Representing Data
Numerical data can be shown three main ways: a dot plot, a histogram, or a box plot. Each one is a picture of the same idea — where the values fall — but each display does a different job, and test questions love to ask you to read all three.
A dot plot shows one dot for each individual value. A histogram groups the values into equal intervals and shows a bar for each interval. A box plot skips the individual values entirely and shows five summary numbers. Pick the display based on how much detail you need.
Three displays, three jobs
Dot plot: one dot per data value, stacked above a number line. You can recover the exact data set from it, so it is best for small sets. To count how many values equal , just count the dots above .
Histogram: the values are grouped into equal intervals, and each bar's height is the count in that interval, with no gaps between bars. You lose the individual values but gain a clear picture of shape for large sets.
Box plot: shows the five-number summary — minimum, first quartile , median, third quartile , maximum. The box covers the middle half of the data, and the whiskers stretch to the extremes.
The dot plot below shows a small data set — one dot per value stacked above the number line, so you can read every value back exactly.
Reading a histogram
Each bar answers one question: how many values landed in this interval? To answer a question like "how many students scored at least 80," add the counts of every bar from to the right. "At least" includes the starting value; "more than" does not.
Because the values are grouped, you can never tell the exact values from a histogram — only how many fell in each interval. If a question asks for the exact maximum from a histogram alone, the honest answer is that you cannot know it.
The five-number summary
Order the data, find the median, then find the median of each half: the lower half's median is and the upper half's median is . Together with the minimum and maximum, those five numbers are everything a box plot draws.
One quarter of the data sits between each pair of neighboring summary numbers. So the box — from to — always holds the middle of the data, no matter how stretched the whiskers look.
The box plot below is drawn from a five-number summary: the two whisker ends mark the minimum and maximum, the box edges mark and , and the line inside the box marks the median.
Worked examples
Example 1: reading a histogram
A histogram of test scores has bars: – with students, – with students, – with students, and – with students. How many students scored at least ?
Answer: students
Example 2: building a five-number summary
Find the five-number summary of .
Answer: Minimum , , median , , maximum
Example 3: choosing the right display
You surveyed 300 people about their commute times and want to show the overall shape of the data. Which display fits best?
Answer: A histogram
Try one yourself
Common questions
What is the difference between a histogram and a bar graph?
A histogram shows numerical data grouped into equal intervals, so the bars touch — the intervals flow into each other. A bar graph shows separate categories (like favorite colors), so the bars have gaps and their order does not matter.
Can I find the mean from a box plot?
No. A box plot shows the minimum, quartiles, median, and maximum — the mean is not one of them. You can read the median from the line inside the box, but the exact mean is not recoverable.
Does 'at least 30' include 30?
Yes. "At least " means or more, so you include the interval that starts at . "More than " would exclude exactly .
How much of the data is inside the box of a box plot?
The middle half — . The box runs from to , and each whisker covers another quarter of the data.
Want the video version?
Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.