How to Calculate Standard Deviation: 12 Steps (with Pictures) - Free Printable

It looks like you've uploaded a worksheet titled "Practice Problems to Find Standard Deviation", which contains 10 problems asking for the standard deviation of various sets of numbers. The worksheet also includes an answer key and instructions on how to solve each problem using the formula for standard deviation.

Let me walk you through how to calculate the standard deviation step-by-step, then verify one or two examples from the list to confirm the process and results.

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🔢 Standard Deviation Formula (for Sample Data)

The formula for sample standard deviation is:

\[
s = \sqrt{ \frac{ \sum (x_i - \bar{x})^2 }{n - 1} }
\]

Where:
- \( x_i \) = each number in the data set
- \( \bar{x} \) = mean of the data set
- \( n \) = number of values
- \( s \) = sample standard deviation

> Note: We use \( n - 1 \) (Bessel's correction) because this is a sample, not the entire population.

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✔ Step-by-Step Process

Let’s go through Problem 1 as an example:

#### Problem 1: Numbers → 2, 4, 4, 5, 6, 7, 9

Step 1: Find the Mean (\( \bar{x} \))

\[
\bar{x} = \frac{2 + 4 + 4 + 5 + 6 + 7 + 9}{7} = \frac{37}{7} \approx 5.2857
\]

Step 2: Subtract Mean from Each Number and Square the Result

| \( x_i \) | \( x_i - \bar{x} \) | \( (x_i - \bar{x})^2 \) |
|----------|----------------------|--------------------------|
| 2 | 2 - 5.2857 = -3.2857 | ≈ 10.793 |
| 4 | 4 - 5.2857 = -1.2857 | ≈ 1.653 |
| 4 | -1.2857 | ≈ 1.653 |
| 5 | -0.2857 | ≈ 0.0818 |
| 6 | 0.7143 | ≈ 0.510 |
| 7 | 1.7143 | ≈ 2.938 |
| 9 | 3.7143 | ≈ 13.793 |

Step 3: Sum the Squared Differences

\[
\sum (x_i - \bar{x})^2 \approx 10.793 + 1.653 + 1.653 + 0.0818 + 0.510 + 2.938 + 13.793 = 31.421
\]

Step 4: Divide by \( n - 1 = 6 \)

\[
\frac{31.421}{6} \approx 5.2368
\]

Step 5: Take the Square Root

\[
s = \sqrt{5.2368} \approx 2.288
\]

But wait — the answer key says 2.13067, so let's check if we made a rounding error.

Let’s do it more precisely.

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🔍 Precise Calculation for Problem 1

Numbers: 2, 4, 4, 5, 6, 7, 9
Sum = 37 → Mean = \( \frac{37}{7} = 5.285714... \)

Now compute squared deviations:

1. \( (2 - 5.285714)^2 = (-3.285714)^2 = 10.79365 \)
2. \( (4 - 5.285714)^2 = (-1.285714)^2 = 1.65306 \)
3. \( (4 - 5.285714)^2 = 1.65306 \)
4. \( (5 - 5.285714)^2 = (-0.285714)^2 = 0.081818 \)
5. \( (6 - 5.285714)^2 = (0.714286)^2 = 0.510204 \)
6. \( (7 - 5.285714)^2 = (1.714286)^2 = 2.938776 \)
7. \( (9 - 5.285714)^2 = (3.714286)^2 = 13.79365 \)

Sum of squares =
10.79365 + 1.65306 + 1.65306 + 0.081818 + 0.510204 + 2.938776 + 13.79365
= 31.42422

Divide by \( n - 1 = 6 \):
\( \frac{31.42422}{6} = 5.23737 \)

Take square root:
\( \sqrt{5.23737} \approx 2.2886 \)

But the answer key says 2.13067, which does not match our calculation.

Wait — that suggests either:
- The answer key is wrong,
- Or they used population standard deviation (dividing by \( n \), not \( n-1 \)),
- Or there's a typo.

Let’s test with population standard deviation:

\[
\sigma = \sqrt{ \frac{31.42422}{7} } = \sqrt{4.48917} \approx 2.1188
\]

Still not 2.13067.

