acalculator

What is the confidence interval?

Enter your sample's mean and standard deviation, or its count of successes, to get the confidence interval and the margin of error.

Your numbers

Interval for a
Common values: 90, 95, 99.
Margin of error (±)
1.015054

The 95% confidence interval is 66.984946 to 69.015054 (68 ± 1.015054).

Lower limit
66.984946
Upper limit
69.015054
Estimate
68
Standard error
0.5
Critical value
2.030108

Margin of error (±): 1.015054. The 95% confidence interval is 66.984946 to 69.015054 (68 ± 1.015054).

Where is the interval?

How to calculate

Computes a confidence interval for a population mean (t or z) or a population proportion at any confidence level, with the margin of error and standard error.

Example with the default inputs (Interval for a Mean, Sample mean (x̄) 68, Standard deviation 3, Standard deviation is From the sample (t), Sample size (n) 36, Confidence level 95%): The 95% confidence interval is 66.984946 to 69.015054 (68 ± 1.015054).

Method: Mean: x̄ ± t × s ÷ √n with n − 1 degrees of freedom (or z × σ ÷ √n for a known σ). Proportion: p̂ ± z × √(p̂(1 − p̂) ÷ n). The critical value has an upper tail of (1 − confidence) ÷ 2.

  • The sample is random, and the sample mean is roughly normal (the data are close to normal or n is large).
  • The proportion interval is the normal-approximation (Wald) interval; it works best when at least 5 successes and 5 failures are counted. Its limits are cut to 0 and 1.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Interval for a Mean, Sample mean (x̄) 68, Standard deviation 3, Standard deviation is From the sample (t), Sample size (n) 36, Confidence level 95% gives Critical value 2.030108, Margin of error (±) 1.015054, Lower limit 66.984946, Upper limit 69.015054.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean. https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm
  2. Interval for a Mean, Sample mean (x̄) 68, Standard deviation 3, Standard deviation is Known σ (z), Sample size (n) 36, Confidence level 90% gives Critical value 1.644854, Margin of error (±) 0.822427, Lower limit 67.177573, Upper limit 68.822427.Source: OpenStax, Introductory Statistics 2e, §8.1 A Single Population Mean using the Normal Distribution. https://openstax.org/books/introductory-statistics-2e/pages/8-1-a-single-population-mean-using-the-normal-distribution, Example 8.2 (z = 1.645, EBM = 0.8225, interval 67.18 to 68.82)
  3. Interval for a Mean, Sample mean (x̄) 9.26146, Standard deviation 0.022789, Standard deviation is From the sample (t), Sample size (n) 195, Confidence level 95% gives Lower limit 9.258241, Upper limit 9.264679.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean. https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm (95% interval 9.258242 to 9.264679)
  4. Interval for a Proportion, Successes (x) 421, Sample size (n) 500, Confidence level 95% gives Estimate 0.842, Lower limit 0.81003, Upper limit 0.87397.Source: OpenStax, Introductory Statistics 2e, §8.3 A Population Proportion. https://openstax.org/books/introductory-statistics-2e/pages/8-3-a-population-proportion, Example 8.10 (p′ = 0.842, interval 0.810 to 0.874)
  5. Interval for a Proportion, Successes (x) 300, Sample size (n) 500, Confidence level 90% gives Estimate 0.6, Margin of error (±) 0.036037.Source: OpenStax, Introductory Statistics 2e, §8.3 A Population Proportion. https://openstax.org/books/introductory-statistics-2e/pages/8-3-a-population-proportion, Example 8.11 (EBP = 0.036, interval 0.564 to 0.636)

How it works

Let C be the confidence level in percent and α = 1 − C ÷ 100.

Mean, standard deviation from the sample (t). With sample mean x̄, sample standard deviation s, and size n:

  • standard error SE = s ÷ √n
  • critical value t* = the t with upper tail α ÷ 2 and n − 1 degrees of freedom
  • margin of error = t* × SE, and the interval is x̄ − margin to x̄ + margin.

Mean, known σ (z). The same with σ in place of s and the standard normal z* (upper tail α ÷ 2) in place of t*.

