Range Calculator

The range is the simplest measure of spread: the distance from the smallest value to the largest. A class whose exam scores run from 64 to 95 has a range of 31 points. It is in the same units as the data and is always 0 or more, and it is 0 only when every value is the same.

Because it uses only two values, the range says nothing about how the values in between are spread out, and a single extreme value changes it completely. It is a quick first look at a data set rather than a full description of it.

The midrange is the value halfway between the two extremes. It is a measure of the centre of the data, not of its spread, and like the range it depends on the two extreme values only. The range rule of thumb estimates the standard deviation s from the range; it is explained below.

Eight students scored 72, 85, 91, 68, 77, 95, 88 and 64. Load example fills in these scores.

1. Sort the scores (optional, but it makes the extremes easy to see): 64, 68, 72, 77, 85, 88, 91, 95.
2. Find the extremes: the minimum is 64 and the maximum is 95.
3. Range = 95 − 64 = 31.
4. Midrange = (64 + 95) / 2 = 79.5.
5. Range rule of thumb: s ≈ 31 / 4 = 7.75.

The sample standard deviation of these scores is 11.39, so the rule of thumb is low. With only eight values it has too little chance to reach the extremes: dividing by 4 works for about 30 values. Now suppose the top score were typed as 195 instead of 95. The range would jump from 31 to 131, while the interquartile range would stay at 19.5 because it ignores the extreme values.

Report the range together with the minimum and the maximum, or with a resistant measure such as the interquartile range: on its own it hides where the values lie. The five number summary gives both extremes and the quartiles in one line, and the outlier calculator shows which values are unusually far from the rest.

For bell-shaped data, almost all values lie within two standard deviations of the mean, so the range covers about four standard deviations and s is roughly range / 4. The rule can also be read the other way: usual values lie between the mean minus 2s and the mean plus 2s.

The rule fits samples of about 30 values. A larger sample is more likely to include values far from the mean, so its range is wider, and a smaller one has a narrower range. For a sample from a normal distribution the expected range is about 2.3 standard deviations for 5 values, 3.1 for 10, 4.1 for 30 and 5.0 for 100. Dividing by 4 therefore underestimates s for small samples and overestimates it for large ones, and it is only an estimate: use the standard deviation calculator for the actual value.

A frequency table of classes hides the individual values, so only an approximate range is possible: the upper limit of the last class minus the lower limit of the first class. The class width calculator shows how classes are chosen.

This page finds the range of a data set, a measure of spread. The range of a function, the set of all the values it can take, is a different idea and is not calculated here.

• An outlier or a typing error changes the range. Check the minimum and the maximum against the source data first.
• The range grows with the sample size. A larger sample is more likely to include extreme values, so ranges of samples of different sizes cannot be compared directly.
• The range ignores the middle of the data. Two data sets with the same range can be spread very differently between the extremes.
• Use one unit. Convert all values to the same unit before finding the range, and report the unit with it.
• The midrange is not the median. It moves with either extreme, while the median depends on the middle of the sorted data.