Midrange Calculator

What the midrange is

The midrange is the halfway point between the smallest and largest value in a set. Add the minimum to the maximum and divide by two. Nothing else in the data enters the arithmetic, which makes it the fastest measure of center to compute by hand and the least forgiving of a single odd reading.

Eight test scores of 62, 71, 74, 78, 81, 85, 88 and 94 give a midrange of (62 + 94) / 2 = 78. The mean of the same scores is 79.125 and the median is 79.5. The three answers sit close together here because the scores are evenly spread.

The midrange is not the middle value. The middle value is the median, and for an even count it is the average of the two central numbers. The midrange is the midpoint of the span, so it depends on how far apart the extremes sit, not on how the values cluster.

That property is why the midrange shows up in quality control and in weather records. It is also why it fails on income, home prices and anything else with a long tail, where one large value can drag the midpoint far above the typical case.

Midrange against mean and median

Putting the three measures on the same data shows what each one protects against. The eight scores above are well behaved. Add a single score of 100, as if one student aced an easier retake, and the three measures move by different amounts.

The midrange moves by half of the change in the extreme, because the extreme is one of the two numbers being averaged. The median barely moves, because adding a ninth value only shifts which value sits in the middle.

Home prices show the split in a way that matters. Seven recent sales of 210,000, 245,000, 268,000, 289,000, 310,000, 335,000 and 1,250,000 dollars give a midrange of 730,000 dollars and a median of 289,000 dollars. The midrange describes a house that does not exist in that neighborhood. The median describes what a buyer typically pays.

Why the extremes carry the answer

Every value except the minimum and maximum could change without touching the midrange. That is the appeal when only the ends are known, such as when a gauge reports a low and a high reading for the day and nothing in between.

The sensitivity runs in one direction: the midrange shifts by exactly half of any shift in an extreme. Raise the top score from 94 to 100 and the midrange rises by 3. Double the largest home price and the midrange rises by half of the increase, which is why a single mansion sale distorts a neighborhood figure.

NIST's statistics handbook lists the mean and the median among the standard measures of location and notes that no single summary fits every data set. The midrange belongs on that list with a narrower job.

Sample size does not fix the problem. Adding more values to the middle of a skewed set leaves the midrange where it was, because the extremes stay put. Only a change at one end moves it, and in a skewed distribution that end is often a recording error or a rare event.

The opposite holds for well spread data. For a sample drawn from a uniform distribution, the midrange is a better estimate of the center than the mean is. Its variance is (b - a) squared divided by 2(n + 1)(n + 2), against (b - a) squared divided by 12n for the mean, so with 10 values from a uniform range the midrange is about 2.2 times more efficient.

Where the midrange is the right summary

Heating and cooling degree days, the numbers behind utility bills and weather normals, are built on the daily mean temperature, which the Climate Prediction Center defines as the average of the daily maximum and minimum. That average is a midrange of the day's two extremes, and it works because a day's temperature curve is roughly symmetric around its midpoint.

Industrial quality control uses the same idea. A midrange chart plots the midpoint of each small subgroup, and it reacts to a shift in the process center with fewer samples than a mean chart needs. The trade is sensitivity to a single bad part, so midrange charts pair with a range chart and are used on subgroups of three to five pieces.

Field work uses it as a shortcut. If a thermometer logs 41 F at dawn and 77 F at mid afternoon, the midrange is 59 F, which is a fair single number for the day when no hourly record exists. A materials inspector who measures 11.8 mm and 12.2 mm on a batch can report 12.0 mm as the midpoint of the tolerance band.

The common thread is a distribution with two clear ends and no long tail. When that holds, the midpoint of the span sits near the center of the data and the two numbers you have are enough.

Where it misleads

Skewed data is the first failure. Income, home prices, hospital charges and time to complete a task all have a long right tail, so the maximum sits far from the bulk of the values and the midrange lands in empty space. A dataset of 12, 18, 21, 25, 30, 34 and 96 gives a midrange of 54 against a median of 25, and no one in that group earns anything near 54.

Top-coded and open-ended data is the second. A survey that records incomes as 200,000 dollars or more leaves the maximum unknown, so the midrange cannot be computed honestly. The same applies to a warranty claim recorded as 36 months or longer.

Recording errors are the third. A single extra zero turns 940 into 9,400 and moves the midrange by 4,230 while leaving the median untouched. Any process that produces occasional outliers should use the median and report the range separately.

The fourth failure is interpretation. A midrange of 78 on a test does not mean the average student scored 78. It means the lowest and highest scores average 78. Confusing the midpoint of the span with the middle of the class is the mistake that gets quoted in meetings.

Range, midhinge and trimean

Three related measures sit next to the midrange on a statistics sheet, and each answers a different question.

The midhinge replaces the extremes with the quartiles, so it ignores the top and bottom quarter of the data. With Tukey hinges, the lower half of the eight scores is 62, 71, 74 and 78, which gives Q1 of 72.5, and the upper half is 81, 85, 88 and 94, which gives Q3 of 86.5. The midhinge is (86.5 + 72.5) / 2 = 79.5.

The trimean weights the median twice and adds the quartiles: 72.5 + 159 + 86.5 = 318, and 318 / 4 = 79.5. It matches the midhinge here because the scores are symmetric. On skewed data the two separate, and the trimean stays closer to the median.

The range is the simplest spread measure and the least stable, since one outlier sets it. Report it with the median rather than alone, and say how many values the set contains.

Worked example: eight quiz scores

A teacher has eight quiz scores of 62, 71, 74, 78, 81, 85, 88 and 94 and wants one sentence for the report card. The steps take under a minute.

Sort the scores, which they already are. The minimum is 62 and the maximum is 94, so the range is 94 - 62 = 32 and the midrange is (62 + 94) / 2 = 78. The sum is 633, so the mean is 633 / 8 = 79.125. With eight values, the median is the average of the fourth and fifth, (78 + 81) / 2 = 79.5.

The honest sentence is that the class average is 79.1, the middle score is 79.5, and scores span 32 points from 62 to 94. The midrange of 78 adds nothing that the other two numbers do not already say, and quoting it alone would hide the spread.

Now suppose one student retakes the quiz and scores 100. The midrange rises to 81 while the median rises to 81 and the mean to 81.4. All three move together because the new value is at the top end. Remove the low score of 62 instead and the midrange climbs to 84, because the minimum is now 71, while the median barely shifts. That difference is the clearest signal that the midrange is a measure of the span, not of the group.