What Standard Deviation and Variance Measure

Variance tells you how spread out your data is from the average. Standard deviation is the square root of variance — it measures the same spread, but in the original units of your data, making it easier to interpret. If you measure test scores, variance gives you a squared number; standard deviation gives you a number in points, which makes more sense when you're talking about how far scores typically fall from the average.

Both numbers start the same way: you find how far each data point sits from the average, square those distances, and average them. Standard deviation then takes one extra step — the square root — to bring the answer back to readable units.

Key Takeaways

  • Variance is the average of squared distances from the mean; standard deviation is the square root of variance.
  • For a full dataset (population variance), divide the sum of squared differences by the total count; for a sample, divide by count minus one.
  • The step-by-step process is the same whether you calculate by hand or use a calculator: find the mean, subtract it from each value, square the results, average them, and take the square root for standard deviation.
  • Most spreadsheet programs and scientific calculators have built-in functions that do this work in one command.

Calculate Variance and Standard Deviation by Hand

Start with a straightforward dataset. Say you have five test scores: 78, 82, 85, 88, and 92. First, find the mean (average): add them up and divide by how many scores there are. (78 + 82 + 85 + 88 + 92) ÷ 5 = 425 ÷ 5 = 85.

Next, subtract the mean from each score and write down the result. This is the distance from the average:

  • 78 − 85 = −7
  • 82 − 85 = −3
  • 85 − 85 = 0
  • 88 − 85 = 3
  • 92 − 85 = 7

Now square each of those differences (multiply each by itself):

  • (−7)² = 49
  • (−3)² = 9
  • (0)² = 0
  • (3)² = 9
  • (7)² = 49

Add up all the squared differences: 49 + 9 + 0 + 9 + 49 = 116. Now divide by the count of data points. Here is where sample and population matter: if these five scores are your entire group of interest (the population), divide by 5. If these five are a sample from a larger group, divide by 4 (count minus one). For this example, assume it's a sample, so 116 ÷ 4 = 29. This is the sample variance.

To get standard deviation, take the square root of variance: √29 ≈ 5.39. This means scores in your sample typically fall about 5.39 points away from the mean of 85.

Population Variance Versus Sample Variance

The only difference between the two is the denominator. Population variance divides by n (the total count). Sample variance divides by n − 1. You use population variance when you have data for an entire group — all students in a class, all employees at a company, all measurements in a complete census. You use sample variance when your data is a subset drawn from a larger group you're trying to learn about.

Why divide by n − 1 for a sample? Because a sample tends to underestimate how spread out the full population really is. Dividing by a smaller number (one less than your count) makes the variance slightly larger, which corrects for this bias. This is called Bessel's correction.

In practice, most real-world calculations use sample variance, because you are usually working with a subset of data rather than a complete population.

Using a Scientific Calculator

Most scientific calculators have a statistics mode. Enter your data points one at a time, pressing a key (often labeled M+, Data, or Σ+) after each entry. Once all values are entered, look for buttons labeled σ (lowercase sigma, for standard deviation) or s (for sample standard deviation). Press the one you need.

Some calculators show both population standard deviation (σ) and sample standard deviation (s) at the same time. Check your calculator's manual to confirm which button corresponds to which formula. The difference is small with large datasets but noticeable with small ones.

A few calculators also display variance directly. If yours does not, you can square the standard deviation result to get variance: if standard deviation is 5.39, then variance is 5.39² ≈ 29.

Using a Spreadsheet Program

Spreadsheet programs like Excel, Google Sheets, and LibreOffice Calc have built-in functions that do all the work in one step. For sample variance, use =VAR() or =VAR.S(). For population variance, use =VARP() or =VAR.P(). For sample standard deviation, use =STDEV() or =STDEV.S(). For population standard deviation, use =STDEVP() or =STDEV.P().

Enter your data in a column or row, then type the function in an empty cell. For example, if your scores are in cells A1 through A5, type =VAR.S(A1:A5) to get sample variance. The program calculates the mean, the squared differences, and the average all at once and returns the result.

Most spreadsheets default to sample variance and standard deviation, which is the right choice for most situations. If you need population values, use the .P versions of the functions.

Common Mistakes to Avoid

The most frequent error is forgetting to square the differences before averaging them. If you average the unsquared differences (−7, −3, 0, 3, 7), they add to zero, which tells you nothing. Squaring them first ensures negative and positive distances both contribute to the spread.

Another common mistake is using the wrong denominator. If you are working with a sample and divide by n instead of n − 1, your variance will be too small. Check whether your data represents a full population or a sample before you calculate.

When using a calculator or spreadsheet, verify you are using the sample version (s or .S) unless you have a specific reason to use the population version. In most real-world work, sample variance is correct.

Why Standard Deviation Matters More Than Variance

Variance is mathematically useful because squaring makes the math cleaner and easier to work with in further calculations. But standard deviation is more practical because it is in the same units as your original data. If you measure height in inches, variance is in square inches (which is hard to visualize), while standard deviation is in inches (which makes sense).

Standard deviation also connects directly to the normal distribution. In a normal distribution, about 68% of data falls within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three. This rule lets you quickly estimate how typical or unusual a single data point is.

Frequently Asked Questions

What is the difference between standard deviation and variance?

Variance is the average of squared distances from the mean. Standard deviation is the square root of variance. They measure the same thing — spread — but standard deviation is in the original units of your data, making it easier to interpret. If variance is 29, standard deviation is about 5.39.

When do I use population variance instead of sample variance?

Use population variance only when your data includes every member of the group you are studying. Use sample variance when your data is a subset. In most real-world situations, you have a sample, so use sample variance (divide by n − 1).

Can I calculate standard deviation without a calculator?

Yes, but it is tedious. Find the mean, subtract it from each value, square each result, add them up, divide by n − 1 (or n), and take the square root. For datasets with more than a few points, a calculator or spreadsheet saves time and reduces arithmetic errors.

Why do I divide by n minus one for a sample?

Dividing by n − 1 instead of n corrects for the fact that a sample tends to underestimate the true spread of the full population. This adjustment, called Bessel's correction, makes sample variance a more accurate estimate of population variance.

What does a high standard deviation mean?

High standard deviation means your data points are spread far from the average. Low standard deviation means they cluster close to the average. A standard deviation of 5 on test scores with a mean of 85 means scores typically fall between 80 and 90.