Fundamentals of Business Mathematics and Statistics · Measures of Central Tendency and Dispersion

Standard Deviation, Variance and Coefficient of Variation

Updated 10 October 2026 · Fact-checked

Standard deviation measures how far values typically lie from their mean. Variance is the average of the squared deviations, and standard deviation is its square root. Coefficient of variation is SD ÷ mean × 100, used to compare consistency: the lower the CV, the more consistent the data. For merged groups, use the combined SD formula.

Understand Standard Deviation, Variance and Coefficient of Variation

An average tells you the centre of the data, but not how spread out the values are. Two shops can both earn an average of ₹50,000 a month. One earns close to ₹50,000 every month. The other swings between ₹10,000 and ₹90,000. You need a measure of spread to see the difference.

Variance is the average of the squared distances of values from the mean. We square the distances because plain deviations from the mean add up to zero. Standard deviation (SD) is the square root of the variance. It brings the answer back to the original unit, such as rupees or marks. This is why SD is easier to interpret than variance.

SD is an absolute measure, so it carries units. You cannot fairly compare SD of two series with different means or units. For that, use the coefficient of variation (CV), which is SD as a percentage of the mean. The series with the lower CV is more consistent (less variable). The series with the higher CV is more variable.

When two groups are merged, you cannot just average their SDs. The combined SD depends on each group's size, its SD, and how far each group's mean lies from the combined mean. Questions in this topic are usually calculations, so speed with a clean method matters.

Key formulas to remember

Variance (ungrouped data): σ² = Σ(x − x̄)² ÷ n = Σx² ÷ n − (x̄)²; The second form is faster when the values are small whole numbers. Foundation questions divide by n.
Standard deviation: σ = √Variance; SD is always zero or positive. It is zero only when all values are equal.
Grouped data (direct): σ² = Σfx² ÷ N − (Σfx ÷ N)², where N = Σf; Use x as the class mid-point for continuous classes.
Step deviation method: d = (x − A) ÷ h; σ = h × √[Σfd² ÷ N − (Σfd ÷ N)²]; A is the assumed mean and h is the common class width. Do not forget to multiply by h at the end.
Effect of change of origin and scale: If y = ax + b, then SD of y = |a| × SD of x, and Variance of y = a² × Variance of x; Adding or subtracting a constant does not change SD. Multiplying by a constant scales SD by |a|.
Coefficient of variation: CV = (σ ÷ x̄) × 100; Lower CV means more consistent. Higher CV means more variable.
Combined mean: x̄₁₂ = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂); Find this first, because the combined SD needs it.
Combined standard deviation: σ₁₂ = √[(n₁σ₁² + n₂σ₂² + n₁d₁² + n₂d₂²) ÷ (n₁ + n₂)], where d₁ = x̄₁ − x̄₁₂ and d₂ = x̄₂ − x̄₁₂; The d values are measured from the combined mean, not from each other.
SD of first n natural numbers: σ = √[(n² − 1) ÷ 12]; A ready-made result for 1, 2, 3, ..., n.

How to solve Standard Deviation, Variance and Coefficient of Variation questions

This method works for ungrouped data, grouped data, and combined-group questions.

1Read what is asked: variance, SD, CV, or combined SD. Note whether the data is ungrouped, grouped, or two groups.
2Find the mean first. For grouped data, find mid-points and use x̄ = Σfx ÷ N.
3For small ungrouped data, compute Σx² and use Σx² ÷ n − (x̄)². For grouped data with a common class width, use step deviation with d = (x − A) ÷ h.
4Work out the variance. Check it is not negative, since a negative variance means an arithmetic error.
5Take the square root to get SD. If the data was step-deviated, multiply by h.
6For CV, compute (SD ÷ mean) × 100. For comparison, the series with the lower CV is more consistent.
7For combined SD, find the combined mean, then d₁ and d₂, then put everything into the combined formula. Square root only at the end.
8Check that the unit and the answer fit the question: SD in original units, variance in squared units, CV in percent.

Quickest way: Option-driven shortcuts for MCQs

When to use it: Use under time pressure, when you have about one minute per question and four options to check against.

If values are shifted by a constant (for example x − 100), SD stays the same. If they are multiplied by a constant, multiply SD by that constant.
For small data, use Σx² ÷ n − (x̄)². Pick convenient numbers to square.
For grouped data with equal class widths, always use step deviation. The numbers stay small.
For CV comparison, do not calculate both exactly if one has both a higher SD and a lower mean. That one has the higher CV.
For combined SD, compute the total under the root and divide by (n₁ + n₂). Check which option is the square root of that value.
Remember that variance is SD squared. If an option equals the SD when the question asks for variance, eliminate it.

