Free maths tool · Level: Ages 15 to 18
Boxplot and spread
Type in your own set of numbers, or drag a single value away from the rest, and watch what happens to the mean, the median and the box.
A boxplot shows five numbers: the minimum, the first quartile Q1, the median, the third quartile Q3 and the maximum. The box runs from Q1 to Q3, so it holds the middle half of the data. The line inside the box is the median. The whiskers reach the furthest value still within one and a half interquartile ranges, and anything beyond that is drawn separately as an outlier.
Drag one value away and watch the difference
The mean and the median usually sit close together. Now drag the rightmost dot far to the right. The mean follows it; the median stays put.
That is where the whole mean-versus-median argument comes from. For salaries, house prices or one forgotten test, the median describes a typical value better, because it only counts the order and not how extreme an outlier happens to be.
Quartiles: which method does your course use?
This tool splits the data at the median, and with an odd count the median belongs to neither half. Q1 is then the median of the lower half. That is the method Belgian textbooks use.
Excel and Google Sheets work differently and interpolate between values, so Q1 can come out slightly different there. If your answer disagrees with a classmate, this is usually why, rather than an arithmetic slip. Check which method your course uses.
Divide by n or by n - 1?
The tool prints both. Dividing by n gives you σ, which you use when your data is the whole group, for instance every mark in your own class.
Dividing by n - 1 gives you s, which you use when your data is a sample from a larger group, for instance twenty students standing in for a whole school. On large sets the two numbers are close. On small sets the gap is noticeable, and then it matters which one you write down.
Does statistics still feel abstract?
A tutor works through your own dataset with you and shows which measure belongs where.
Frequently asked questions
- How do you make a boxplot?
- Put the data in order and find the minimum, Q1, the median, Q3 and the maximum. Draw a box from Q1 to Q3 with a line at the median, then whiskers out to the most extreme values that are not outliers.
- How do you read a boxplot?
- The box holds the middle 50 per cent of the data. A long box means a lot of spread in the middle. If the median line is not centred in the box, the distribution is skewed. Separate points beyond the whiskers are outliers.
- What is the interquartile range?
- It is Q3 minus Q1, which is the width of the box. It measures spread without being thrown off by a single extreme value, unlike the range and the standard deviation.
- When does a value count as an outlier?
- By the usual convention, when it falls more than one and a half interquartile ranges below Q1 or above Q3. That is a convention rather than a law: it means a value stands out, not that it is wrong.
- What is the difference between σ and s?
- σ divides by n and belongs to a complete population. s divides by n - 1 and belongs to a sample. The n - 1 corrects for the fact that you do not know the true mean and estimated it from the same data.