2.4 The Midpoint

Find the Midpoint of Two Numbers

Statistics often replaces a whole stretch of the number line with the single number at its center. When data are grouped into classes, the class midpoint stands in for every value in the class: when you estimate a mean from grouped data or draw a frequency polygon, a class of heights from 60 to 62 inches is represented by 61. The symmetric confidence intervals used in this lesson are reported as a point estimate plus or minus a margin of error. In both cases the center is the midpoint, and finding it takes one addition and one division.

The midpoint is the same as the mean of the two numbers. The distance from the midpoint to either number is half the distance between the two numbers, so neither end is closer to it than the other. The same two steps work when one or both numbers are negative or are decimals: add them by the rules for signed numbers, then divide the sum by 2.

Find Class Midpoints

When data are grouped into classes, each class is written as a lower value and an upper value, such as 59.95–61.95. To draw a frequency polygon or to estimate a mean from grouped data, every value in a class is represented by the class midpoint: the midpoint of the class's lower and upper values.

Class midpoints show up in two places in an introductory statistics course. A frequency polygon plots one point for each class, directly above its class midpoint. And when only a grouped frequency table is available, the mean is estimated by treating every value in a class as if it were equal to the class midpoint.

Some tables write each class with whole-number limits and leave a gap between one class and the next, such as 0–2, 3–5, 6–8. The same add-and-halve rule applies: add the class's lower and upper values and divide by 2. For the class 10–19, the class midpoint is $\frac{10+19}{2}=\frac{29}{2}=14.5$.

Find the Center and Half-Width of an Interval

The symmetric confidence intervals used in this lesson can be written as two endpoints in parentheses, such as $(67.18,68.82)$. For these intervals, the midpoint is the point estimate, the estimate the interval was built around. Half the width of the interval is the margin of error (your statistics text also calls it the error bound): how far the interval reaches on each side of the point estimate.

For a confidence interval symmetric about its point estimate, the interval can also be written as the point estimate plus or minus the margin of error. The interval $(67.18,68.82)$ is the same as $68\pm 0.82$: it starts $0.82$ below $68$ and ends $0.82$ above it.

Practice Makes Perfect

Find the Midpoint of Two Numbers

In the following exercises, find the midpoint of the two numbers.

Find Class Midpoints

In the following exercises, find the class midpoint.

Find the Center and Half-Width of an Interval

In the following exercises, each confidence interval is symmetric about its point estimate. Find the point estimate and the margin of error.