Standard Deviation Examples- Statistical Calculations
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.
You see this metric everywhere—in finance, science, quality control, sports analytics. It's one of the most practical statistical tools you can use.
Population vs. Sample Standard Deviation
Before calculating anything, you need to know which type you're working with:
- Population standard deviation — you have data for every single member of the group. Divide by N.
- Sample standard deviation — you're working with a subset. Divide by N-1 (Bessel's correction).
Most real-world situations use sample standard deviation. You almost never have data for an entire population.
The Formula (Don't Panic)
Population standard deviation:
σ = √[Σ(xi - μ)² / N]
Sample standard deviation:
s = √[Σ(xi - x̄)² / (N-1)]
Where:
σ or s = standard deviation
xi = each individual value
μ or x̄ = the mean
N = number of values
Σ = sum of
Step-by-Step Calculation Example
Let's say you tracked daily sales at a small shop for 5 days:
$120, $130, $125, $140, $135
Step 1: Calculate the Mean
(120 + 130 + 125 + 140 + 135) / 5 = 650 / 5 = $130
Step 2: Find Each Deviation from the Mean
- 120 - 130 = -10
- 130 - 130 = 0
- 125 - 130 = -5
- 140 - 130 = 10
- 135 - 130 = 5
Step 3: Square Each Deviation
- (-10)² = 100
- 0² = 0
- (-5)² = 25
- 10² = 100
- 5² = 25
Step 4: Sum the Squared Deviations
100 + 0 + 25 + 100 + 25 = 250
Step 5: Divide by N (or N-1)
If this is your entire dataset (population): 250 / 5 = 50
If this is a sample: 250 / 4 = 62.5
Step 6: Take the Square Root
Population: √50 = 7.07
Sample: √62.5 = 7.91
Your standard deviation is roughly $7-$8. Most days you'll make within $8 of your $130 average.
Real-World Example: Test Scores
Two classrooms took the same exam. Here's how to compare them:
| Class | Scores | Mean | Std Dev | Interpretation |
|---|---|---|---|---|
| A | 70, 72, 73, 71, 74 | 72 | 1.58 | Consistent scores |
| B | 60, 80, 55, 90, 75 | 72 | 13.96 | Widely varied performance |
Both classes averaged 72. But Class A clustered tightly around that score. Class B had students way above and below average. Same mean, completely different situations.
Quick Reference: When to Use Which Calculation
| Scenario | Use | Divide By |
|---|---|---|
| All employees in a company | Population | N |
| Survey respondents (subset) | Sample | N-1 |
| Entire batch of products tested | Population | N |
| Quality control sample check | Sample | N-1 |
| Temperature readings for one city | Population | N |
| Stock prices over 30 random days | Sample | N-1 |
Common Mistakes That Ruin Your Calculation
- Using population formula on sample data — This underestimates variability. Use N-1 for samples.
- Forgetting to square negative deviations — They cancel out otherwise. Always square first.
- Rounding too early — Keep full precision until the final step.
- Confusing variance with standard deviation — Variance is the squared value. Standard deviation is the square root of variance.
How to Calculate Standard Deviation in Excel or Google Sheets
Skip the manual math for large datasets:
- =STDEV.P(range) — Population standard deviation
- =STDEV.S(range) — Sample standard deviation
That's it. Let the spreadsheet do the heavy lifting.
What Standard Deviation Tells You
In a normal distribution:
- ~68% of values fall within 1 standard deviation of the mean
- ~95% fall within 2 standard deviations
- ~99.7% fall within 3 standard deviations
If your data isn't roughly bell-shaped, these percentages don't apply. Check your distribution first.
The Bottom Line
Standard deviation quantifies spread. It tells you whether your data points are clustered together or scattered widely. Calculate it correctly based on whether you're working with a full population or a sample. Use the right formula. Don't overthink it.