The formula
Population: σ = √( Σ(x − μ)² / N )
Sample: s = √( Σ(x − x̄)² / (n − 1) )
The steps are the same either way: find the mean, subtract it from each value, square each difference, average those squares, then take the square root.
Variance is the same thing without the final square root. Standard deviation is usually more useful because it is in the same units as your data.
Why n − 1 for a sample
A sample tends to underestimate the spread of the population it came from. Dividing by n − 1 instead of n corrects for this — it is called Bessel's correction. Use the sample formula unless your data genuinely covers every member of the group you care about.
Worked example
Data: 4, 8, 6, 10, 12
| Step | Working |
|---|---|
| Mean | 40 ÷ 5 = 8 |
| Differences | −4, 0, −2, 2, 4 |
| Squared | 16, 0, 4, 4, 16 → sum 40 |
| Sample variance | 40 ÷ 4 = 10 |
| Sample SD | √10 = 3.16 |
| Population SD | √(40 ÷ 5) = 2.83 |
Reading standard deviation
For roughly bell-shaped data, the empirical rule is a useful guide:
| Range | Approximate share of data |
|---|---|
| Mean ± 1 SD | 68% |
| Mean ± 2 SD | 95% |
| Mean ± 3 SD | 99.7% |
If a class averages 65 marks with an SD of 10, about two thirds of students scored between 55 and 75, and almost all between 45 and 85.
In investing
Standard deviation of returns is the standard measure of volatility. A fund returning 12% with an SD of 8% behaves very differently from one returning 12% with an SD of 22% — same average, wildly different experience. Higher SD means larger swings in both directions, which is why it is worth checking alongside CAGR rather than looking at returns alone.
Common mistakes
- Using the population formula on sample data, which understates the spread.
- Reporting variance when standard deviation is what the reader needs — variance is in squared units.
- Comparing standard deviations across datasets with very different means. Use the coefficient of variation (SD ÷ mean) instead.
- Applying the 68–95–99.7 rule to data that is clearly not bell-shaped.
- Quoting an average with no measure of spread at all.
Frequently Asked Questions
What does standard deviation tell me?
How spread out values are around the mean. A small standard deviation means values cluster tightly; a large one means they are widely dispersed.
What is the difference between sample and population standard deviation?
The sample version divides by n − 1 instead of n, correcting for the tendency of a sample to understate the spread of the wider population.
Which one should I use?
The sample formula, unless your data covers every member of the group you are describing. Most real datasets are samples.
What is variance?
Standard deviation squared. It is used in further statistical work but is harder to interpret directly because it is in squared units.
What is the 68-95-99.7 rule?
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three.
How is standard deviation used in investing?
As the standard measure of volatility. Two funds with the same average return but different standard deviations offer very different experiences to the investor.
Can standard deviation be negative?
No. It is a square root of squared values, so it is always zero or positive. Zero means every value is identical.
Nivesh mein SD ka matlab
Fund ke returns ka standard deviation uski volatility batata hai — utaar-chadhaav kitna hai.
| Fund | औsat return | SD | Aam saal ka range |
|---|---|---|---|
| Debt fund | 7% | 3% | 4% se 10% |
| Large cap | 12% | 15% | −3% se 27% |
| Small cap | 15% | 25% | −10% se 40% |
Do funds ka average return ek jaisa ho lekin SD alag, to kam SD wale mein safar zyada aaram se katta hai. Lambi avadhi ke liye zyada SD sehna pad sakta hai, lekin agar 5 saal mein paisa chahiye to kam SD behtar hai.
Kitna data chahiye
- 5 se kam values — SD nikal to jayega lekin uska matlab kam hai.
- 30 ya zyada — yahan se SD bharosemand hone lagta hai.
- Nivesh ke liye — kam se kam 3 saal ka maasik data, warna ek achha ya bura daur poori tasveer bigaad deta hai.
Aur ek baat — SD tabhi seedha matlab rakhta hai jab data lagbhag ghanti ke aakaar (bell curve) mein ho. Aamdani jaise data mein, jahan kuch bahut badi values hoti hain, SD gumrah kar sakta hai.
More questions
Fund ka standard deviation kitna hona chahiye?
Ye aapki zaroorat par nirbhar hai. Debt funds mein 2–4%, large cap mein 13–16%, small cap mein 22–28% aam hai. Kam samay ka lakshya ho to kam SD behtar.
SD nikalne ke liye kitne numbers chahiye?
Kam se kam 30 values se bharosemand nateeja milta hai. Nivesh ke liye kam se kam 3 saal ka maasik data dekhiye.
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