# What is the standard error?

Computes the standard error of the mean from a list of numbers or from the sample standard deviation and sample size, or the standard error of a sample proportion, with a 95% confidence interval.

- Page: https://www.acalculator.org/statistics/standard-error-calculator
- JSON spec: https://www.acalculator.org/statistics/standard-error-calculator.json
- Version: 0a3af42b1773

## Default answer

Example with the default inputs (What do you have? My numbers, Your numbers [12, 15, 9, 14, 10]): The standard error is 1.14018.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mode | What do you have? | A list of numbers, the sample standard deviation and sample size, or a sample proportion and sample size. |
| data | Your numbers | The sample, separated by commas, spaces, semicolons, or new lines. |
| sd | Sample standard deviation (s) | The sample standard deviation, computed with n − 1. |
| mean | Sample mean (optional) | The sample mean. With it, the page also gives the 95% confidence interval for the mean. |
| p | Sample proportion | The share of the sample with the trait, as a percent (45 means 45%). |
| n | Sample size (n) | How many values or people are in the sample. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| se | Standard error | The standard error: s ÷ √n for a mean, √(p × (1 − p) ÷ n) for a proportion (a decimal, 0.025 = 2.5 points). |
| xbar | Mean | The sample mean: the sum divided by n. |
| sd | Sample standard deviation | The square root of the sum of squared differences from the mean divided by n − 1. |
| n | Sample size (n) | How many values are in the sample. |
| t | t value (95%) | The Student t value with n − 1 degrees of freedom that leaves 2.5% in each tail. |
| margin | Margin of error (95%) | For a mean: t × standard error. For a proportion: z × standard error, z = 1.959964 rounded (normal approximation), as a decimal (0.05 = 5 points). |
| low | 95% interval, low end | The estimate minus the margin of error; for a proportion, a decimal (0.40 = 40%). |
| high | 95% interval, high end | The estimate plus the margin of error; for a proportion, a decimal (0.40 = 40%). |

## Method

SE of the mean = s ÷ √n, with s = √(Σ(x − x̄)² ÷ (n − 1)); 95% interval = x̄ ± t(0.975, n − 1) × SE. SE of a proportion = √(p(1 − p) ÷ n); 95% interval = p ± z × SE, z = 1.959964 (rounded).

## Assumptions

- The sample is a simple random sample, and the values are independent.
- The standard deviation is the sample standard deviation (divide by n − 1).
- The confidence interval for a mean uses Student’s t with n − 1 degrees of freedom, which assumes roughly normal data or a large sample.
- The interval for a proportion is the normal approximation (Wald interval); it is poor when n × p or n × (1 − p) is below about 5.
- No finite population correction: the population is taken as much larger than the sample.

## Worked examples

1. mode = data, data = 12 or 15 gives se = 1.140175, mean = 12, sd = 2.54951, t = 2.776445, low = 8.834366, high = 15.165634. Source: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.2, Confidence Limits for the Mean (https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm).
2. mode = summary, sd = 6, n = 36, mean = 50 gives se = 1, t = 2.030108, margin = 2.030108, low = 47.969892, high = 52.030108. Source: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.2, Confidence Limits for the Mean (https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm).
3. mode = proportion, p = 45%, n = 400 gives se = 0.024875, margin = 0.048753, low = 0.401247, high = 0.498753. Source: NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.4.1, Confidence intervals for a proportion (https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm).

## FAQ

### What is the standard error?

It is the standard deviation of an estimate: how much a sample mean (or a sample proportion) would change from one random sample to the next. A small standard error means the estimate is precise.

### How do I calculate the standard error of the mean?

Find the sample standard deviation s (divide the sum of squared differences from the mean by n − 1, then take the square root), and divide it by the square root of the sample size: SE = s ÷ √n. For s = 6 and n = 36, SE = 6 ÷ 6 = 1.

### What is the difference between standard deviation and standard error?

The standard deviation describes the spread of the data. The standard error describes the spread of the mean of samples of that size. The standard error is the standard deviation divided by √n, so it shrinks as the sample grows, while the standard deviation does not.

### How do I find the standard error of a proportion?

Use SE = √(p × (1 − p) ÷ n), with p as a decimal. If 45% of 400 people say yes, SE = √(0.45 × 0.55 ÷ 400) = 0.0249, or about 2.5 percentage points.

### How is the 95% confidence interval worked out?

For a mean, it is the mean ± t × SE, where t is the Student t value with n − 1 degrees of freedom that leaves 2.5% in each tail (2.776 for n = 5). For a proportion, it is p ± 1.96 × SE.

### How large a sample do I need to halve the standard error?

Four times as large. The standard error falls with the square root of n, so 4 times the sample halves it, and 100 times the sample cuts it to a tenth.

### Why does the calculator need at least 2 numbers?

The sample standard deviation divides by n − 1, so one number gives 0 ÷ 0. One value tells you nothing about the spread, so there is no standard error.

## Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.2, Confidence Limits for the Mean (s ÷ √N and the t interval), retrieved 2026-10-01. https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm
- NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.6, Measures of Scale (sample standard deviation), retrieved 2026-10-01. https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm
- NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.4.1, Confidence intervals for a proportion (normal approximation), retrieved 2026-10-01. https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm