75, while in the other, half the students scored in the 30s and half scored near 100. The mean is identical, but "how spread out" the scores are is completely different. That spread is called dispersion.
The classic way to see dispersion at a glance is a box plot. Line the data up in order of size, then find five landmark numbers: the smallest value (minimum), the largest value (maximum), the exact middle value (median), and the median of the lower half (the first quartile, Q1) and the median of the upper half (the third quartile, Q3). Draw a box from these five numbers, and the wider the box (Q3−Q1, the interquartile range), the more spread out the middle half of the data is. Just glancing at a box plot tells you right away whether the data is tightly clustered or widely scattered.
Where a box plot shows how spread out a single batch of data is, a scatter plot shows what kind of relationship exists between two different sets of data (variables). For example, take paired values like "hours studied" and "test score," plot one on the x-axis and the other on the y-axis, and mark each pair as a point. If the points generally trend up and to the right, that means one value tends to increase as the other does — a positive correlation. If they trend down and to the right, one value tends to decrease as the other increases — a negative correlation. If the points are scattered with no clear direction, there's no correlation.
Here's something important: having a correlation doesn't necessarily mean one thing "causes" the other. Ice cream sales and drowning incidents, for instance, both rise together in summer, giving them a positive correlation — but ice cream doesn't cause drownings. Both change together because of a shared underlying cause: hot weather. It's worth remembering that spotting a relationship with a scatter plot and pinning down that relationship's true cause are two different problems.
Box plots and scatter plots are extremely practical tools in statistics. They compress information that would be hard to grasp from a long list of numbers into a single picture. Whenever a news story or report says something like "there's a big gap" or "there seems to be a relationship," a graph like one of these is usually lurking behind the claim.
On our activity page, you can organize randomly generated data into a box plot and check the five key numbers from minimum to maximum for yourself, and try a quiz where you look at a scatter plot and guess whether it shows a positive correlation, a negative correlation, or no correlation.