1.3 - Dispersion and standard deviation

Spread and the range

Two groups can have the same mean while their participants' scores differ greatly in consistency.

For original illustrative recall scores, group A has 7, 8, 8, 8, 9 words and group B has 4, 6, 8, 10, 12 words. Both have mean 8 words. Group A's scores cluster near eight; group B's scores are more spread out. Dispersion describes this spread rather than where the centre lies.

Group A's range is $9-7=2$ words; group B's is $12-4=8$ words. Use the subtraction, not the number of distinct scores or an interval with an extra one added. The range is easy to calculate but depends only on the two extreme values. It cannot tell us whether most remaining scores cluster together, and one unusual value can enlarge it sharply.

If the lowest score in a plot is approximately 5 seconds and the highest about 25 seconds, estimate the range as about 20 seconds. Report that it is an estimate when exact raw scores are unavailable. A spread measure requires an interpretable ordered scale: subtracting codes for nominal categories such as music = 2 and quiet = 1 does not produce a meaningful range of preference.

Calculating variance and standard deviation

Variance and standard deviation use every score's distance from the mean.

The variance in the population-form descriptive calculation is the mean of the squared deviations from the mean. Squaring prevents positive and negative deviations cancelling. The standard deviation (SD) is the square root of that variance and returns the spread measure to the original score units.

OCR's descriptive-statistics teaching guide uses this population-form calculation, dividing by $N$. A different formula used to estimate population variability from a sample divides by $N-1$. Follow the formula and convention supplied for a task; do not silently switch denominators or a calculator's population/sample setting.

For group B, mean 8 words, work through each observation:

The deviations sum to zero but the squared deviations sum to 40. Variance is $40/5=8$ squared words. SD is $\sqrt{8}=2.8284\ldots$, or 2.83 words to two decimal places. Square each individual deviation before adding: squaring the sum of the deviations would incorrectly produce zero.

Keep unrounded intermediate values where possible and round the final answer as requested. If the original measurement is seconds, variance is in squared seconds and SD in seconds. SD is never negative because it is the non-negative square root of a non-negative variance.

Using dispersion to interpret results

A spread measure helps explain how well a mean represents individual observations.

For group A, squared deviations about mean 8 are 1, 0, 0, 0 and 1. Variance is $2/5=0.4$, and SD is $\sqrt{0.4}=0.63$ words to two decimal places. Group B's SD of 2.83 words indicates much wider spread around the same mean. This supports a claim about the consistency of these recorded scores; it does not show that group A's instrument is more reliable. Reliability requires evidence that a measurement produces consistent results under an appropriate repeat or comparison.

The range is quick and useful for a broad view of the extremes. SD provides a fuller summary because every score contributes, though extreme scores still affect it: large deviations are squared. Neither is immune to outliers. Comparing SDs makes most sense for the same variable measured in the same units; an SD of 2 seconds cannot directly be judged smaller than an SD of 200 milliseconds without converting units.

Zero SD means every recorded score is identical. A large SD does not mean the mean was calculated incorrectly. Equally, identical means and SDs do not guarantee identical distributions, so inspect raw data or a suitable display when shape matters. Summaries describe the sample; they do not on their own establish significance or explain the cause of a difference.