Proportion. With x successes in n: p̂ = x ÷ n, SE = √(p̂ × (1 − p̂) ÷ n), margin = z* × SE, and the interval is p̂ − margin to p̂ + margin, with the lower limit raised to 0 and the upper limit lowered to 1 if they pass those bounds (the margin of error itself is not cut). This is the normal-approximation (Wald) interval.

The page shows the margin of error, the lower and upper limits, the estimate (x̄ or p̂), the standard error, and the critical value. For a proportion with fewer than 5 successes or fewer than 5 failures (x < 5 or n − x < 5), it also shows the caution "Fewer than 5 successes or 5 failures: this interval is rough."; with 0 or n successes the Wald interval has zero width.

Rules

  • The confidence level is from 1% to 99.9999%.
  • The standard deviation must be above 0. A t interval needs n of at least 2; a z interval allows n = 1.
  • For a proportion, the successes are a whole number from 0 to n.
  • α = (100 − C) ÷ 100 is worked out exactly from the typed confidence level. The critical values come from the inverse of the t and normal distributions, good to at least 11 significant digits at any degrees of freedom (checked against mpmath). From 1,000 degrees of freedom up, t* is found by Newton's method on the t tail computed as the integral of the t density.
  • Results show up to 6 decimal places (6 significant digits below 0.0001), with halves rounded up.

Worked examples by hand

The default (x̄ = 68, s = 3, n = 36, 95%, t). SE = 3 ÷ 6 = 0.5. With 35 degrees of freedom t* = 2.030108, so the margin is 2.030108 × 0.5 = 1.015054 and the interval is 66.984946 to 69.015054.

Known σ (OpenStax Example 8.2: x̄ = 68, σ = 3, n = 36, 90%). z* = 1.644854 (tables: 1.645). Margin = 1.644854 × 0.5 = 0.822427; interval 67.177573 to 68.822427 (OpenStax: 67.18 to 68.82).

NIST data (x̄ = 9.261460, s = 0.022789, n = 195, 95%). SE = 0.022789 ÷ √195 = 0.001632; t* = 1.972268 with 194 degrees of freedom; margin = 0.003219; interval 9.258241 to 9.264679 (NIST: 9.258242 to 9.264679, from the unrounded data).

Proportion (OpenStax Example 8.10: 421 of 500, 95%). p̂ = 0.842; SE = √(0.842 × 0.158 ÷ 500) = 0.016312; z* = 1.959964; margin = 0.03197; interval 0.81003 to 0.87397 (OpenStax: 0.810 to 0.874).

Proportion at 90% (OpenStax Example 8.11: 300 of 500). p̂ = 0.6; SE = √(0.24 ÷ 500) = 0.021909; margin = 1.644854 × 0.021909 = 0.036037; interval 0.563963 to 0.636037.

Other questions people ask

What is a confidence interval?

It is a range of values, worked out from a sample, that is likely to contain the true population value. A 95% confidence interval comes from a method that captures the true value in 95% of samples. It does not mean the true value moves: any one interval either contains it or not.

How do I calculate a 95% confidence interval for a mean?

Take the sample mean plus or minus the critical value times the standard error: x̄ ± t × s ÷ √n, with n − 1 degrees of freedom. For x̄ = 68, s = 3, and n = 36, t = 2.0301, the margin is 2.0301 × 3 ÷ 6 = 1.015, and the interval is 66.985 to 69.015.

Should I use t or z?

Use t when you have the sample standard deviation, which is nearly always. Use z only when the population standard deviation σ is known from elsewhere. With large samples the two give almost the same interval.

What is the margin of error?

It is half the width of the interval: the critical value times the standard error. The interval is the estimate plus or minus the margin of error. Polls often report it, such as ±3 points.

Why is a 99% interval wider than a 95% interval?

To be more confident of catching the true value, the interval must cover more ground. The critical z rises from 1.96 at 95% to 2.576 at 99%, so the interval is about 31% wider.

How do I make the interval narrower?

Use a larger sample. The standard error shrinks with √n, so four times as many observations halve the margin of error. The sample size calculator finds the n for a margin you choose.

How is the interval for a proportion found?

With x successes in n, the sample proportion is p̂ = x ÷ n and the interval is p̂ ± z × √(p̂(1 − p̂) ÷ n). For 421 of 500 at 95%, p̂ = 0.842 and the interval is 0.810 to 0.874.