Quantitative Aptitude · Measures of Central Tendency and Dispersion

Standard Deviation and Variance for CA Foundation

Updated 1 October 2026 · Fact-checked

Variance is the average of the squared deviations of observations from their mean. Standard deviation is its square root. To solve questions, find the mean, square the deviations (or use Σfx² ÷ N − mean²), divide by N, then take the root. Shifting the origin does not change SD, but scaling does.

Understand Standard Deviation and Variance

The mean tells you the centre of the data. It does not tell you how spread out the data is. Two classes can both average 50 marks, yet in one everyone scores 48 to 52 and in the other scores range from 10 to 90. Dispersion measures this spread.

Variance is the mean of the squared deviations from the mean. You square the deviations because plain deviations from the mean add up to zero. Standard deviation (SD), written σ, is the positive square root of variance. It is in the same units as the data, so it is easier to interpret. Variance is in squared units.

A small SD means the values cluster close to the mean. A large SD means they are scattered. SD is zero only when all observations are equal. It is never negative.

Two properties are tested again and again. Change of origin (adding or subtracting a constant from every value) does not change SD or variance. Change of scale (multiplying or dividing every value by a constant) multiplies SD by the absolute value of that constant, and variance by its square.

When two groups are merged, the combined SD is not the average of the two SDs. It also depends on how far each group's mean sits from the combined mean. That is why the combined SD formula has extra d² terms.

Key formulas to remember

Variance (ungrouped): σ² = Σ(x − x̄)² ÷ N; N is the number of observations. CA Foundation questions normally divide by N.
Shortcut form of variance: σ² = Σx² ÷ N − (x̄)²; Use this when the mean is not a whole number. It avoids many subtractions.
Standard deviation: σ = √variance; Always the positive root.
Grouped data (frequency distribution): σ² = Σf(x − x̄)² ÷ N = Σfx² ÷ N − (Σfx ÷ N)²; N = Σf. For classes, x is the class mark (mid-value).
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. Multiply by h at the end for SD, and by h² for variance.
Change of origin and scale: If y = a + bx, then σy = |b| × σx and variance of y = b² × variance of x; a (the origin shift) has no effect. A negative b does not make SD negative.
Combined mean: x̄₁₂ = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂); Find this first for the combined SD.
Combined standard deviation: σ₁₂ = √[(n₁σ₁² + n₂σ₂² + n₁d₁² + n₂d₂²) ÷ (n₁ + n₂)], where d₁ = x̄₁ − x̄₁₂ and d₂ = x̄₂ − x̄₁₂; Works for two groups. If the two means are equal, d₁ = d₂ = 0.
First n natural numbers: Variance of 1, 2, 3, …, n = (n² − 1) ÷ 12; The same holds for any n equally spaced values with common difference 1.
Coefficient of variation: CV = (σ ÷ x̄) × 100; Used to compare the spread of two series. A lower CV means more consistency.

How to solve Standard Deviation and Variance questions

This method works for ungrouped data, grouped data and property-based questions.

1Read what is asked: variance or SD. Many wrong answers come from giving one when the other is asked.
2Check whether the question is about a property (a change of origin or scale, or combining groups). If it is, jump to the property rule or the combined formula and skip the full table.
3For raw data, find the mean. Then compute Σ(x − x̄)² or Σx² and apply the formula.
4For grouped data, take class marks as x. Choose A near the middle class, find d = (x − A) ÷ h, and build columns for f, d, fd and fd².
5Compute Σfd ÷ N and Σfd² ÷ N. Then variance in d-units = Σfd² ÷ N − (Σfd ÷ N)².
6Convert back: multiply by h² for variance, or take the root and multiply by h for SD.
7For combined SD, find the combined mean first. Then find d₁ and d₂ and substitute into the combined formula.
8Check the answer: SD must be positive, and it can never exceed half the range (σ ≤ range ÷ 2), which holds for every dataset. If it looks too large, recheck the squares.

Quickest way: Shortcuts and option elimination for SD and variance

When to use it: Use these in the MCQ paper when the data is small or the question tests a property. Each shortcut saves a minute or more.

Subtract a convenient number from every value first. Since SD does not change with origin, the smaller numbers make squaring quick.
If all values share a common factor, divide it out, find SD, then multiply back by the factor.
For y = a + bx, write the answer directly: σy = |b|σx. Do not touch the data.
For consecutive integers, use variance = (n² − 1) ÷ 12 directly.
For a combined SD with equal group sizes, the combined variance equals the average of the two variances plus the square of half the difference between the means.
Eliminate options: the combined SD is at least the square root of the weighted average of the two group variances. Discard any option below that value, and discard any negative option.
If a grouped-data table is long and time is short, attempt it later after the quicker questions. A wrong answer costs 0.25 marks, so do not guess blindly.

