# Confidence interval calculator

> Confidence interval calculator for a mean (t or z), a proportion (Wilson or Wald) or a difference of two means, with the margin of error and both bounds.

Version interactive : https://www.calcopenly.com/fr/statistics/confidence-interval-calculator
Sujet : Calculatrices de statistiques et de probabilités

A confidence interval is a point estimate plus or minus a margin of error, and the margin is a critical value times the standard error. For a mean the standard error is s/√n and the critical value comes from Student's t with n − 1 degrees of freedom, or from the normal distribution when σ is known. For a proportion the Wilson score interval is the default, with the Wald formula p̂ ± z√(p̂(1 − p̂)/n) offered for comparison.

Survey results, lab measurements and A/B tests are reported this way. The default sample of 36 with mean 72.4 and standard deviation 8.1 gives SE = 1.35, t = 2.030 on 35 df and a 95% interval of 69.66 to 75.14, or 72.4 ± 2.74.

The 95% describes the procedure: across many samples, about 95% of intervals built this way contain the true value. Any single interval either contains it or does not.

## Données

- **Interval for** (options : Mean, σ unknown (t), Mean, σ known (z), Proportion, Difference of two means)
- **Enter** (options : Summary statistics, Raw data)
- **Sample 1**
- **Sample 2**
- **Sample 1 mean**
- **Sample standard deviation s₁**
- **Sample size n₁**
- **Known population σ**
- **Sample 2 mean**
- **Sample standard deviation s₂**
- **Sample size n₂**
- **Variances** (options : Unequal (Welch), Equal (pooled))
- **Successes x**
- **Sample size n**
- **Method** (options : Wilson score, Wald (p̂ ± z·SE))
- **Confidence level**

## Résultats

- Margin of error (±) — résultat principal
- Lower bound
- Upper bound
- Point estimate
- Standard error
- Critical value
- Degrees of freedom

## Formule

$$
\bar x \pm t_{1-\alpha/2,\,n-1}\frac{s}{\sqrt n},\qquad \text{Wilson: } \frac{\hat p + \frac{z^2}{2n} \pm z\sqrt{\frac{\hat p(1-\hat p)}{n} + \frac{z^2}{4n^2}}}{1 + z^2/n}
$$

## Exemples détaillés

### Mean with sample SD, n = 36 (defaults)

- Interval for: Mean, σ unknown (t)
- Enter: Summary statistics
- Sample 1 mean: 72.4
- Sample standard deviation s₁: 8.1
- Sample size n₁: 36
- Confidence level: 95%
- **Critical value: 2.030108**
- **Margin of error (±): 2.740646**
- **Lower bound: 69.659354**
- **Upper bound: 75.140646**
- Source de vérification : t₀.₉₇₅,₃₅ = 2.030 (t table); Python bisection on the A&S 26.7.3 closed form gives 2.0301079283; margin = t·8.1/6

### Mean with known σ

- Interval for: Mean, σ known (z)
- Enter: Summary statistics
- Sample 1 mean: 72.4
- Sample size n₁: 36
- Known population σ: 8
- Confidence level: 95%
- **Critical value: 1.959964**
- **Margin of error (±): 2.613285**
- Source de vérification : z₀.₉₇₅ = 1.959964 (z table; Python NormalDist().inv_cdf(0.975)); margin = z·8/6

### Proportion 540/1000, Wilson

- Interval for: Proportion
- Successes x: 540
- Sample size n: 1000
- Method: Wilson score
- Confidence level: 95%
- **Lower bound: 0.509015**
- **Upper bound: 0.570679**
- **Margin of error (±): 0.030832**
- Source de vérification : Wilson score formula evaluated in Python with z = NormalDist().inv_cdf(0.975)

### Proportion 540/1000, Wald

- Interval for: Proportion
- Successes x: 540
- Sample size n: 1000
- Method: Wald (p̂ ± z·SE)
- Confidence level: 95%
- **Lower bound: 0.50911**
- **Upper bound: 0.57089**
- **Margin of error (±): 0.03089**
- Source de vérification : p̂ ± z√(p̂(1−p̂)/n) in Python