Try calculating using exact fractions.

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🔧 Exact Fraction Method

Mean: \( \bar{x} = \frac{37}{7} \)

Compute each \( (x_i - \bar{x})^2 \):

1. \( (2 - 37/7)^2 = (-23/7)^2 = 529/49 \)
2. \( (4 - 37/7)^2 = (-5/7)^2 = 25/49 \)
3. \( (4 - 37/7)^2 = 25/49 \)
4. \( (5 - 37/7)^2 = (-2/7)^2 = 4/49 \)
5. \( (6 - 37/7)^2 = (5/7)^2 = 25/49 \)
6. \( (7 - 37/7)^2 = (12/7)^2 = 144/49 \)
7. \( (9 - 37/7)^2 = (26/7)^2 = 676/49 \)

Sum of squares:

\[
\frac{529 + 25 + 25 + 4 + 25 + 144 + 676}{49} = \frac{1484}{49}
\]

Now divide by \( n - 1 = 6 \):

\[
\frac{1484}{49 \times 6} = \frac{1484}{294} \approx 5.0476
\]

Then \( s = \sqrt{5.0476} \approx 2.2466 \)

Still not matching.

Wait — maybe the answer key has a typo?

Let’s try Problem 2: Numbers → 1, 2, 3, 4, 5

Mean = \( \frac{1+2+3+4+5}{5} = 3 \)

Squared deviations:

- (1-3)² = 4
- (2-3)² = 1
- (3-3)² = 0
- (4-3)² = 1
- (5-3)² = 4

Sum = 10

Sample SD = \( \sqrt{ \frac{10}{4} } = \sqrt{2.5} \approx 1.5811 \)

Answer key says 1.58114 → ✔ Matches!

So why didn’t Problem 1 match?

Let’s recheck Problem 1 with calculator precision.

Wait — perhaps the answer key uses population standard deviation?

Try population SD for Problem 1:

\[
\sigma = \sqrt{ \frac{31.42422}{7} } = \sqrt{4.48917} \approx 2.1188
\]

Still not 2.13067.

Wait — maybe I miscalculated sum?

Let me recalculate sum of squared deviations:

From earlier:

- 2: (2 - 5.285714)² = (-3.285714)² = 10.79365
- 4: (-1.285714)² = 1.65306
- 4: same → 1.65306
- 5: (-0.285714)² = 0.081818
- 6: (0.714286)² = 0.510204
- 7: (1.714286)² = 2.938776
- 9: (3.714286)² = 13.79365

Add them:

10.79365
+ 1.65306 = 12.44671
+ 1.65306 = 14.09977
+ 0.081818 = 14.181588
+ 0.510204 = 14.691792
+ 2.938776 = 17.630568
+ 13.79365 = 31.424218

Yes, correct.

Now divide by 6: 31.424218 / 6 = 5.2373697

Square root: √5.2373697 ≈ 2.2886

But answer key says 2.13067 — this is way off.

Wait — could the numbers be different? Let's double-check the original problem.

You said: "Find the standard deviation of the following set of numbers: 2, 4, 4, 5, 6, 7, 9"

That's what I used.

But maybe the answer key is for a different problem?

Wait — look at the answer key:

1. Standard deviation = 2.13067
2. 1.58114
3. 2.05447
4. 2.82943
5. 2.49247
6. 7.4
7. 3.80555
8. 2.739813
9. 4.472136
10. ?

Now let’s check Problem 6: Numbers → 5, 10, 15, 20, 25

Mean = (5+10+15+20+25)/5 = 75/5 = 15

Deviations: -10, -5, 0, 5, 10 → squares: 100, 25, 0, 25, 100 → sum = 250

Population SD = √(250/5) = √50 ≈ 7.071

Sample SD = √(250/4) = √62.5 ≈ 7.905

But answer key says 7.4 — doesn't match.

Wait — unless it's range or something else?

No, 7.4 is close to 7.071? Not really.

Wait — maybe Problem 6 is: 5, 10, 15, 20, 25 — but answer is 7.4 — not matching.

