Midrange Calculator

midrange = (min + max) / 2

min

The smallest value in the data set. Only this one and the largest value enter the calculation, so every value in between can change without moving the result

max

The largest value in the data set. The pair of extremes is worth checking before you trust the answer, since a single mistyped digit lands straight in the result

midrange

The midpoint of the extremes, in the same units as the data. It always lies between the smallest and largest values, and it equals both of them when the list holds a single value

mean

The arithmetic average, printed beside the midrange for comparison. The gap between the two shows how much of the data sits away from the middle of the extremes

The midrange is the right summary when the extremes are the subject: the midpoint of a day's high and low temperature, the middle of a river's highest and lowest level over a year, the centre of the best and worst score in a competition, the midpoint of a bid and an asking price. In each of those the two endpoints are the data — you are not sampling from a larger population, you are describing a range that has two ends. It is the wrong choice when the interior of the data matters, because it discards everything except two points and therefore has a higher variance than the mean or the median on any real sample. It is also not the same as the median: for 2, 4, 4, 4, 5, 5, 7 and 9 the median is 4.5 and the midrange is 5.5, and the two only coincide for symmetric data. Use it as a quick estimate of the centre when a rough number will do — it can be read off a table of maximum and minimum readings without any further arithmetic.

1. Eight values: the midrange is 5.5, the mean is 5

   1. The smallest value is 2 and the largest is 9
   2. midrange = (2 + 9) / 2 = 5.5
   3. The mean of the same list is 40 / 8 = 5 — half a unit lower, because six of the eight values sit at or below it

   The two numbers disagree, and that is the useful part. The midrange only sees that the data starts at 2 and ends at 9; the mean also sees that the data is bunched toward the bottom. Neither is wrong — they answer different questions, and printing both means you can tell at a glance whether the data is evenly spread between its extremes or crowded to one side.

2. One value changed: the midrange more than doubles

   1. Only the largest value changed, from 9 to 90 — the other seven are identical to the first example
   2. midrange = (2 + 90) / 2 = 46
   3. The mean of the same list is 121 / 8 = 15.125, up from 5

   The midrange moved by more than forty units while the mean moved by ten. The midrange absorbs half of any change to an extreme value, with nothing to dilute it; the mean spreads the same change over every value in the list. If one reading in your data is unreliable, the midrange will be wrong by half of that error, which is why it is reserved for cases where the extremes are the point.

3. Temperatures below and above freezing

   1. The coldest reading is −10 and the warmest is 40
   2. midrange = (−10 + 40) / 2 = 30 / 2 = 15
   3. Negative values need no special handling: they are added to the maximum, so a list that crosses zero behaves exactly like one that does not

   A daily midrange reported by a weather service is this number: the midpoint of the day's low and high, not the average of the hourly readings. That convention is worth knowing, because the two differ whenever the temperature spent more time near one end of its range than the other.

The midrange is not robust, and the reason is structural rather than a matter of degree: it is computed from the two most extreme points, so it has no defence against a mistyped digit, an instrument glitch or a genuinely unusual observation. It is typically restricted to situations in which the behaviour at the extreme points is the relevant thing. It is also not the median, despite the similar name in some languages — the median is the middle value of the sorted list and the midrange is the midpoint of the extremes, and they agree only for data that is symmetric about its centre. With a single value the midrange, the mean, the smallest and the largest values are all that one number. The list is capped at 200 values, and a token such as 1,500 is refused rather than guessed at, because a comma between digits is a decimal point in some countries and a thousands separator in others. Nothing on this page flags outliers; the extremes that define the midrange are exactly the values such a rule would be testing.

How do you find the midrange?

Add the smallest value to the largest and divide by two: midrange = (min + max) / 2. For the list 2, 4, 4, 4, 5, 5, 7, 9 that is (2 + 9) / 2 = 5.5. No other value in the data set takes part, so you do not need to sort the list — you need the two extremes, and a quick scan is usually enough to find them. The calculator above prints the minimum and maximum it used so the sum can be checked.

Is the midrange a measure of centre or of spread?

It is a measure of location — a way of saying roughly where the data sits, like the mean and the median. It is easy to mistake for a measure of spread because the two extremes that define it also define the range, and because the two formulas are one operation apart: the range is max − min and the midrange is (max + min) / 2. They answer different questions. The range says how wide the data is; the midrange says where its middle is. The standard reference lists it with the measures of location and not with the measures of scale.

Why is the midrange not robust?

Because it is built from the two most extreme values in the sample and from nothing else, so any error in either of them goes straight into the answer — and only half of it is absorbed. Change 9 to 90 in the example and the midrange jumps from 5.5 to 46, while the mean only moves from 5 to 15.125. That sensitivity is worth having when the extremes are what you are measuring, such as a daily high and low, and it is a liability when they are not.

Is the midrange the same as the median?

No. The median is the middle value of the sorted list, so with eight values it is the average of the fourth and fifth: (4 + 5) / 2 = 4.5. The midrange is the midpoint of the two extremes: (2 + 9) / 2 = 5.5. They coincide only when the data is symmetric about its centre, and the gap between them is itself informative — here the midrange sits above the median, which means the upper half of the data is stretched further from the centre than the lower half.

When would I actually use the midrange?

When the two endpoints are the data, rather than a sample of something larger. A weather summary that reports the day's high and low is describing a midrange. So is a range check on a manufacturing tolerance, where the specification gives a minimum and a maximum and you want their centre. It is also a useful first estimate when only the extremes have been recorded: if a table lists the cheapest and the dearest quote, the midrange is the midpoint of that spread, available without knowing anything about the prices in between.

What if all the values are the same, or there is only one?

Both cases work and both give that same value. With the single value 42, the smallest and largest are both 42, so the midrange, the mean, the minimum and the maximum are all 42 — nothing about this calculation needs a second observation. Likewise for 3, 3, 3, 3, where the extremes are equal and the midrange is 3. There is no minimum number of values here, unlike the variance and standard deviation, which need a divisor and therefore need more than one.

• Measures of Location — NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.1 (lists the midrange among the alternative measures of location as (smallest + largest)/2, and states that it is not robust because it is based on the two most extreme points) — National Institute of Standards and Technology
• Measures of Scale — NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.6 (the companion list of spread measures, where the midrange does not appear — the reason this page treats it as a measure of location) — National Institute of Standards and Technology
• GB/T 3358.1-2009, Statistics — Vocabulary and symbols, Part 1 (the Chinese national vocabulary standard covering the location measures; the record page notes that no online full text is offered) — State Administration for Market Regulation, China