### Difference of means, Welch

- Interval for: Difference of two means
- Enter: Summary statistics
- Sample 1 mean: 72.4
- Sample standard deviation s₁: 8.1
- Sample size n₁: 36
- Sample 2 mean: 68.9
- Sample standard deviation s₂: 9.4
- Sample size n₂: 40
- Variances: Unequal (Welch)
- Confidence level: 95%
- **Point estimate: 3.5**
- **Degrees of freedom: 73.8705**
- **Margin of error (±): 4.000864**
- Source de vérification : Welch–Satterthwaite df in Python fractions; t quantile by bisection on Gauss–Legendre quadrature of the t density (pyref.t_sf_numeric)

### Mean from raw data, 90%

- Interval for: Mean, σ unknown (t)
- Enter: Raw data
- Sample 1: 68, 75, 71, 80, 66, 73, 77, 70, 74, 69
- Confidence level: 90%
- **Point estimate: 72.3**
- **Margin of error (±): 2.505254**
- Source de vérification : Python statistics.mean/stdev; t₀.₉₅,₉ = 1.833 (t table), 1.8331129327 by bisection on the A&S closed form

## Questions

### What does a 95% confidence interval mean?

The method captures the true value in 95% of repeated samples. For the default data the interval is 69.66 to 75.14, but there is not a 95% probability that the true mean lies in that particular range: the true mean is fixed, and this interval either contains it or not. Nor does the interval hold 95% of individual values, which spread far wider (SD 8.1); that needs a prediction or tolerance interval.

### How do you calculate the margin of error?

Multiply the critical value by the standard error. For a mean, E = t × s/√n: with s = 8.1 and n = 36, E = 2.030 × 1.35 = 2.74. For a proportion, E ≈ z√(p̂(1 − p̂)/n): 54% of 1,000 respondents gives 1.96 × 0.0158 ≈ 0.031, or ±3.1 percentage points. Because n sits under a square root, quadrupling the sample size halves the margin.

### When should I use a z interval instead of a t interval?

Only when the population standard deviation σ is known, which is rare outside textbook problems and long-running process data. With σ estimated from the sample, use t. Its 95% critical value is larger for small samples, 2.262 for n = 10 against 1.960 for z, and approaches z as n grows: 2.030 at n = 36 and 1.984 at n = 101.

### Why use the Wilson interval for a proportion?

The Wald interval p̂ ± z·SE covers the true proportion less often than stated when n is small or p̂ is near 0 or 1; Brown, Cai and DasGupta (2001) found its coverage can fall far below 95% even with hundreds of observations. Wilson stays close to the stated level and never leaves the range 0 to 1. With 0 successes in 20 trials Wald gives the zero-width interval [0, 0], while Wilson gives 0 to 0.161.

### If a confidence interval for a difference excludes 0, is the result significant?

Yes, for the matching test: a 95% interval for a difference in means excludes 0 exactly when a two-sided test at α = 0.05 using the same method (Welch or pooled) rejects no difference. The default two-group summaries give 3.5 ± 4.0, from −0.5 to 7.5, which includes 0, so the difference is not significant at 5%. The interval also shows how large the effect could plausibly be, which a p-value does not.

### Quelle est la précision de « Confidence interval calculator » ?

La précision dépend de vos données et des hypothèses de la méthode. Le calcul décimal utilise 50 chiffres significatifs, mais les estimations, méthodes numériques et données sources peuvent être moins précises ; l’arrondi affiché ne supprime pas ces limites. Exemples résolus vérifiés à partir de sources indépendantes : 7. Par exemple, « Mean with sample SD, n = 36 (defaults) » est vérifié à l’aide de t₀.₉₇₅,₃₅ = 2.030 (t table); Python bisection on the A&S 26.7.3 closed form gives 2.0301079283; margin = t·8.1/6.

### D’où vient cette méthode ?

NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2.1 Confidence limits for the mean; NIST/SEMATECH e-Handbook, §7.2.4.1 Confidence intervals for a proportion (Wilson and normal approximation); Brown, Cai & DasGupta (2001). Interval estimation for a binomial proportion. Statistical Science 16(2), 101–133.

## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2.1 Confidence limits for the mean](https://www.itl.nist.gov/div898/handbook/prc/section2/prc221.htm)
- [NIST/SEMATECH e-Handbook, §7.2.4.1 Confidence intervals for a proportion (Wilson and normal approximation)](https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm)
- Brown, Cai & DasGupta (2001). Interval estimation for a binomial proportion. Statistical Science 16(2), 101–133