Standard Deviation Example and Calculation

What Standard Deviation Actually Is

Standard deviation measures how spread out numbers are from their average. That's it. A low standard deviation means numbers cluster close together. A high standard deviation means they're all over the place.

You see it everywhere—in finance, science, quality control. Most people who claim to understand it actually don't. This article will fix that.

Population vs Sample Standard Deviation

There are two versions. Using the wrong one is a common mistake.

• Population standard deviation (σ) — used when you have every single data point. Divide by N.
• Sample standard deviation (s) — used when your data is just a sample from a larger group. Divide by N-1.

In most real-world situations, you're working with samples. So you'll probably use the sample formula.

The Formulas

Population formula:

σ = √[Σ(xi - μ)² / N]

Sample formula:

s = √[Σ(xi - x̄)² / (N-1)]

Where:

• xi = each individual value
• μ or x̄ = the mean (average)
• N = number of data points
• Σ = sum of

Step-by-Step Calculation Example

Let's say you tracked daily sales for five days: $100, $150, $200, $175, $125

Step 1: Find the Mean

Add them up: 100 + 150 + 200 + 175 + 125 = 750

Divide by 5: 750 / 5 = $150

Step 2: Find Each Deviation from the Mean

• 100 - 150 = -50
• 150 - 150 = 0
• 200 - 150 = +50
• 175 - 150 = +25
• 125 - 150 = -25

Step 3: Square Each Deviation

• (-50)² = 2,500
• (0)² = 0
• (50)² = 2,500
• (25)² = 625
• (-25)² = 625

Step 4: Sum the Squared Deviations

2,500 + 0 + 2,500 + 625 + 625 = 6,250

Step 5: Divide by N (or N-1)

Population: 6,250 / 5 = 1,250

Sample: 6,250 / 4 = 1,562.5

Step 6: Take the Square Root

Population: √1,250 = $35.36

Sample: √1,562.5 = $39.53

Your standard deviation is roughly $35-40 depending on which formula you use.

Quick Comparison Table

What the Number Actually Tells You

A standard deviation of $35 on $150 average sales means most days fall within roughly $115 to $185. That's the mean ± one standard deviation.

For normal distributions:

• 68% of data falls within ±1 standard deviation
• 95% falls within ±2 standard deviations
• 99.7% falls within ±3 standard deviations

Most real data isn't perfectly normal, but this rule holds well enough for practical purposes.

How to Calculate in Excel or Google Sheets

You don't need to do this by hand. Use the built-in functions:

• =STDEV.P(range) — population standard deviation
• =STDEV.S(range) — sample standard deviation

Select your data, pick the right function, done.

Common Mistakes

• Using population formula on samples — gives you a biased, usually underestimated value
• Confusing variance with standard deviation — variance is the squared value before you take the square root
• Forgetting to square the deviations — negative and positive values would cancel out otherwise

When Standard Deviation Is Useless

Standard deviation fails badly with skewed data or outliers. A single extreme value inflates it dramatically. If your data has a heavy tail, consider using median absolute deviation instead.

It's also meaningless for categorical data. You can't calculate standard deviation on "red, blue, red, green."

Getting Started Checklist

• □ Identify if you have a full population or a sample
• □ Calculate the mean first
• □ Subtract the mean from each value
• □ Square each result
• □ Sum all squared values
• □ Divide by N (population) or N-1 (sample)
• □ Take the square root
• □ Interpret in context of your data

That's the whole process. Memorize it or bookmark this page—your call.