Mean, Median, Mode and Range Calculator

Which of the three tabs you need

The shape of the data decides which averages are even possible, so the tool asks for the shape first. A raw list and a frequency table both give exact answers. A grouped table cannot, and the results are labelled as estimates rather than quietly presented as facts.

On the raw list tab you can paste straight from a question — commas, spaces and line breaks all work. Anything that is not a number is counted and reported rather than silently treated as a zero, which is the failure mode of most calculators that accept pasted text.

On the two table tabs, add rows as you need them. A row with a blank or zero frequency is left out of every total, and a class whose upper bound is not above its lower bound is ignored instead of producing a negative width.

• Raw list — a set of values written out one by one, such as test marks for a class.
• Frequency table — an x column and an f column, for when the same value repeats.
• Grouped data — class intervals such as 10 ≤ x < 20, where the individual values have been lost.

What each figure is actually doing

The mean is the total divided by how many values there are. From a list that is Σx ÷ n; from a frequency table it is Σfx ÷ Σf, because each value has to be counted as many times as it occurs. Both totals are printed, because in a table question the marks are for Σfx and Σf, not for the final division.

The median is found by position, not by eye. The position is (n + 1) ÷ 2 in the ordered data. When n is odd that lands on a value; when n is even it lands between two, and the median is the mean of that pair. The tool prints the position first and the value second, in that order, because that is the order the method runs in.

The mode is the value that occurs most often — the value, never the frequency. If every distinct value occurs equally often there is no mode at all, which the tool says plainly instead of listing everything. The range is largest minus smallest, and it is a measure of spread rather than an average.

Quartiles use the (n + 1) ÷ 4 and 3(n + 1) ÷ 4 positions, and where a position falls between two values the tool takes the value part-way between them and says it has. This convention is not universal, so the note under the results names it — on a set of seven or eight values, the n ÷ 4 convention taught in some textbooks gives a different answer, and neither is wrong in itself.

Why the grouped mean is only an estimate

Once data has been grouped, the original values are gone. All that survives is how many values fell in each class. There is no arithmetic that can recover a total from that, so the mean of grouped data cannot be calculated — only estimated.

The estimate works by assuming every value in a class sits at the midpoint of that class. Twelve values in 10 ≤ x < 20 are treated as twelve values of 15. That is exactly right only if the values happen to be evenly spread within each class, which real data rarely is, and it is why every mark scheme asks for an "estimate of the mean" rather than the mean.

For the same reason this tool will not print a range or an interquartile range for grouped data. It gives the widest the range could possibly be, from the outermost class bounds, and it names the class each quartile falls into. Turning a class into an estimated value needs linear interpolation or a cumulative frequency diagram, and this tool does not draw one rather than inventing a figure that looks precise.

Where the marks are usually lost

Almost every dropped mark on this topic is one of five things, and four of them happen before the arithmetic starts. The tool prints the intermediate columns partly so that a student can see which step went wrong rather than only that the answer did.

The most expensive of them is dividing by the number of rows in a frequency table instead of by Σf. A table with five rows and forty-two values has n = 42, and dividing by 5 gives an answer that is roughly eight times too large without looking obviously absurd.

• Dividing Σfx by the number of rows rather than by Σf.
• Finding the median without putting the data in order first.
• Giving the frequency as the mode instead of the value that has that frequency.
• Giving a frequency as the modal class, when the answer is the class interval itself.
• Calling the grouped answer a mean rather than an estimate of the mean, which loses the mark in the wording alone.