var() in R: Sample Variance With NA and Covariance Handling

The var() function in base R computes the sample variance of a numeric vector using the unbiased n-1 denominator. Pass na.rm = TRUE to ignore missing values, supply a second vector for covariance, or pass a matrix to get a full covariance matrix.

var(x)                               # sample variance of a vector
var(x, na.rm = TRUE)                 # ignore NA values
var(x) * (length(x) - 1) / length(x) # population variance (n denominator)
var(x, y)                            # covariance between two vectors
var(mtcars)                          # covariance matrix of a data frame
sqrt(var(x))                         # standard deviation (same as sd(x))
diag(var(mtcars))                    # per-column variances from a frame

What var() does in one sentence

var() returns the sum of squared deviations from the mean divided by n-1 as a single numeric value. It accepts numeric, integer, and logical vectors (where TRUE = 1). For matrices and data frames, it returns the full sample covariance matrix instead of a single number.

The n-1 denominator (Bessel's correction) makes the estimate unbiased when x is a sample drawn from a larger population. R uses sample variance by default; population variance requires a manual adjustment.

Syntax

var(x, y = NULL, na.rm = FALSE, use) takes a vector or matrix plus optional second vector and missing-value controls. When y is supplied, the function returns covariance; when x is a matrix or data frame, it returns a covariance matrix.

The four arguments:

• x: numeric vector, matrix, or data frame
• y: optional second numeric vector for covariance; default NULL
• na.rm: if TRUE, drop NA before computing; default FALSE
• use: how to handle NA in matrix input; one of "everything", "all.obs", "complete.obs", "na.or.complete", "pairwise.complete.obs"

Five common patterns

1. Plain sample variance

The mean is 3, the squared deviations are (1-3)^2, (2-3)^2, ..., (5-3)^2 = 4, 1, 0, 1, 4. Their sum is 10, divided by n-1 = 4 gives 2.5.

2. Ignore missing values

Without na.rm = TRUE, any NA poisons the result. With it, the function drops NA and computes variance from the remaining four values.

3. Population variance instead of sample variance

R has no built-in population variance function. Multiply the sample variance by (n - 1) / n whenever your data represents the entire population, not a sample drawn from one.

4. Covariance between two vectors

When you pass a second vector, var() returns the sample covariance between the two. This is identical to cov(x, y) and measures how the two variables move together. The negative value confirms that higher horsepower goes with lower miles per gallon in mtcars.

5. Covariance matrix of a data frame

For a matrix or data frame, var() returns a full covariance matrix: variances on the diagonal, pairwise covariances off-diagonal. To get just the per-column variances, use diag(var(df)) or sapply(df, var).

var vs sd vs cov vs population variance

Pick the spread function that matches your need. The table compares var(), sd(), cov(), and the manual population variance.

Sample variance is the right default for almost every applied analysis. Switch to population variance only when you genuinely have the full population (not a sample of it). Use sd() when you want a spread measure on the original scale.

Common pitfalls

Pitfall 1: var() returns NA when any element is NA. Always set na.rm = TRUE for real-world data. If you forget, summaries silently propagate NA and break tables and plots downstream.

Pitfall 2: var() uses the n-1 denominator, not n. R reports the unbiased sample estimate. If your textbook formula has n in the denominator, you are computing population variance; multiply the R result by (n - 1) / n.

Pitfall 3: var(df) returns a covariance matrix, not column-wise variances. To get a vector of per-column variances, use sapply(df, var) or diag(var(df)). Beginners often pass a data frame expecting a per-column summary and get a square matrix instead.

Pitfall 4: var() of a single value returns NA. With one observation, n - 1 = 0 and the division is undefined. Check length(x) > 1 before computing if your data may shrink to a single row after filtering.

Try it yourself

Related base R spread functions

After mastering var(), look at:

• sd(): sample standard deviation, the square root of var()
• cov() and cor(): covariance and correlation between two vectors
• range(), IQR(), mad(): alternative spread measures
• summary(): five-number summary plus mean in one call
• apply(m, 2, var): per-column variances of a matrix
• aggregate() and dplyr::summarise(): group-wise variances

For a full breakdown of spread, central tendency, and shape measures, see the descriptive statistics in R guide. The official base R documentation lives at the R stats package reference.

FAQ

How do I compute variance in R while ignoring missing values?

Pass na.rm = TRUE: var(x, na.rm = TRUE). By default, var() returns NA if any element of x is NA, which silently propagates into downstream summaries. Always set na.rm = TRUE when working with real-world data, or impute missing values first using a domain-appropriate rule before computing the variance.

What is the difference between var() and sd() in R?

var() returns the sample variance (average squared deviation from the mean, with n-1 denominator); sd() returns the sample standard deviation, which is the square root of the variance. They measure the same spread, but sd() is in the same units as the original data while var() is in squared units. Use sd() for reporting, var() when you need the squared quantity for downstream formulas.

Does var() in R use the n or n-1 denominator?

R's var() always uses n - 1 (Bessel's correction), giving the unbiased sample variance estimator. For population variance with the n denominator, multiply by (n - 1) / n, or compute mean((x - mean(x))^2) directly. The choice matters most in small samples where the difference between dividing by n and n-1 is meaningful.

Why does var() return a matrix when I pass a data frame?

Because var() is overloaded: a vector input returns a scalar variance, a matrix or data frame input returns a full sample covariance matrix with variances on the diagonal and pairwise covariances off-diagonal. To get a simple vector of per-column variances, use sapply(df, var) or diag(var(df)) to extract the diagonal of the covariance matrix.

How do I compute the variance of every column in a data frame?

Use sapply(df, var) for a named numeric vector of per-column variances. For numeric-only frames you can also extract the diagonal of the covariance matrix with diag(var(df)). The tidyverse equivalent is dplyr::summarise(across(where(is.numeric), var, na.rm = TRUE)) for a one-line column-wise variance with NA handling.