Calculating Standard Deviation- Practice Problems and Solutions
What Standard Deviation Actually Is
Standard deviation measures how spread out numbers are from their average. That's it. Nothing fancy.
If your data points cluster tightly around the mean, your standard deviation is small. If they're scattered all over the place, it's large. This tells you whether your data is consistent or all over the map.
Investors use it to gauge risk. Scientists use it to validate experiments. Teachers use it to grade on a curve. If you're in statistics, finance, or research, you'll need to calculate this by hand until it's automatic.
The Formula (And Why It Looks Scarier Than It Is)
For a population:
σ = √[Σ(x - μ)² / N]
For a sample:
s = √[Σ(x - x̄)² / (n - 1)]
Break it down and it's just five steps:
- Find the mean (average) of your data
- Subtract the mean from each value (these are called deviations)
- Square each deviation
- Find the average of those squared deviations
- Take the square root
The sample formula uses n - 1 instead of n. This corrects for the fact that a sample usually underestimates the true population spread. Use n - 1 unless you're working with every single member of a population.
Step-by-Step Calculation
Let's use this data set: 4, 8, 6, 5, 3
Step 1: Calculate the Mean
Add them up: 4 + 8 + 6 + 5 + 3 = 26
Divide by how many: 26 ÷ 5 = 5.2
Step 2: Find Each Deviation
Subtract the mean from each value:
- 4 - 5.2 = -1.2
- 8 - 5.2 = 2.8
- 6 - 5.2 = 0.8
- 5 - 5.2 = -0.2
- 3 - 5.2 = -2.2
Step 3: Square Each Deviation
- (-1.2)² = 1.44
- (2.8)² = 7.84
- (0.8)² = 0.64
- (-0.2)² = 0.04
- (-2.2)² = 4.84
Step 4: Average the Squared Deviations
Add them: 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8
Divide by n (5 values): 14.8 ÷ 5 = 2.96
Step 5: Take the Square Root
√2.96 = 1.72
That's your standard deviation. The data spreads about 1.72 units from the mean on average.
Practice Problems
Work through these yourself before checking the solutions. That's how you actually learn this.
Problem 1: Test Scores 📊
A class of 6 students scored: 70, 75, 80, 85, 90, 95
Calculate the population standard deviation.
Show Solution
Step 1: Mean
70 + 75 + 80 + 85 + 90 + 95 = 495
495 ÷ 6 = 82.5
Step 2: Deviations
- 70 - 82.5 = -12.5
- 75 - 82.5 = -7.5
- 80 - 82.5 = -2.5
- 85 - 82.5 = 2.5
- 90 - 82.5 = 7.5
- 95 - 82.5 = 12.5
Step 3: Squared Deviations
- 156.25
- 56.25
- 6.25
- 6.25
- 56.25
- 156.25
Step 4: Variance
156.25 + 56.25 + 6.25 + 6.25 + 56.25 + 156.25 = 437.5
437.5 ÷ 6 = 72.92
Step 5: Standard Deviation
√72.92 = 8.54
Problem 2: Daily Sales 💰
A store tracked daily sales for 5 days: $200, $350, $400, $600, $450
Calculate the sample standard deviation.
Show Solution
Step 1: Mean
200 + 350 + 400 + 600 + 450 = 2000
2000 ÷ 5 = $400
Step 2: Deviations
- 200 - 400 = -200
- 350 - 400 = -50
- 400 - 400 = 0
- 600 - 400 = 200
- 450 - 400 = 50
Step 3: Squared Deviations
- 40,000
- 2,500
- 0
- 40,000
- 2,500
Step 4: Sample Variance
40,000 + 2,500 + 0 + 40,000 + 2,500 = 85,000
85,000 ÷ (5 - 1) = 85,000 ÷ 4 = 21,250
Step 5: Standard Deviation
√21,250 = $145.77
Problem 3: Small Data Set ⚠️
Heights of 4 plants (in cm): 12, 14, 15, 17
Calculate the sample standard deviation.
Show Solution
Step 1: Mean
12 + 14 + 15 + 17 = 58
58 ÷ 4 = 14.5
Step 2: Deviations
- 12 - 14.5 = -2.5
- 14 - 14.5 = -0.5
- 15 - 14.5 = 0.5
- 17 - 14.5 = 2.5
Step 3: Squared Deviations
- 6.25
- 0.25
- 0.25
- 6.25
Step 4: Sample Variance
6.25 + 0.25 + 0.25 + 6.25 = 13
13 ÷ (4 - 1) = 13 ÷ 3 = 4.33
Step 5: Standard Deviation
√4.33 = 2.08 cm
Population vs Sample: When to Use Which
This trips up a lot of people. Here's the difference:
| Type | Formula | When to Use |
|---|---|---|
| Population SD (σ) | Divide by N | You have data from every member of the group |
| Sample SD (s) | Divide by n-1 | Your data is a sample from a larger group |
Example: You measure every single employee in a company. That's population data. You survey 500 customers out of 50,000. That's sample data.
The n-1 in the sample formula (called Bessel's correction) compensates for the fact that samples tend to cluster around the true mean. Without it, you'd consistently underestimate the real spread.
Common Mistakes to Avoid
- Using population formula on sample data. Your answer will be wrong. Always check what you're actually measuring.
- Forgetting to square the deviations. The negatives and positives cancel out if you don't. That's why you square first.
- Rounding too early. Keep full precision until the final answer. Rounding at each step compounds errors.
- Confusing variance with standard deviation. Variance is the average of squared deviations. Standard deviation is the square root of that. They are not the same.
- Misidentifying outliers. One extreme value can inflate your standard deviation dramatically. Check your data for errors.
Quick Reference: The Five Steps
| Step | Action | Example (data: 2, 4, 4, 6) |
|---|---|---|
| 1 | Calculate the mean | (2+4+4+6) ÷ 4 = 4 |
| 2 | Subtract mean from each value | -2, 0, 0, 2 |
| 3 | Square each deviation | 4, 0, 0, 4 |
| 4 | Average the squares (divide by n or n-1) | 8 ÷ 4 = 2 (or 8 ÷ 3 = 2.67) |
| 5 | Square root of the result | √2 = 1.41 (or √2.67 = 1.63) |
When Standard Deviation Is Useless
Standard deviation assumes your data follows a normal distribution (bell curve). If your data is heavily skewed, has multiple peaks, or contains extreme outliers, standard deviation won't give you an accurate picture.
In those cases, use:
- Interquartile range (IQR) — ignores outliers entirely
- Median absolute deviation — more robust to extreme values
- Range — simple but sensitive to outliers
Know your data's shape before you choose your metric.
Final Take
Standard deviation is a tool. It measures spread. That's all. The math is straightforward once you stop being intimidated by the formula.
Memorize the five steps. Practice with real numbers. Know when to use n vs n-1. That's the job done.