Common mistakes in Standard Deviation and Variance

Giving the variance when SD is asked, or the reverse.
You stop after Σfd² ÷ N − (Σfd ÷ N)² and forget the final step.
Fix: Circle the word variance or SD in the question. Take the square root at the end only when SD is asked.
Multiplying by h instead of h² for variance in the step deviation method.
You remember 'multiply by h' from the SD formula and apply it to everything.
Fix: SD scales by h. Variance scales by h². Decide which you have before multiplying.
Subtracting or adding the origin shift to the SD, for example saying SD of (x + 5) is σ + 5.
You treat SD like the mean, which does shift with origin.
Fix: Remember that SD measures spread, and adding a constant moves all values together. Spread does not change.
Writing the combined SD as the average of the two SDs.
It feels natural, as with simple averages.
Fix: Use the full combined formula with d₁² and d₂². The gap between the group means adds spread.
Computing Σx² ÷ N and forgetting to subtract the square of the mean.
You stop halfway through the shortcut formula.
Fix: Write the formula as Σx² ÷ N − (x̄)² and fill in both parts before calculating.
Taking d₁ and d₂ in the combined formula as the difference between the two group means.
You mix up the distance from each mean to the combined mean with the distance between the two means.
Fix: Compute x̄₁₂ first. Then d₁ = x̄₁ − x̄₁₂ and d₂ = x̄₂ − x̄₁₂.

Worked examples

Example 1

The classes 0–10, 10–20, 20–30, 30–40, 40–50 have frequencies 1, 6, 6, 6, 1 respectively. The variance of the distribution is: (a) 10 (b) 50 (c) 100 (d) 200

Show the solution

Class marks are 5, 15, 25, 35, 45. Take A = 25 and h = 10, so d = −2, −1, 0, 1, 2.
N = 1 + 6 + 6 + 6 + 1 = 20.
fd = −2, −6, 0, 6, 2, so Σfd = 0.
fd² = 4, 6, 0, 6, 4, so Σfd² = 20.
Variance in d-units = 20 ÷ 20 − 0² = 1.
Variance = h² × 1 = 100 × 1 = 100. The SD is 10, which is option (a), a trap.

Answer: (c) 100

Example 2

The standard deviation of x is 5. If y = 2x − 3, the variance of y is: (a) 25 (b) 47 (c) 97 (d) 100

Show the solution

For y = a + bx, variance of y = b² × variance of x. Here a = −3 and b = 2.
Variance of x = 5² = 25.
Variance of y = 2² × 25 = 4 × 25 = 100.
The −3 is a change of origin and does not affect variance. Options 47 and 97 wrongly subtract 3.

Answer: (d) 100

Example 3

Two groups each have 50 observations. Group A has mean 44 and SD 8. Group B has mean 56 and SD 8. The combined standard deviation is: (a) 8 (b) 10 (c) 12 (d) 14

Show the solution

Combined mean = (50 × 44 + 50 × 56) ÷ 100 = (2,200 + 2,800) ÷ 100 = 50.
d₁ = 44 − 50 = −6, so d₁² = 36. d₂ = 56 − 50 = 6, so d₂² = 36.
n₁σ₁² = 50 × 64 = 3,200. n₂σ₂² = 50 × 64 = 3,200.
n₁d₁² = 50 × 36 = 1,800. n₂d₂² = 1,800.
Sum = 3,200 + 3,200 + 1,800 + 1,800 = 10,000.
Combined variance = 10,000 ÷ 100 = 100.
Combined SD = √100 = 10. Option (a) 8 comes from ignoring the gap between the means.

Answer: (b) 10

Exam tips

Questions on properties (origin and scale) are the fastest marks. Learn σy = |b|σx and the matching variance rule so you can answer in seconds.
Check whether the question asks for variance or SD, and whether options include both versions of the same calculation as traps.
Use the step deviation method for any grouped table with equal class widths. It keeps the numbers small and reduces arithmetic errors.
For combined SD, write the combined mean first on the paper. Most errors come from using the wrong d values.
Practise squares up to 30 and common square roots. Time is lost on arithmetic more often than on concepts.

Practice questions from Measures of Central Tendency and Dispersion

Standard Deviation and Variance: frequently asked questions

What is the difference between variance and standard deviation?

Variance is the average of squared deviations from the mean. Standard deviation is the square root of variance. SD is in the same units as the data, while variance is in squared units.

Does adding a constant to every observation change the SD?

No. Adding or subtracting a constant shifts every value equally, so the spread stays the same. Multiplying or dividing by a constant does change it. SD is multiplied by the absolute value of that constant.

How do I calculate variance for grouped data?

Take class marks as x and use σ² = Σfx² ÷ N − (Σfx ÷ N)². For equal class widths, use the step deviation method with d = (x − A) ÷ h and multiply the variance in d-units by h².

Can standard deviation be negative?

No. It is the positive square root of variance, and variance is a mean of squares. SD is zero only when every observation is the same.

Is the combined SD the average of the two SDs?

No. You must also account for how far each group mean is from the combined mean. The combined formula includes n₁d₁² and n₂d₂² for that reason. The result is at least as large as the simple weighted spread within the groups.