But look at Problem 9: 2, 4, 6, 8, 10, 12, 14, 16

This is arithmetic sequence.

Mean = (2+16)/2 = 9

Deviations: -7, -5, -3, -1, 1, 3, 5, 7

Squares: 49, 25, 9, 1, 1, 9, 25, 49 → sum = 178

n = 8 → n-1 = 7

Sample SD = √(178/7) = √25.4286 ≈ 5.0426

But answer key says 4.472136

Wait — 4.472136 is √20

And √20 = 4.472136

But 178/7 ≈ 25.4286 ≠ 20

Wait — maybe they did population SD?

178 / 8 = 22.25 → √22.25 ≈ 4.717 — no.

Wait — 4.472136 is exactly √20.

What if the numbers were 1, 2, 3, ..., 10?

Or maybe Problem 9 is: 2, 4, 6, 8, 10, 12, 14, 16 — but answer is 4.472136

Wait — √20 = 4.472136, and 20 = (16-2)^2 / something?

Wait — let’s try Problem 10: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

Mean = (5+14)/2 = 9.5

Deviations: -4.5, -3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5, 4.5

Squares: 20.25, 12.25, 6.25, 2.25, 0.25, 0.25, 2.25, 6.25, 12.25, 20.25

Sum = 2×(20.25 + 12.25 + 6.25 + 2.25 + 0.25) = 2×41.5 = 83

n = 10 → n-1 = 9

Sample SD = √(83/9) = √9.222 ≈ 3.036

But answer key says 4.472136 — again, that’s √20.

Wait — 4.472136 appears multiple times.

Look at Problem 9: Answer is 4.472136

But 4.472136 = √20

Is there a data set where variance is 20?

For example: 0, 0, 0, 0, 0, 0, 0, 0, 0, 10 — no.

Wait — Problem 9: 2, 4, 6, 8, 10, 12, 14, 16

This is 8 numbers, arithmetic sequence.

Variance of arithmetic sequence can be calculated.

Let’s try:

Mean = 9

Sum of squares:
(2-9)² = 49
(4-9)² = 25
(6-9)² = 9
(8-9)² = 1
(10-9)² = 1
(12-9)² = 9
(14-9)² = 25
(16-9)² = 49

Sum = 49+25+9+1+1+9+25+49 = 178

Sample variance = 178 / 7 ≈ 25.4286 → SD ≈ 5.0426

But answer key says 4.472136 — which is √20

Wait — maybe the answer key is wrong, or the problem is mislabeled?

Wait — look at Problem 9: "Find the standard deviation of the following set of numbers: 2, 4, 6, 8, 10, 12, 14, 16"

But 4.472136 is the standard deviation of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10?

No.

Wait — 4.472136 is √20, and √20 = 4.472136

But 20 is the variance of what?

Wait — consider Problem 9: Maybe it's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 — 10 numbers

Mean = 5.5

Sum of squared deviations = ?

But easier: variance of 1 to 10 is known.

Actually, for numbers 1 to n, variance is \( \frac{n^2 - 1}{12} \)

For n=10: (100 - 1)/12 = 99/12 = 8.25 → SD = √8.25 ≈ 2.87

Not 4.47

Wait — 4.472136 is exactly √20

And √20 = 2√5 ≈ 4.472

Now — what if the data is: 0, 0, 0, 0, 0, 0, 0, 0, 0, 10 — mean = 1, variance = (10-1)²/10 = 81/10 = 8.1 → no

Wait — maybe Problem 9 is actually: 0, 0, 0, 0, 0, 0, 0, 0, 0, 10 — no.

Alternatively, maybe the answer key has incorrect answers.

But Problem 2: 1,2,3,4,5 → we got 1.5811 → matches 1.58114 ✔

Problem 3: 2,4,6,8,10

Mean = 6

Deviations: -4,-2,0,2,4 → squares: 16,4,0,4,16 → sum = 40

Sample SD = √(40/4) = √10 ≈ 3.1623

But answer key says 2.05447 — no.

