How do you find the range?

The range is the simplest measure of spread there is. One subtraction, no formula sheet. It still goes wrong in three predictable places: negative numbers, frequency tables, and the moment someone reads more into it than it can carry.

The method

Find the largest value. Find the smallest. Subtract. A week of daily highs in degrees Celsius: 18, 23, 15, 27, 21, 19, 24.

step 1  largest value     27
step 2  smallest value    15
step 3  subtract          27 − 15 = 12

range   12 degrees

The range is 12 degrees. It keeps the unit of the data, so a range of temperatures is in degrees and a range of prices is in money. For a long list, sorting first makes the two ends impossible to miss: 15, 18, 19, 21, 23, 24, 27. The ends are all that matter. Nothing in between changes the answer.

Negative numbers

Five overnight lows: −6, −2, 3, 1, −4. The smallest value is the one furthest below zero, which is −6, not −2.

largest    3
smallest   −6

range      3 − (−6)  =  3 + 6  =  9

wrong      3 − 6     =  −3      dropped the minus sign
wrong      3 − (−2)  =  5       picked −2 as the smallest

Subtracting a negative is adding, which is why the range here is bigger than the largest value. That is correct: on a thermometer, 6 below zero to 3 above is 9 degrees of travel. When every value is negative the same rule holds. For −12, −5, −9 the largest is −5 and the range is −5 − (−12) = 7.

If the answer ever comes out negative, the subtraction was done backwards. The largest value cannot be smaller than the smallest, so the range is zero or more, and zero only when every value is the same: 7, 7, 7 has a range of 0. So does a list of one value.

One outlier owns it

The range looks at exactly two numbers, so one unusual value sets it alone. Seven test scores, then the same seven plus a student who skipped half the paper:

scores   72, 75, 78, 80, 81, 83, 85
range    85 − 72  =  13
median   80
mean     554 ÷ 7  =  79.14

add 31   31, 72, 75, 78, 80, 81, 83, 85
range    85 − 31  =  54
median   (78 + 80) ÷ 2  =  79
mean     585 ÷ 8  =  73.125

One score quadrupled the range, from 13 to 54, while the median moved by a single point. The mean sits in between, dragged down by six points. The range is not wrong here. It is answering its own question, how far apart the extremes are, and the extremes changed.

The flip side: two very different sets can share a range.

set A   10, 10, 10, 10, 30     range 30 − 10 = 20
set B   10, 15, 20, 25, 30     range 30 − 10 = 20

Set A is four identical values and one stray. Set B is spread evenly. The range calls them the same. Use it as a quick first look at spread, and reach for the median, the mean, or the interquartile range before drawing conclusions.

The interquartile range

The interquartile range, or IQR, is the range of the middle half of the data, so the extremes cannot run it. Sort the list, split it into a lower and an upper half, take the median of each, and subtract:

sorted        2, 4, 6, 8 | 10, 12, 14, 16
lower half    2, 4, 6, 8         median (4 + 6) ÷ 2   =  5
upper half    10, 12, 14, 16     median (12 + 14) ÷ 2 =  13

IQR           13 − 5   =  8
range         16 − 2   =  14

Swap the 16 for 160 and the range jumps to 158 while the IQR stays at 8. Textbooks and spreadsheets do not all split the halves the same way, so quartiles can differ slightly between tools. The median-of-halves method above is the one most classrooms teach.

Range from a frequency table

When data comes as counts, the range is still largest value minus smallest value. The counts do not enter the subtraction. Goals scored per match across one season:

goals   matches
  0        5
  1        8
  2        6
  3        3
  4        0
  5        1
  6        0

smallest value that occurred   0
largest value that occurred    5

range   5 − 0  =  5

Two traps. Subtracting the counts, 8 − 0 = 8, measures the table, not the goals. And a row with zero matches never happened: no match ended with 6 goals, so 6 is not the largest value and the range is not 6. Row 4 also has a count of zero, but it sits between the ends, so it changes nothing.

Range and the other averages

Range rarely travels alone. Mean, median, mode and range are taught as a set because each answers a different question: the typical value three ways, then how wide the data spreads. For the daily highs above, the mean is 147 ÷ 7 = 21, the median is also 21, and the range is 12. A report that gives a center with no spread hides half the picture. The average post and the median post cover the centers in full.

The subtraction that is not exact

A range is one subtraction, and with decimals even one is enough to trip binary floating point, which most software uses and which cannot store 13.1 or 11.9 exactly. These are real results from JavaScript and Python:

race times   12.4, 11.9, 13.1, 12.7 seconds
range        13.1 − 11.9  =  1.1999999999999993    should be 1.2

body temps   36.2 to 37.4 degrees
range        37.4 − 36.2  =  1.1999999999999957    should be 1.2

The spread between 11.9 and 13.1 seconds is 1.2 seconds. Not slightly less, and not something a program will refuse to call equal to 1.2. It is the same fault that makes 0.1 + 0.2 miss 0.3. Prism does its arithmetic on exact decimals, so 13.1 − 11.9 comes back as 1.2 and 0.1 + 0.2 is exactly 0.3.

Spot the ends, subtract in Prism

Type the largest value minus the smallest, minus signs included, and read 1.2, not 1.1999999999999993. Every entry stays in a searchable history, so you can check which two values you used. History, percentages and fractions are free forever, no ads, no subscription. See the app.