Midpoint of x coordinates:
formula and worked example

What the midpoint of two x values represents

The midpoint of two x coordinates is the single position that sits exactly halfway between them on a horizontal axis. If you draw a segment from one point to the other and fold it in half, the fold lands on this value.

It is useful anywhere you need a centre point: placing a label in the middle of a bar on a chart, finding the centre of a room along one wall, locating the centre of a class interval when you plot a histogram, or splitting a range into two equal halves for a search.

It only handles one axis. A midpoint in two dimensions needs the same calculation done separately on the y values, and the pair of results together is the actual midpoint.

The formula and its two inputs

Add the two coordinates and divide by two. In other words, you are averaging them.

• X1: the coordinate of the first endpoint.
• X2: the coordinate of the second endpoint.
• Result: the x value halfway between them, in the same unit and on the same axis.

The order of the inputs does not matter, because addition is order-free. The two values must also be measured from the same origin, otherwise the average is not a real position. If you know the midpoint and one end, the same relation run backwards gives the other end: double the midpoint and subtract the known value.

Worked example: halfway between -4 and 10

Take endpoints at x = -4 and x = 10, one on each side of zero.

1. 1Add the coordinates: -4 + 10 = 6.
2. 2Divide by two: 6 ÷ 2 = 3.
3. 3Check the distances: 3 − (−4) = 7 and 10 − 3 = 7, so both sides match.

That last check is worth doing whenever negative numbers are involved. A correct midpoint is always equally far from each endpoint.

How to sanity-check and use the midpoint

A midpoint must fall between the two endpoints, so if your answer is outside that range, the arithmetic is wrong. The total length of the segment is the difference, here 14, and the midpoint is half of that length in from either end.

The midpoint is not necessarily a typical or central value of any data that sits between the points. It is purely geometric. If three or more values are involved, the average of all of them is a different quantity from the midpoint of the extremes.

For a range such as a salary band or a temperature window, the midpoint is a quick summary of the middle of the range, but it says nothing about where most of the values actually fall.

How a change in X1 shifts the midpoint

The midpoint responds to each endpoint with a weight of one half. Moving X1 by some amount moves the midpoint by half that amount in the same direction.

The row looks odd at first: a 10% increase turned -4 into -4.4, which is further left on the axis, so the midpoint dropped from 3 to 2.8. A percentage change applies to the magnitude, and with negative numbers that points the opposite way to what the word increase suggests.

Errors that turn up with midpoints

• Dropping the minus sign on a negative coordinate and adding 4 + 10 instead of -4 + 10.
• Dividing by the number of units rather than by two, or subtracting instead of adding.
• Treating the result as a two-dimensional point when only the x values were used.
• Averaging coordinates that come from different origins or different scales, such as one in metres and one in centimetres.

A short example you can adapt

Imagine marking the centre of a shelf that runs from the 40 cm mark to the 190 cm mark on a wall ruler, to place a bracket.

1. 1Add the two marks: 40 + 190 = 230.
2. 2Halve it: 115 cm is the centre.
3. 3Check it: 115 − 40 = 75 and 190 − 115 = 75, so the bracket is level between the ends.

Replace the numbers with any pair and the same three steps hold, including when one end sits at a negative coordinate. If the bracket must sit off-centre, say one third of the way along, the same idea extends: multiply the width by the fraction wanted and add it to the smaller coordinate.