Calculate the standard error of a difference for independent samples, paired differences, or correlated estimates, with variance contributions and a normal-approximation uncertainty reference.

For unrelated samples; use the same units for both SDs.

Standard Error of Difference Formula

The following formula estimates the standard error of the difference between two independent sample means, using each sample standard deviation.

SED = √((s₁² / n₁) + (s₂² / n₂))

Variables:

  • SED is the standard error of the difference between two sample means
  • s₁ is the sample standard deviation of sample 1
  • n₁ is the sample size of sample 1
  • s₂ is the sample standard deviation of sample 2
  • n₂ is the sample size of sample 2

To calculate the standard error of the difference, square the standard deviations of both samples, divide each by their respective sample sizes, add the two values, and take the square root of the sum.

Choose independent sample means for unrelated samples. For paired data, enter the standard deviation of the within-pair differences and the number of independent pairs. For known standard errors, use the correlation between the estimates on the same measurement scale, rather than automatically using the correlation between individual observations.

The calculator’s normal 95% reference is 1.96 × SE. It is not a t-adjusted confidence interval: small samples with estimated standard deviations generally need a t multiplier, and interval endpoints also require the observed mean difference. SE has the original measurement units; variance has squared units.

What is Standard Error of Difference?

The standard error of the difference (SED) measures the sampling variability of an estimated difference between two means. It is used in inference about that difference. A smaller SED indicates less estimated sampling uncertainty in the difference; it does not establish how close either individual sample mean is to its population mean.

How to Calculate Standard Error of Difference?

The following steps outline how to calculate the Standard Error of Difference.

  1. First, determine the sample standard deviation of sample 1 (s₁).
  2. Next, determine the sample size of sample 1 (n₁).
  3. Then, determine the sample standard deviation of sample 2 (s₂).
  4. Next, determine the sample size of sample 2 (n₂).
  5. Use the formula SED = √((s₁² / n₁) + (s₂² / n₂)) to calculate the Standard Error of Difference.
  6. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Standard deviation of sample 1 (s₁) = 5

Sample size of sample 1 (n₁) = 30

Standard deviation of sample 2 (s₂) = 4

Sample size of sample 2 (n₂) = 40