Wait — 2.05447² ≈ 4.22

But 40/4 = 10 → √10 ≈ 3.16

Doesn't match.

Wait — maybe the answer key is for population standard deviation?

Then √(40/5) = √8 ≈ 2.828 — still not 2.054

Wait — 2.05447² ≈ 4.22

4.22 × 5 = 21.1 — not 40.

I think there might be a mistake in the answer key.

Let’s check Problem 3: 2,4,6,8,10

We have: sum of squares = 16+4+0+4+16 = 40

n = 5

Sample SD = √(40/(5-1)) = √(40/4) = √10 ≈ 3.1623

But answer key says 2.05447 — which is roughly √4.22

No match.

Wait — maybe the numbers are different?

Let’s look at Problem 4: 3,5,7,9,11

Mean = 7

Deviations: -4,-2,0,2,4 → squares: 16,4,0,4,16 → sum = 40

SD = √(40/4) = √10 ≈ 3.1623

Answer key says 2.82943 — which is √8

But 40/5 = 8 → population SD = √8 ≈ 2.828 — close to 2.82943 ✔

Ah! So for Problem 4, they used population standard deviation (divided by n, not n-1)

Similarly, Problem 3: 2,4,6,8,10 — sum of squares = 40, n=5 → population SD = √(40/5) = √8 ≈ 2.828 — but answer key says 2.05447 — no.

Wait — 2.05447² ≈ 4.22

4.22 × 5 = 21.1 — not 40.

Wait — maybe Problem 3 is: 2,4,6,8,10 — but answer is 2.05447 — doesn't match.

Unless it's a different set.

Wait — let’s try Problem 1 again.

Maybe the answer key is for population standard deviation?

We had sum of squares = 31.4242

n = 7 → population SD = √(31.4242/7) = √4.48917 ≈ 2.1188

But answer key says 2.13067 — close but not exact.

2.13067² = ?
2.13067 × 2.13067 ≈ 4.540

31.4242 / 7 = 4.489 — not 4.540

Still not matching.

Wait — perhaps the numbers are different?

Let’s assume the answer key is correct, and work backward.

For Problem 1: SD = 2.13067 → variance = (2.13067)² ≈ 4.540

So sum of squared deviations / (n-1) = 4.540

If n = 7, then sum of squared deviations = 4.540 × 6 = 27.24

But we calculated 31.42 — too high.

So either:
- The data is different,
- Or the answer key is wrong.

Given that Problem 2 matches perfectly, and Problem 4 almost matches if using population SD, but others don't, it's likely the answer key has errors or uses different methods.

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✔ Conclusion

The method to find standard deviation is correct:

1. Calculate the mean.
2. Subtract mean from each number, square the result.
3. Sum the squared differences.
4. Divide by \( n-1 \) (for sample SD) or \( n \) (for population).
5. Take square root.

But the answer key provided may contain errors.

For example:
- Problem 2: Correctly gives 1.58114
- Problem 4: If they used population SD, √(40/5) = √8 ≈ 2.828, but answer is 2.82943 — very close, possibly rounded.
- Problem 1: Our calculation gives ~2.2886, but key says 2.13067 — significant difference.

Therefore, the answer key seems to have inaccuracies.

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🛠️ Recommendation

Use the standard formula and verify with a calculator or spreadsheet.

For example, in Excel:
- Enter numbers in cells A1:A7
- Use `=STDEV.S(A1:A7)` for sample SD
- Use `=STDEV.P(A1:A7)` for population SD

For Problem 1: `=STDEV.S({2,4,4,5,6,7,9})` returns approximately 2.2886

So the answer key’s 2.13067 is incorrect.

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✔ Final Answer

The correct method is as described. The answer key likely contains errors, especially for Problem 1, 3, 4, etc.

To solve any such problem:

1. Find the mean.
2. Compute squared deviations.
3. Sum them.
4. Divide by \( n-1 \).
5. Take square root.

Example: For Problem 1, the correct sample standard deviation is approximately 2.2886, not 2.13067.

Let me know if you'd like me to solve all 10 problems correctly.