Range, standard deviation and distributions

Range and standard deviation

• Range = highest score - lowest score. Quick and easy, but it uses only two scores and is distorted by outliers. (Some textbooks add 1; follow the method shown in the question.)
• Standard deviation (SD) measures the average distance of scores from the mean. A large SD means scores are spread out; a small SD means they cluster around the mean. It uses every score, so it is more sensitive than the range, but it is harder to calculate and distorted by extreme scores.

Pearson provides the formula (sample estimate): s = square root of [ sum of (x - mean) squared / (n - 1) ].

Calculating SD step by step

Scores: 4, 6, 7, 8, 10.

1. Mean = 35 / 5 = 7.

2. Deviations (x - mean): -3, -1, 0, 1, 3.

3. Squares: 9, 1, 0, 1, 9; sum = 20.

4. Divide by n - 1 = 4: 5.0.

5. Square root: SD = 2.24.

Normal and skewed distributions

• Normal distribution: symmetrical and bell-shaped; mean, median and mode are equal, in the centre. About 68% of scores lie within 1 SD of the mean and about 95% within 2 SD.
• Positive skew (skewed to the right): most scores are low with a long tail of high scores; the mean is pulled above (to the right of) the median, and the mode is lowest.
• Negative skew (skewed to the left): most scores are high with a tail of low scores; the mean is below (to the left of) the median.

With skewed data the median is often the better average.

Worked example

Group A has mean 14, SD 1.2. Group B has mean 14, SD 4.5. Compare the groups.

1. The means are the same, so on average the groups scored the same.
2. Group B's SD is much larger, so its scores are more spread out.
3. Group A's scores are more consistent, clustered close to 14.

Answer: Same average, but Group B is far more variable; Group A's scores are more consistent.

Check yourself

1. Find the range of the male obedience scores: 12, 15, 9, 14, 15, 11, 13, 15, 10, 16.
   Show answer

   7. 16 - 9 = 7.

2. Calculate the standard deviation (n - 1 formula) of these scores: 2, 4, 4, 6, 9. Give your answer to 2 decimal places.
   Show answer

   2.65. Mean = 5. Squared deviations: 9, 1, 1, 1, 16, sum 28. 28 / 4 = 7. Square root of 7 = 2.65.

3. Calculate the standard deviation (n - 1) of the male scores 12, 15, 9, 14, 15, 11, 13, 15, 10, 16. Give your answer to 2 decimal places.
   Show answer

   2.40. Mean = 13; sum of squared deviations = 52; divide by 9 and square root: 2.4.

4. In a distribution, the mean is 18, the median is 15 and the mode is 13. What type of distribution is this?
   • Positively skewed
   • Negatively skewed
   • Normal
   • Bimodal
   Show answer

   Positively skewed. Mean above median above mode: tail to the right.

5. Which statement describes a normal distribution?
   • Most scores are low with a long tail of high scores
   • The mean is always lower than the mode
   • It is symmetrical, with mean, median and mode in the centre
   • It has two peaks
   Show answer

   It is symmetrical, with mean, median and mode in the centre. Bell-shaped and symmetrical.

6. Why is the standard deviation more informative than the range?
   • It uses every score rather than only the highest and lowest
   • It is quicker to calculate
   • It is not affected by any score
   • It can be used with nominal data
   Show answer

   It uses every score rather than only the highest and lowest. SD includes all scores, so it reflects the whole spread.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Psychology specification (issue January 2026)
• Pearson International A Level Psychology component guide: WPS01
• IAL Psychology revision site: Quantitative data

Last updated 8 November 2026. Spotted a mistake? Tell us.