Standard Error Calculator
Calculate the standard error of the mean, the standard error of a proportion, or compute standard error directly from raw data. Includes a confidence interval preview, a small sampling-distribution visual, quick picks, and clear step-by-step output.
Background
Standard error measures how much a sample-based estimate is expected to vary from sample to sample. A smaller standard error means the estimate is typically more precise. For means, standard error depends on the sample standard deviation and sample size. For proportions, it depends on the sample proportion and the number of observations.
How to use this calculator
- Choose Standard error of the mean, Standard error of a proportion, or From raw data.
- Enter the required values for your mode. In proportion mode, you can enter either p̂ directly or successes and trials.
- Optionally choose a confidence level to preview a confidence interval around the estimate.
- Click Calculate to see the standard error, optional CI preview, interpretation, and step-by-step solution.
How this calculator works
- Standard error of the mean: uses SE = s / √n.
- Standard error of a proportion: uses SE = √(p̂(1−p̂)/n).
- Raw data mode: first computes the sample mean and sample standard deviation, then applies SE = s / √n.
- For mean and raw-data modes, the CI preview automatically uses t when n < 30 and z when n ≥ 30.
- For proportion mode, the CI preview stays z-based using p̂ ± z*·SE.
- The result area also includes a tiny SE vs SD explainer so students can quickly distinguish precision of the estimate from spread of the data.
- A smaller standard error means the estimate is generally more precise from sample to sample.
Formula & Equations Used
Standard error of the mean: SE = s / √n
Standard error of a proportion: SE = √(p̂(1−p̂)/n)
Sample mean: x̄ = (Σx) / n
Sample standard deviation: s = √(Σ(x − x̄)² / (n − 1))
z-based confidence interval preview: estimate ± z*·SE
t-based confidence interval preview: estimate ± t*·SE, with df = n − 1
Example Problem & Step-by-Step Solution
Example 1 — Standard error of the mean
A sample has standard deviation s = 12 and sample size n = 36. Find the standard error.
- Use SE = s / √n.
- Substitute: SE = 12 / √36.
- √36 = 6.
- SE = 12 / 6 = 2.
Example 2 — Standard error of a proportion
A survey finds p̂ = 0.42 from n = 200. Find the standard error.
- Use SE = √(p̂(1−p̂)/n).
- Substitute: SE = √(0.42·0.58/200).
- Compute the value inside the square root.
- The result is the standard error of the sample proportion.
Example 3 — From raw data
Use the dataset 4, 6, 7, 9, 10 to find the standard error.
- Compute the sample mean.
- Compute the sample standard deviation using n−1.
- Use SE = s / √n.
- The result is the standard error of the mean for that dataset.
Frequently Asked Questions
Q: Is standard error the same as standard deviation?
No. Standard deviation describes variability in the data, while standard error describes variability in the sample-based estimate from sample to sample.
Q: What happens to standard error when sample size increases?
Standard error usually gets smaller as sample size increases, because dividing by √n makes the estimate more stable.
Q: Why does the raw-data mode use sample standard deviation instead of population standard deviation?
Because raw-data mode is designed for sample data, so it uses the sample standard deviation formula with n−1.
Q: What does the confidence interval preview mean?
It shows an interval around the estimate. Mean-based previews automatically use either estimate ± t*·SE or estimate ± z*·SE depending on sample size, while proportion previews use estimate ± z*·SE.
