Understanding Standard Deviation with Examples

What Is Standard Deviation?

Standard deviation is a measure of how spread out the values in a dataset are around the mean (average). A small standard deviation means values cluster tightly around the mean; a large standard deviation means values are widely scattered.

It answers the fundamental question: "How typical is the average?" If you're told a class averages 75 on an exam, a standard deviation of 3 (most scores between 72–78) tells a completely different story than a standard deviation of 20 (scores ranging from 35 to 100+).

Standard deviation is represented by the Greek letter σ (sigma) for a population and s for a sample.

The Formulas

Population Standard Deviation (σ)

Use this when you have data for an entire population — every possible member of the group you're studying.

$$ \sigma = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}} $$

Sample Standard Deviation (s)

Use this when your data is a sample drawn from a larger population — which is true for almost all real-world data collection.

$$ s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}} $$

The critical difference: Sample formula divides by (n − 1) instead of n. This is called Bessel's correction — it compensates for the fact that a sample systematically underestimates population variance when dividing by n. Using (n − 1) gives an unbiased estimator of the true population standard deviation.

Variance is simply the square of standard deviation — the intermediate value before taking the square root:

$$ \text{Variance} (\sigma^2 \text{ or } s^2) = \frac{\sum(x_i - \mu)^2}{N} \quad \text{or} \quad \frac{\sum(x_i - \bar{x})^2}{n-1} $$

Step-by-Step Guide: Worked Example

Dataset: [4, 7, 13, 2, 1] — Treat this as a sample (n = 5).

Step 1: Calculate the Mean

$$ \bar{x} = \frac{4 + 7 + 13 + 2 + 1}{5} = \frac{27}{5} = 5.4 $$

Step 2: Find Each Deviation from the Mean

$$ (4 - 5.4) = -1.4 \qquad (7 - 5.4) = 1.6 \qquad (13 - 5.4) = 7.6 $$ $$ (2 - 5.4) = -3.4 \qquad (1 - 5.4) = -4.4 $$

Step 3: Square Each Deviation

$$ (-1.4)^2 = 1.96 \qquad (1.6)^2 = 2.56 \qquad (7.6)^2 = 57.76 $$ $$ (-3.4)^2 = 11.56 \qquad (-4.4)^2 = 19.36 $$

Step 4: Sum the Squared Deviations

$$ \sum(x_i - \bar{x})^2 = 1.96 + 2.56 + 57.76 + 11.56 + 19.36 = \textbf{93.2} $$

Step 5: Divide by (n − 1) for Sample Variance

$$ s^2 = \frac{93.2}{5-1} = \frac{93.2}{4} = \textbf{23.3} $$

Step 6: Take the Square Root

$$ s = \sqrt{23.3} \approx \textbf{4.83} $$

Interpretation: The average value in this dataset is 5.4, and a typical data point deviates from that mean by approximately ±4.83 units. Notice that the value 13 is a large outlier — it contributes 57.76 out of 93.2 total squared deviation (62% of total variance by itself).

For comparison, the population standard deviation would be: $$ \sigma = \sqrt{\frac{93.2}{5}} = \sqrt{18.64} \approx 4.32 $$

The Empirical Rule (68-95-99.7 Rule)

For datasets that follow a normal (bell-curve) distribution, standard deviation defines precise probability intervals:

Example: Adult male heights in the US are normally distributed with μ = 69.1 inches and σ = 2.9 inches:

• 68% of men are between 66.2 and 72.0 inches (5'6" – 6'0")
• 95% are between 63.3 and 74.9 inches (5'3" – 6'3")
• 99.7% are between 60.4 and 77.8 inches (5'0" – 6'6")

Z-Score: Standard Deviation in Context

A z-score expresses any individual value in terms of how many standard deviations it lies above or below the mean:

$$ z = \frac{x - \mu}{\sigma} $$

Example: A man who is 74 inches tall (6'2"): $$ z = \frac{74 - 69.1}{2.9} = \frac{4.9}{2.9} \approx 1.69 $$

He is 1.69 standard deviations above average — taller than approximately 95.5% of men.

When to Use Population vs. Sample Formula

Rule of thumb: If there are values that exist but weren't measured, use the sample formula.

Key Concepts

Frequently Asked Questions

Why is standard deviation more useful than variance? Variance is in squared units (e.g., kg², inches²), making it hard to interpret alongside the original data. Standard deviation returns the spread to the original unit of measurement, making it directly comparable to the mean and individual values.

Can standard deviation be zero or negative? Standard deviation can equal zero — this means every value in the dataset is identical (no spread whatsoever). It can never be negative, because it's a square root of a sum of squares, which is always non-negative.

Does standard deviation work for non-normal distributions? Yes — standard deviation can be calculated for any dataset. However, the empirical rule (68-95-99.7%) only applies to normally distributed data. For skewed distributions, the standard deviation still measures spread, but the percentages at each interval differ.

What's a "good" standard deviation? There is no universal good or bad value — it depends entirely on the context and scale of measurement. A standard deviation of $5 is excellent for a restaurant meal but enormous for a gumball machine. Always interpret standard deviation relative to the mean, using the coefficient of variation (CV = σ/μ × 100%) for scale-independent comparison.