One-Variable Data: Distributions and Measures

Compute a frequency-weighted mean

Suppose scores 2, 4, and 7 have frequencies 3, 2, and 1. There are 6 observations, and their total is $2(3)+4(2)+7(1)=21$. The mean is $\frac{21}{6}=3.5$.

Averaging the three displayed scores would ignore their frequencies. Expand a short table into the list $2,2,2,4,4,7$ if the weighted calculation feels unclear.

Find the median after ordering

For $2,2,2,4,4,7$, the middle positions are the third and fourth, so the median is $\frac{2+4}{2}=3$. With an odd count, use the one middle observation.

The median depends on positions, not the average of the smallest and largest values. In a frequency table, use cumulative counts to locate the middle observation or pair.

Compare spread separately from center

The sets $4,5,6$ and $1,5,9$ both have mean and median 5. The second has a larger range and standard deviation because its observations are farther from the mean.

Standard deviation describes distance from the mean. You can compare it qualitatively when one set is visibly more concentrated; equal means do not imply equal spread.

Predict what changing every value does

Adding 10 to every observation raises the mean and median by 10 while leaving the range and standard deviation unchanged. Multiplying every observation by 3 multiplies the mean, median, range, and standard deviation by 3.

Adding one new outlier is a different operation. It can change the mean substantially without shifting the median by the same amount. Recompute positions and totals instead of applying a whole-set rule.

Check your understanding

Practice this skill

1. Count observations before calculating a mean or median.
2. Compare center and spread as separate features.
3. Practice transformations and new-observation questions separately.

Print the one-variable data: distributions and measures worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.