Common mistakes in Standard Deviation, Variance and Coefficient of Variation

Giving the variance as the final answer when SD is asked, or the reverse.
Students stop before the square root, or take the root of an SD that is already the answer.
Fix: Underline what is asked. Variance = σ². SD = √variance. Check the last line of working against the question.
Forgetting to multiply by h in the step deviation method.
The d values look like the real data, so students treat the result as the final SD.
Fix: Write σ = h × √[...] before you start. Remember that d is a scaled version of x, so the SD must be scaled back.
Taking the average of the two SDs as the combined SD.
It feels natural, just like averaging two means.
Fix: Always use the combined formula. The group means differing from the combined mean adds extra spread through the d² terms.
Measuring d₁ and d₂ as x̄₁ − x̄₂ in the combined SD.
Students mix up the two differences.
Fix: Compute the combined mean first. Then d₁ = x̄₁ − x̄₁₂ and d₂ = x̄₂ − x̄₁₂. Squaring removes the sign.
Calling the series with the higher CV more consistent.
Students link a bigger number with a better result.
Fix: CV measures variability. Lower CV means more consistent, so choose the smaller CV.
Believing that adding a constant to every value changes the SD.
Students confuse the effect on the mean with the effect on spread.
Fix: Adding or subtracting a constant shifts every value and the mean equally, so spread is unchanged. Only multiplication or division changes SD.

Worked examples

Example 1

The monthly sales (in ₹ thousands) of a shop in eight months are 2, 4, 4, 4, 5, 5, 7, 9. Find the variance, standard deviation and coefficient of variation.

Show the solution

n = 8. Sum of values = 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40. Mean = 40 ÷ 8 = 5.
Deviations from the mean: −3, −1, −1, −1, 0, 0, 2, 4.
Squares of deviations: 9, 1, 1, 1, 0, 0, 4, 16. Their sum = 32.
Variance = 32 ÷ 8 = 4.
SD = √4 = 2.
CV = (2 ÷ 5) × 100 = 40%.

Answer: Variance = 4, SD = 2 (₹ thousand), CV = 40%.

Example 2

Group A has 40 workers with mean wage ₹50 hundred per day and SD 5. Group B has 60 workers with mean wage ₹60 hundred per day and SD 5. Find the combined standard deviation of the 100 workers.

Show the solution

Combined mean = (40 × 50 + 60 × 60) ÷ 100 = (2,000 + 3,600) ÷ 100 = 56.
d₁ = 50 − 56 = −6, so d₁² = 36. d₂ = 60 − 56 = 4, so d₂² = 16.
n₁σ₁² = 40 × 25 = 1,000. n₂σ₂² = 60 × 25 = 1,500.
n₁d₁² = 40 × 36 = 1,440. n₂d₂² = 60 × 16 = 960.
Total = 1,000 + 1,500 + 1,440 + 960 = 4,900.
Combined variance = 4,900 ÷ 100 = 49.
Combined SD = √49 = 7.

Answer: Combined SD = 7 (in the same unit, ₹ hundred per day). Note that it is larger than 5, because the two group means differ.

Exam tips

Most questions are one-step or two-step calculations. Do not start long tables unless the data is grouped.
Watch for questions that give a transformation such as y = 3x + 5. The answer is |3| × SD of x, and the constant 5 has no effect.
CV comparison questions often give means and SDs directly. Compute CV = SD ÷ mean × 100 for each and pick the lower for consistency.
Combined SD questions are formula-driven. Write the combined mean first, then the d values, so that you do not mix them up.
There is no negative marking, so always mark an option. If stuck, eliminate options that give variance when SD is asked, or SD smaller than both group SDs when means differ.

Practice questions from Measures of Central Tendency and Dispersion

Standard Deviation, Variance and Coefficient of Variation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Standard Deviation, Variance and Coefficient of Variation: frequently asked questions

What is the difference between variance and standard deviation?

Variance is the average of squared deviations from the mean, so its unit is squared. Standard deviation is the square root of variance and has the same unit as the data. Both measure spread, but SD is easier to interpret.

How do I use coefficient of variation to compare consistency?

Calculate CV = (SD ÷ mean) × 100 for each series. The series with the lower CV is more consistent. CV works even when the series have different means or units, because it is a relative measure.

Why is the combined standard deviation not the average of the two SDs?

The combined spread also depends on how far each group's mean is from the overall mean. The d² terms in the formula account for this. Without them, you would understate the spread.

When should I use the step deviation method?

Use it for grouped data when the class widths are equal and the mid-points are large or awkward. Use d = (x − A) ÷ h to keep numbers small. At the end, multiply the SD by h.