8 min readConcept sheetOften a 5- or 10-mark answer, with a sum

Range and standard deviation

Dispersion shows how spread out the values are around the average. Range uses only the two extremes; standard deviation uses every value.

RangeLargest value − smallest value

SD symbolσ (population), s (sample)

SD formulaσ = √[Σ(x − x̄)² ÷ n]

SD is zero whenAll values are the same

Formulas

RangeR = L − S (L = largest, S = smallest value)

Coefficient of range(L − S) ÷ (L + S)

Standard deviation (college texts)σ = √[Σ(x − x̄)² ÷ n]

Sample standard deviations = √[Σ(x − x̄)² ÷ (n − 1)]

VarianceSquare of the SD (σ²)

Coefficient of variationCV = (σ ÷ x̄) × 100, in %

Merits and demerits

Measure	Merits	Demerits
Range	Simplest to understand and calculate; quick check in quality control, such as the weights of food packets	Uses only two values; one extreme value changes it; says nothing about the values in between; not used in further statistics
Standard deviation	Uses every value; rigidly defined; changes least from sample to sample; basis of further tests such as correlation and the t test	Harder to calculate; squaring gives extra weight to extreme values; not easy for non-experts to understand

Example: SD of protein intake

Problem

Daily protein intake (g) of 6 women: 42, 48, 50, 54, 46, 60. Find the range and the standard deviation.

Working

Range = 60 − 42 = 18 g. Coefficient of range = 18 ÷ (60 + 42) = 18 ÷ 102 = 0.18 (rounded to 2 decimals).
Mean: Σx = 300 and n = 6, so x̄ = 300 ÷ 6 = 50 g.
x = 42: d = 42 − 50 = −8, d² = 64
x = 48: d = 48 − 50 = −2, d² = 4
x = 50: d = 50 − 50 = 0, d² = 0
x = 54: d = 54 − 50 = 4, d² = 16
x = 46: d = 46 − 50 = −4, d² = 16
x = 60: d = 60 − 50 = 10, d² = 100
Check: Σd = −8 − 2 + 0 + 4 − 4 + 10 = 0. Then Σd² = 64 + 4 + 0 + 16 + 16 + 100 = 200.
σ = √(200 ÷ 6) = √33.33 = 5.77 g (rounded to 2 decimals).
If the question asks for the sample SD: s = √(200 ÷ 5) = √40 = 6.32 g.

Answer

Range = 18 g. SD (σ, dividing by n) = 5.77 g; sample SD (s, dividing by n − 1) = 6.32 g. Values rounded to 2 decimals.

Divide by n or by n − 1?

Population SD (σ): divide by nUsed when the data are the whole group you are describing. Most college texts, such as S.P. Gupta, use n in SD sums.

Sample SD (s): divide by n − 1Used when the data are a sample used to estimate the population SD. Biostatistics books and the t test use n − 1.

In the exam, write which one you used. With a large n the two answers are almost the same.

In nutrition research

Nutrition studies report results as mean ± SD, usually the sample SD (n − 1) that SPSS or Excel gives: 50 ± 6.32 g in the worked example. A small SD means the group is alike; a large SD means intakes vary widely, so one diet plan may not suit everyone. Range gives a quick check, such as the spread in weights of packed food portions.

Exam angle

2 marks: “Define standard deviation.” SD is the square root of the average of the squared deviations of the values from their mean. It shows how spread out the values are.

5 marks: “Calculate the range and standard deviation of fruit servings per week: 2, 4, 6, 8, 10.” Range = 10 − 2 = 8. Mean = 6; deviations −4, −2, 0, 2, 4; Σd² = 40. σ = √(40 ÷ 5) = √8 = 2.83 (2 decimals). With n − 1, s = √10 = 3.16.

10 marks: “Explain range and standard deviation with their merits, demerits and an example.” Define dispersion. Give the formula, merits and demerits of each. Solve a 6-value SD sum with a deviation table, compare the two measures and add a nutrition use.

Remember it

SD recipe: mean → deviations → square → average → square root.