# 95% Confidence Interval Calculator

URL: https://speedycalculator.com/calculators/statistics/95-confidence-interval/
Last updated: 2026-10-30

> The 95% confidence interval calculator finds the range that contains the population mean with 95% confidence from your sample mean, standard deviation and sample size. It uses z* = 1.96 when σ is known and the t distribution otherwise, and also handles proportions.

## Inputs

- Calculate a confidence interval for
- Sample mean (x̄)
- Spread you know
- Standard deviation or standard error
- Sample proportion (p̂) (%)
- Sample size (n)
- Confidence level
- Custom confidence level (%)

## Results

- 95% confidence interval
- Lower bound
- Upper bound
- Margin of error (±)
- Standard error
- Critical value
- Critical value from
- Interval width
- Wilson score interval
- Normal approximation check
- Sampling distribution with the 95% interval shaded

## Formula

```
95% CI = x̄ ± 1.96 × σ ÷ √n (σ known) · x̄ ± t*₀.₉₇₅,ₙ₋₁ × s ÷ √n (s from the sample) · α = 0.05, 2.5% in each tail
```

## Explanation

Ninety-five percent is the default confidence level in most of science, medicine and polling, and it pairs naturally with the 5% significance level used in hypothesis tests: a value outside the 95% interval would be rejected by a two-sided test at α = 0.05. The interval leaves 2.5% in each tail of the normal curve, which gives the famous critical value 1.96 – often rounded to 2, which leads to the handy rule "estimate plus or minus two standard errors".

For a mean, the interval is x̄ ± 1.96 σ/√n when the population standard deviation is known. In practice σ is almost always estimated from the same sample, so the calculator switches to the t distribution, whose 95% critical value is larger for small samples – 2.262 for n = 10, 2.045 for n = 30 – and approaches 1.96 as n grows.

The preset level is 95%, but you can change it, switch between mean and proportion, or enter a standard error directly if a paper or software output already gives you one.

### How to use the 95% Confidence Interval Calculator

- **Calculate a confidence interval for**
- **Sample mean (x̄)**
- **Spread you know**
- **Standard deviation or standard error**
- **Sample proportion (p̂)**
- **Sample size (n)**
- **Confidence level**
- **Custom confidence level**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Divide the standard deviation by √n to get the standard error, multiply by 1.96 (or by t* with n − 1 degrees of freedom when s is a sample statistic) to get the margin of error, and take the mean minus and plus that margin.

### Worked example

Sixty-four commuters report a mean one-way commute of 27.3 minutes with s = 9.8 minutes. The standard error is 9.8 ÷ 8 = 1.225 min. With 63 degrees of freedom t* = 1.9983, so the margin of error is 2.45 min and the **95% confidence interval is 24.85 to 29.75 minutes**. Using z* = 1.96 would give almost the same 24.90 to 29.70 minutes, because with 64 observations t and z barely differ.

### The 95% interval and the 5% test

A two-sided hypothesis test at α = 0.05 rejects H₀: μ = μ₀ exactly when μ₀ lies outside the 95% interval. In the commuting example, a claimed average of 30 minutes would be rejected (30 is above 29.75), while 28 minutes would not.

## Worked example

Inputs — Calculate a confidence interval for: A population mean; Sample mean (x̄): 27.3; Spread you know: Sample standard deviation s (t interval); Standard deviation or standard error: 9.8; Sample size (n): 64; Confidence level: 95%.

Results — **95% confidence interval = 24.852 to 29.748 (27.3 ± 2.448)**; Lower bound = 24.852; Upper bound = 29.748; Margin of error (±) = 2.448; Standard error = 1.225; Critical value = 1.9983.

## Key terms

- **95% confidence interval**: Interval from a procedure that captures the true value in 95% of samples.
- **1.96**: The 97.5th percentile of the standard normal distribution.
- **Two-standard-error rule**: Approximating a 95% interval as estimate ± 2 SE.

## FAQ

**Why is 1.96 used for a 95% confidence interval?**
95% of a standard normal distribution lies between −1.96 and 1.96, leaving 2.5% in each tail.

**How do I calculate a 95% confidence interval by hand?**
Mean ± 1.96 × SD ÷ √n. With mean 27.3 min, SD 9.8 and n = 64: SE = 1.225, margin = 2.40, interval 24.90 to 29.70 (z) or 24.85 to 29.75 with t* = 1.998.

**Is it correct to say there is a 95% chance the true mean is in my interval?**
Strictly no – the true mean is fixed. The 95% describes the method: 95 out of 100 such intervals would capture it.

**What does it mean if two 95% intervals overlap?**
Not necessarily that the difference is non-significant. Compare the groups with a proper two-sample test or an interval for the difference.

**What is the 95% confidence interval for a proportion?**
p̂ ± 1.96 √(p̂(1 − p̂)/n). For 44% of 64 people: 44% ± 12.2 percentage points.

**How large a sample do I need for a ±2 margin?**
n = (1.96 × σ ÷ 2)². With σ = 9.8 that is 92.2, so 93 observations.


## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)
- [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e)
- [Penn State STAT 500 – Applied Statistics](https://online.stat.psu.edu/stat500/)