9.1.10, 1.2.5, 2.2.11, D.1.6, D.1.11, D.1.14 - Spread and distributions
Learn why an average needs a measure of spread, how to calculate the sample standard deviation with Pearson's supplied formula, and how normal and skewed distributions affect interpretation. These skills connect descriptive statistics in social and cognitive research with the mathematical requirements for dispersion and distribution shape.
Range and variability
Two samples can have the same mean while their participants' scores differ greatly in consistency.
Consider invented recall scores, measured in words: sample A contains 4, 4, 6, 6, and sample B contains 2, 4, 6, 8. Both total 20 words across four people and therefore have a mean of 5 words. Sample A is tightly clustered around that mean; sample B is more spread out. A mean alone hides this difference.
Dispersion
Dispersion describes how spread out scores are. The range is the highest score minus the lowest score. Standard deviation describes the spread of scores around their mean using all the observations.
Sample A's range is words; sample B's is words. A larger range therefore describes more separation between the extremes here. Range is quick to calculate and retains the score's unit, but it ignores every value between the minimum and maximum.
For example, 2, 2, 2, 8 and 2, 4, 6, 8 both have range 6. Their internal patterns differ, showing why equal ranges do not establish equal distributions. Nor can a larger range automatically be interpreted as lower reliability: it may reflect genuine individual differences, a larger sample with more opportunity to include extremes, measurement noise or several of these factors. Reliability concerns consistency of measurement, not simply a small spread across different people.
An extreme observation can greatly alter range. Check whether such a value is an error, but retain a valid value and explain its influence. Standard deviation gives a fuller account because all scores contribute, although it too is sensitive to extreme values.
Calculating sample standard deviation
Sample standard deviation combines each score's distance from the sample mean into one measure in the original score units.
Pearson supplies the sample estimate formula in Appendix 4. You need to use it competently rather than memorise it. The denominator is , not , because this formula estimates population variability using a sample whose mean has also been estimated from those observations. Calculators often offer both sample and population standard deviations; select the sample version for this formula.
Here is each score, is the sample mean, and is the number of scores. Sum the squared deviations, divide by , then take the square root.
For the invented recall scores 2, 4, 4, 6 words, the total is 16 and the mean is words. Work through every observation, including repeated values:
| Score | Deviation | Squared deviation |
|---|---|---|
| 2 | −2 | 4 |
| 4 | 0 | 0 |
| 4 | 0 | 0 |
| 6 | 2 | 4 |
| Total | 0 | 8 |
The deviations sum to zero because positive and negative distances cancel. Squaring makes each contribution nonnegative, preventing that cancellation from concealing spread. The sum of squared deviations is 8, and , so the sample variance is squared words. Finally,
The square root returns the measure to words rather than squared words. Keep the unrounded intermediate value: rounding too early needlessly changes the final answer. Standard deviation cannot be negative; identical scores give zero. A value of zero means no observed spread, not a guarantee that the measurement is valid. With only one score the sample formula cannot be used because its denominator would be zero.
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