Standard Deviation and Variance Explained (With a Step-by-Step Example)

Quick Answer

Variance.
The average of the squared differences from the mean. It tells you how spread out your data is, but the units are squared (which makes it hard to interpret directly).

Standard deviation.
The square root of the variance. Same information as variance but in the original units of your data, which makes it directly interpretable. This is what you report in research papers.

How to choose.
You almost always report the standard deviation. Variance shows up in calculations (regression, ANOVA, t-tests all use it internally) but rarely in your results section.

What These Measure

The mean tells you the center of your data. Standard deviation and variance tell you how spread out your data is around that center. Two datasets can have the same mean and look completely different. One can be tightly clustered. The other can be scattered widely. Standard deviation is how you quantify that difference.

Consider two classes that both averaged 85 on an exam. In Class A, every student scored between 82 and 88. In Class B, scores ranged from 60 to 100. Same mean, very different spread. Standard deviation captures that difference in one number. A small standard deviation means the data clusters tightly around the mean. A large one means it spreads out.

Variance

Variance is calculated first because standard deviation is just its square root. The variance formula has four parts: subtract the mean from each value, square the differences, sum them, and divide.

Formula

s² = Σ (x − M)² / (n − 1)

Where Σ means "sum of," x is each value, M is the mean, and n is the sample size.

Note the n − 1 in the denominator. This is what makes it a sample variance rather than a population variance. The subtraction corrects for the fact that you're estimating the variability of a population from a sample. Most software defaults to n − 1 because sample data is what researchers usually have.

Why we square the differences

Squaring serves two purposes. First, it makes negative differences positive, so they don't cancel out positive ones when summed. Second, it gives more weight to larger differences. A value 10 units from the mean contributes much more to the variance than a value 2 units away (100 squared units vs. 4 squared units).

The drawback is that variance is in squared units. If your data is in dollars, the variance is in squared dollars. That's not meaningful to readers. Which is exactly why we usually report standard deviation instead.

Standard Deviation

Standard deviation is the square root of the variance. The square root undoes the squaring, bringing the measure back to the original units of your data.

Formula

SD = √[Σ (x − M)² / (n − 1)]

Take the square root of the variance to get the standard deviation.

In APA format, standard deviation is reported as SD. Capital letters, both italicized. The population standard deviation uses the Greek letter σ (sigma), but sample data uses SD.

Worked Example: Calculating Both by Hand

Let's use the same test scores from the previous article on mean, median, and mode:

72, 78, 81, 85, 85, 87, 88, 90, 92, 94

The mean was M = 85.2.

Step 1: Subtract the mean from each value

For each value, calculate x − M:

72 − 85.2 = −13.2

78 − 85.2 = −7.2

81 − 85.2 = −4.2

85 − 85.2 = −0.2

87 − 85.2 = 1.8

88 − 85.2 = 2.8

90 − 85.2 = 4.8

92 − 85.2 = 6.8

94 − 85.2 = 8.8

Notice that the deviations sum to zero. They always do, because that's how the mean works. This is why we need to square them.

Step 2: Square each deviation

(−13.2)² = 174.24

(−7.2)² = 51.84

(−4.2)² = 17.64

(−0.2)² = 0.04

(1.8)² = 3.24

(2.8)² = 7.84

(4.8)² = 23.04

(6.8)² = 46.24

(8.8)² = 77.44

Step 3: Sum the squared deviations

Add them all up:

174.24 + 51.84 + 17.64 + 0.04 + 0.04 + 3.24 + 7.84 + 23.04 + 46.24 + 77.44 = 401.60

Step 4: Divide by n − 1 to get the variance

With 10 values, n − 1 = 9.

Variance: s² = 401.60 / 9 = 44.62

The variance is 44.62 squared points. The units are awkward. That's why we go one more step.

Step 5: Take the square root to get the standard deviation

Standard deviation: SD = √44.62 = 6.68

The standard deviation is 6.68 points. Now we have a number in the original units of our data (test points), which we can interpret directly.

How to Interpret the Standard Deviation

For data that follows a roughly normal distribution, standard deviation has specific interpretations.

About 68% of values fall within one standard deviation of the mean.
About 95% of values fall within two standard deviations of the mean.
About 99.7% of values fall within three standard deviations of the mean.

This is the "68-95-99.7 rule," also called the empirical rule. Applied to the test scores above (M = 85.2, SD = 6.68), we'd expect:

68% of scores between 78.5 and 91.9
95% of scores between 71.8 and 98.6
99.7% of scores between 65.2 and 105.2

Looking at the actual data, 8 of the 10 scores fall in the 78.5-to-91.9 range (80%), and all 10 fall within two standard deviations. The empirical rule holds approximately even for small samples.

Population vs. Sample Standard Deviation

There are two versions of the formula, and the difference matters when you write up results.

Sample standard deviation divides by n − 1 and uses the symbol SD or s. This is what you almost always calculate because you almost always have sample data.

Population standard deviation divides by N and uses the Greek letter σ (sigma). You'd use this only when you have data on the entire population, which is rare in research.

