acalculator

What is the standard error?

Paste a sample, or type its standard deviation and size, or a proportion. The standard error calculator gives the standard error and a 95% confidence interval as you type.

Your numbers

What do you have?
Read as: 12; 15; 9; 14; 10
Standard error
1.14018

The standard error is 1.14018.

Mean
12
Sample standard deviation
2.54951
Sample size (n)
5
t value (95%)
2.77645
Margin of error (95%)
3.16563
95% interval, low end
8.83437
95% interval, high end
15.1656

Standard error: 1.14018. The standard error is 1.14018.

How to calculate

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.

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

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).

  • 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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. What do you have? My numbers, Your numbers 12, 15, 9, 14, 10 gives Standard error 1.140175, Sample mean (optional) 12, Sample standard deviation (s) 2.54951, t value (95%) 2.776445, 95% interval, low end 8.834366, 95% interval, high end 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. What do you have? SD and n, Sample standard deviation (s) 6, Sample size (n) 36, Sample mean (optional) 50 gives Standard error 1, t value (95%) 2.030108, Margin of error (95%) 2.030108, 95% interval, low end 47.969892, 95% interval, high end 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. What do you have? A proportion, Sample proportion 45%, Sample size (n) 400 gives Standard error 0.024875, Margin of error (95%) 0.048753, 95% interval, low end 0.401247, 95% interval, high end 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)

How it works

Pick what you have. The calculator uses one of three inputs.

My numbers. Type the sample (2 to 1,000 numbers). With n numbers x₁ … xₙ:

  • mean x̄ = (x₁ + … + xₙ) ÷ n
  • sample standard deviation s = √( Σ(x − x̄)² ÷ (n − 1) )
  • standard error of the mean SE = s ÷ √n

SD and n. Type the sample standard deviation s (0 or more, computed with n − 1) and the sample size n (a whole number, 2 or more). SE = s ÷ √n. The sample mean is optional; with it, the page also gives the interval.

A proportion. Type the sample proportion p as a percent (0 to 100) and the sample size n (2 or more). With p as a decimal, SE = √( p × (1 − p) ÷ n ).

The 95% confidence interval

  • For a mean: t = the Student t value with n − 1 degrees of freedom that leaves 2.5% in the upper tail (t(0.975, n − 1)), computed to about 15 significant figures at any n (below 1,000 degrees of freedom from the t distribution function; from 1,000 up by integrating the t density), margin of error = t × SE, interval = x̄ − margin to x̄ + margin.
  • For a proportion: z = 1.959964 (rounded; the page uses the full standard normal value that leaves 2.5% in the upper tail, 1.959963984540054), margin of error = z × SE, interval = p − margin to p + margin, with p, the margin and the interval ends as decimals (0.40 means 40%). This is the normal approximation (Wald interval); it can run below 0 or above 1 for small samples, and the page shows the ends as computed.

Rules

  • A list needs at least 2 numbers; one number gives no answer.
  • The standard deviation must be 0 or more, and n a whole number from 2 to 1,000,000,000.
  • No finite population correction is applied: the population is taken as much larger than the sample.
  • Every step runs on the full values; only the display is rounded.

What the calculator shows

  • Standard error, sample standard deviation, t value, margin of error and the interval ends: 6 significant figures, rounded half up.
  • Mean: at most 6 decimal places, rounded half up, with thousands separators. Sample size: a whole number.
  • The t value, mean and sample standard deviation are left out for a proportion. The mean and interval are left out in "SD and n" mode when no mean is given.

Worked examples by hand

Five test scores: 12, 15, 9, 14, 10. The mean is 60 ÷ 5 = 12. The differences from the mean are 0, 3, −3, 2, −2, and their squares add up to 0 + 9 + 9 + 4 + 4 = 26. s = √(26 ÷ 4) = √6.5 = 2.54951. SE = 2.54951 ÷ √5 = 1.14018. With 4 degrees of freedom, t = 2.77645, so the margin is 2.77645 × 1.14018 = 3.16563 and the 95% interval is 8.83437 to 15.1656.

s = 6, n = 36, mean 50. SE = 6 ÷ √36 = 1. With 35 degrees of freedom, t = 2.03011, so the interval is 50 ± 2.03011: 47.9699 to 52.0301.

45% of 400. SE = √(0.45 × 0.55 ÷ 400) = √0.00061875 = 0.0248747. The margin is 1.959964 × 0.0248747 = 0.0487535, so the interval is 0.401247 to 0.498753 (40.1% to 49.9%).

Other questions people ask

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.