Calculate Variance Of A Variable Stata

Use this premium variance calculator to enter raw values, choose sample or population variance, and instantly review the mean, standard deviation, sum of squared deviations, and a visual chart. Below the tool, you will also find an expert guide showing how to calculate variance of a variable in Stata with the right commands, interpretation tips, and workflow examples.

Variance Calculator

Paste numbers separated by commas, spaces, semicolons, or new lines. Example: 12, 15, 14, 10, 9, 18

Chart and Formula Summary

Variance formula

• Population variance: σ² = Σ(xᵢ – μ)² / n
• Sample variance: s² = Σ(xᵢ – x̄)² / (n – 1)
• Standard deviation is the square root of variance.
• In Stata, the quickest route is usually summarize variable.

Expert Guide: How to Calculate Variance of a Variable in Stata

Variance is one of the most important measures of dispersion in applied statistics. If the mean tells you where your data are centered, the variance tells you how widely those data are spread around that center. In Stata, calculating variance can be extremely fast, but doing it correctly requires understanding what Stata reports, how missing values are handled, when to use sample variance versus population variance, and how to verify output from summary commands. This guide explains the full process for anyone who needs to calculate variance of a variable in Stata with confidence.

What variance means in practical data analysis

Variance measures the average squared distance of observations from the mean. Squaring the deviations ensures that positive and negative differences do not cancel out. A low variance indicates that values cluster tightly around the mean. A high variance indicates that values are more dispersed. In economics, variance can help describe volatility in income or prices. In biostatistics, it helps quantify the spread of lab measurements or patient outcomes. In education research, it captures how much student performance differs within a class or district.

Because variance uses squared units, it can look less intuitive than standard deviation. However, it is foundational for many procedures you use in Stata, including regression, ANOVA, hypothesis tests, and confidence intervals. Understanding how variance is calculated helps you interpret Stata output more accurately and avoid mistakes when comparing variables with different scales.

Key idea: Stata usually reports the sample standard deviation in the default summarize output. To get the variance, you can square the reported standard deviation, or use returned results directly after the command.

The basic Stata command for variance

The most common command is:

summarize myvariable

When you run this command, Stata returns the number of observations, mean, standard deviation, minimum, and maximum. The default standard deviation shown by summarize is based on the sample formula, which divides by n - 1. If you want the sample variance, square that standard deviation.

For example, if Stata reports a standard deviation of 4.5000, then the sample variance is:

4.5^2 = 20.25

You can also extract the returned standard deviation after running summarize:

summarize myvariable display r(sd)^2

This is one of the cleanest ways to calculate variance in Stata because it uses the exact returned result rather than manual copying from the Results window.

Sample variance versus population variance in Stata

The distinction matters. In most real-world research, your dataset is treated as a sample from a larger population. In that setting, sample variance is the correct measure for inferential work. The sample formula divides by n - 1 to correct bias in estimating the population variance.

• Sample variance: use when your observed data represent only a subset of a larger population.
• Population variance: use when your data include the full population of interest and you want a purely descriptive measure.

Stata’s default summary output aligns with sample statistics. If you need population variance, you can compute it manually once you know the sample size and mean. One approach is to first generate squared deviations from the mean and then divide by n instead of n - 1.

summarize myvariable scalar mean_x = r(mean) generate sqdev = (myvariable – mean_x)^2 summarize sqdev display r(sum) / r(N)

That final displayed value is the population variance. If instead you divide by r(N) - 1, you obtain the sample variance.

A step by step example with real numbers

Suppose your variable contains these seven values: 12, 15, 14, 10, 9, 18, and 16. The mean is 13.4286. The squared deviations sum to approximately 57.7143. The sample variance is:

57.7143 / (7 – 1) = 9.6190

The population variance is:

57.7143 / 7 = 8.2449

Here is how those values compare:

Statistic	Value	Interpretation
Observations	7	Total number of usable data points
Mean	13.4286	Average value of the variable
Sum of squared deviations	57.7143	Total squared spread around the mean
Sample variance	9.6190	Preferred for most inferential analysis
Population variance	8.2449	Used when the data represent the full population
Sample standard deviation	3.1015	Square root of sample variance

In Stata, if these values were stored in a variable called score, the workflow might look like this:

clear input score 12 15 14 10 9 18 16 end summarize score display r(sd)^2

Using tabstat and other Stata commands

While summarize is the default choice, Stata offers several alternatives that are useful in larger workflows:

1. tabstat for custom summary statistics
2. collapse for dataset reduction to summary values
3. egen with grouped operations for panel or subgroup work
4. quietly summarize when you want returned results without printing output

For example, with tabstat you can request variance directly if supported by your workflow and version:

tabstat myvariable, statistics(n mean sd variance min max)

If you want variance by categories, such as test score variance by classroom, you can use:

tabstat score, by(classroom) statistics(n mean sd variance)

This is especially useful in education, health, or labor datasets where understanding within-group variability is often as important as comparing means.

Calculating variance by group in Stata

Researchers often need variance not just for one variable, but for that variable across groups such as sex, treatment status, region, school, or year. Stata can handle that efficiently. Here is a common pattern:

bysort region: summarize income

That command prints separate summary statistics for each region. If you want a more compact and publication-friendly table, use tabstat:

tabstat income, by(region) statistics(n mean sd variance)

Consider the following example with actual summary values from a hypothetical household income study:

Region	N	Mean Income	Standard Deviation	Sample Variance
North	120	54,200	11,800	139,240,000
South	135	49,700	10,400	108,160,000
East	98	57,100	13,500	182,250,000
West	110	60,300	12,100	146,410,000

In this comparison, the East region has the highest variance, indicating the greatest income spread. Even if two regions have similar means, their variances can reveal very different patterns of inequality or heterogeneity.

Missing values and data cleaning considerations

One of the easiest ways to get a misleading variance estimate is to overlook missing or miscoded values. In Stata, numeric missing values such as ., .a, and related forms are treated as missing. Most summary commands automatically exclude them, but miscoded placeholders such as 999, 9999, or -1 are not automatically excluded unless you recode them first.

Before calculating variance, check your variable carefully:

codebook myvariable summarize myvariable, detail tabulate myvariable if myvariable==999

If 999 is a missing-code placeholder, recode it:

replace myvariable = . if myvariable == 999

Only after this step should you calculate variance. Otherwise, one or two invalid outliers can inflate the variance dramatically and distort every downstream result.

How to verify returned results after summarize

Stata stores many values in memory after commands run. After summarize, you can inspect them using:

return list

You will usually see values like r(N), r(mean), r(sd), r(min), r(max), and others. This allows you to write reproducible code. For example:

quietly summarize wage scalar wage_var = r(sd)^2 display wage_var

This is preferable to manual calculations because it reduces transcription errors and works smoothly inside do-files and automated reporting scripts.

Interpreting variance in context

Variance has no universal threshold for “high” or “low.” Interpretation depends on the scale of the original variable. A variance of 25 might be large for blood pressure changes measured over a short interval, but tiny for annual household income. Always interpret variance together with the mean, standard deviation, range, and subject-matter context.

• If variance is near zero, observations are tightly clustered.
• If variance is large, observations are widely dispersed.
• If variance differs strongly across groups, that can signal heterogeneity, measurement issues, or unequal risk.
• For skewed data, supplement variance with plots and percentiles.

In Stata, combining numerical summaries with graphics improves interpretation. Histograms, box plots, and kernel density estimates can show whether a large variance comes from broad spread, long tails, or a few influential outliers.

Recommended Stata workflow for accurate variance analysis

1. Inspect the variable for coding issues and impossible values.
2. Review missing data patterns.
3. Run summarize variable to obtain mean and standard deviation.
4. Square r(sd) to obtain sample variance.
5. If needed, compute population variance manually using squared deviations divided by n.
6. Use tabstat or grouped summaries for subgroup comparison.
7. Add graphical checks to interpret the numerical variance meaningfully.

This process is simple, reproducible, and suitable for academic research, institutional reporting, and policy analysis.

Authoritative resources for further study

If you want to deepen your understanding of variance, statistical interpretation, and software workflows, these authoritative references are useful:

• NIST Engineering Statistics Handbook from the U.S. National Institute of Standards and Technology.
• UCLA Statistical Methods and Data Analytics Stata Resources.
• Princeton University Data and Statistical Services Training Resources.

These sources are especially valuable when you need more than a quick command answer and want a reliable explanation of statistical assumptions, interpretation, and implementation details.

Final takeaway

To calculate variance of a variable in Stata, the most direct path is usually to run summarize and square the returned standard deviation. That gives you the sample variance, which is the standard choice in most research settings. When you need population variance, calculate squared deviations and divide by the total number of observations. For grouped analysis, use tabstat or bysort. Most importantly, always validate your data before interpreting variance, because coding errors and outliers can alter the result dramatically.

Use the calculator above to verify your manual work, understand the difference between sample and population variance, and create a quick visual check of data spread before moving into Stata.