Why the difference? When you estimate a population parameter from a sample, you lose one degree of freedom calculating the mean first. Dividing by n − 1 instead of n corrects for that loss. Without the correction, sample variance would systematically underestimate population variance.

Most statistical software (R, SPSS, Python's pandas, Excel's STDEV function) defaults to the n − 1 version. If you ever need the population version explicitly, you'll find a separate function (STDEVP in Excel, ddof=0 in pandas).

When Standard Deviation Misleads

Standard deviation assumes your data is reasonably symmetric. Two situations break this assumption.

Skewed data

For heavily skewed data (income, home prices, reaction times), the standard deviation gets distorted by extreme values. A small number of large values inflates the squared deviations, pushing the standard deviation higher than it would be for symmetric data with the same typical spread. For skewed data, the median and interquartile range (IQR) often describe the distribution better than the mean and standard deviation.

Outliers

A single extreme outlier can substantially change the standard deviation. The mean and standard deviation are both sensitive to outliers. The median is not, which is why outlier-prone data often gets reported with the median instead of the mean.

Always inspect your data before relying on the standard deviation. A histogram or box plot quickly reveals whether your data is reasonably symmetric.

Reporting in APA Format

APA conventions for standard deviation are specific.

Always pair mean with standard deviation. A mean alone tells readers the center but not the spread. APA requires both whenever you describe a sample. Example:

Participants scored above the population average (M = 85.2, SD = 6.68).

Use italicized SD. Both letters capitalized, both italicized. The leading zero rule doesn't apply because standard deviation can exceed 1. Write SD = 6.68, not SD = .68 (unless the actual value is less than 1, in which case keep the leading zero: SD = 0.34).

Variance is rarely reported in text. If you need to mention it, use s² (italicized lowercase). Example:

The variance among condition means was substantial (s² = 44.62).

Tables. For tables of descriptive statistics, standard deviation appears in its own column, usually after the mean and before any inferential test statistics.

Common Mistakes

Reporting variance instead of standard deviation. Variance is in squared units, which readers can't interpret intuitively. Always convert to standard deviation for reporting.
Forgetting to pair mean with SD. APA requires both. A mean without an SD doesn't tell readers anything about variability.
Using population SD when you have sample data. Use the n − 1 version for sample data. Statistical software defaults to this, but if you calculate by hand, double-check.
Reporting SD without checking for skew. If your data is heavily skewed, the standard deviation may misrepresent the spread. Use the median and IQR for skewed data.
Confusing standard deviation with standard error. Standard deviation describes the spread in your data. Standard error describes the precision of your estimate of the mean. They're related but different. Standard error = SD / √n.

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Frequently Asked Questions

What's the difference between variance and standard deviation?

Standard deviation is the square root of the variance. Both measure how spread out your data is, but variance is in squared units and standard deviation is in the original units. If your data is in dollars, the variance is in squared dollars (which is meaningless to most readers), and the standard deviation is in dollars (which readers can interpret directly). Variance is used in calculations like ANOVA and regression. Standard deviation is what you report in your results section.

Why does the formula divide by n − 1 instead of n?

Dividing by n − 1 corrects a bias that arises when you estimate a population parameter from a sample. When you calculate the variance, you first calculate the mean. That uses up one "degree of freedom" in the data. Dividing by n − 1 instead of n accounts for that. Without this correction, sample variance would systematically underestimate population variance. The n − 1 version is called the "unbiased estimator." Most statistical software uses it by default.

What does a "high" or "low" standard deviation mean?

There's no absolute threshold. "High" and "low" depend on the scale of your data and the typical variation in your field. A standard deviation of 5 might be enormous on a 7-point Likert scale but tiny on an exam scored out of 100. Compare your standard deviation to the mean (a useful ratio is the coefficient of variation, SD/M). Compare it to other studies in your field. Compare it to the theoretical maximum for your measure. Context determines whether your spread is small, moderate, or large.

Can standard deviation be negative?

No. Standard deviation is the square root of variance, and variance is the average of squared values. Squared values are always positive, so variance is always positive (or zero, if all your data points are identical). Taking the square root of a positive number gives another positive number. If your software ever returns a negative standard deviation, something has gone wrong in the calculation or display.

Should I report variance or standard deviation in my paper?

Standard deviation, almost always. Variance is harder to interpret because it's in squared units. Standard deviation is in the original units of your variable, which readers can understand directly. The exception is when you're decomposing variance for ANOVA or other tests that work with variance components. In those cases, the variance itself is the relevant quantity. For descriptive statistics in your results section, report standard deviation.

What's the difference between standard deviation and standard error?

Standard deviation describes the spread of your data. Standard error describes the precision of your estimate of the mean. They're related: standard error equals standard deviation divided by the square root of n. As your sample size grows, standard error shrinks (your estimate of the mean gets more precise), but standard deviation stays roughly the same (the underlying spread doesn't change). Confidence intervals use standard error, not standard deviation. Reporting descriptive statistics uses standard deviation.

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