# 90% Confidence Interval Calculator

URL: https://speedycalculator.com/calculators/statistics/90-confidence-interval/
Last updated: 2026-10-30

> The 90% confidence interval calculator turns a sample mean, standard deviation and sample size into the range that covers the population mean with 90% confidence, using the critical value z* = 1.645 (or the matching t* for a sample standard deviation). Proportions are supported too.

## 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

- 90% 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 90% interval shaded

## Formula

```
90% CI = x̄ ± 1.645 × σ ÷ √n (σ known) · x̄ ± t*₀.₉₅,ₙ₋₁ × s ÷ √n (s from the sample) · α = 0.10, 5% in each tail
```

## Explanation

A 90% confidence interval trades a little certainty for a narrower, more precise range. It leaves 5% of probability in each tail of the sampling distribution, so its normal critical value is z* = 1.645 rather than the 1.96 used at 95%. The interval is therefore about 16% narrower than the 95% interval from the same data.

Ninety percent intervals are common in quality control, engineering tolerances, pilot studies and business forecasting, where decisions are made often and a slightly higher miss rate is acceptable. They also have a neat link to testing: a two-sided 90% interval corresponds to a one-sided test at the 5% level, so it is the interval used in equivalence testing (the "two one-sided tests" approach) for comparing drugs or processes.

Enter your summary statistics; the confidence level is preset to 90% but you can change it. When the standard deviation comes from the sample itself the calculator uses the t distribution, whose 90% critical value is a little larger than 1.645 for small samples (1.833 with 9 degrees of freedom, 1.690 with 35).

### How to use the 90% 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

Compute the standard error s ÷ √n, multiply it by the 90% critical value (z* = 1.645, or t* with n − 1 degrees of freedom when s is a sample statistic) and add the result to and subtract it from the sample mean.

### Worked example

A filling machine is checked by weighing 36 cereal boxes: mean 502.4 g, s = 4.6 g. The standard error is 4.6 ÷ 6 = 0.767 g. With 35 degrees of freedom the 90% critical value is t* = 1.6896, so the margin of error is 1.295 g and the **90% confidence interval is 501.10 g to 503.70 g**. The whole interval lies above the 500 g label weight. With the normal value 1.645 instead, the interval would be 501.14 to 503.66 g.

### 90% compared with other levels

Levelz*Width relative to 95%80%1.281665%90%1.644984%95%1.9600100%99%2.5758131%

## Worked example

Inputs — Calculate a confidence interval for: A population mean; Sample mean (x̄): 502.4; Spread you know: Sample standard deviation s (t interval); Standard deviation or standard error: 4.6; Sample size (n): 36; Confidence level: 90%.

Results — **90% confidence interval = 501.1047 to 503.6953 (502.4 ± 1.2953)**; Lower bound = 501.1047; Upper bound = 503.6953; Margin of error (±) = 1.2953; Standard error = 0.7667; Critical value = 1.6896.

## Key terms

- **90% confidence**: Ten percent of intervals built this way miss the true value, 5% on each side.
- **z* = 1.645**: The 95th percentile of the standard normal distribution.
- **Equivalence test**: A test that two treatments differ by less than a set margin, often based on a 90% interval.

## FAQ

**What is the z-score for a 90% confidence interval?**
z* = 1.645 (more precisely 1.6449), because 5% of the standard normal distribution lies above it and 5% below −1.645.

**How do I calculate a 90% confidence interval?**
x̄ ± 1.645 × σ ÷ √n. For x̄ = 50, σ = 10, n = 25: 50 ± 1.645 × 2 = 46.71 to 53.29.

**Is a 90% interval narrower than a 95% interval?**
Yes. Its critical value is 1.645 instead of 1.96, so it is about 16% narrower for the same data.

**When should I use 90% confidence?**
In exploratory work, quality control, forecasting and equivalence testing, where a 10% chance of missing the true value is acceptable.

**What t value is used for 90% with a small sample?**
t* with n − 1 degrees of freedom at the 95th percentile, e.g. 1.833 for n = 10 and 1.699 for n = 30.

**How does a 90% interval relate to hypothesis tests?**
A value outside the two-sided 90% interval would be rejected by a two-sided test at α = 0.10, or a one-sided test at α = 0.05 in that direction.


## 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)