# 95% Confidence Interval Calculator

Free 95% confidence interval calculator. Calculate the 95% CI for a population mean from sample mean, standard deviation, and sample size using z or t critical values.

## What this calculates

Calculate a 95% confidence interval for a population mean from your sample statistics. Automatically selects the z-score (for large samples) or t-score (for small samples) method.

## Inputs

- **Sample Mean (x̄)** — The average value of your sample data.
- **Standard Deviation (s or σ)** — min 0.0001 — Sample or population standard deviation.
- **Sample Size (n)** — min 2 — The number of observations in the sample.
- **Method** — options: Auto (z if n >= 30, t otherwise), Z-score (known σ or large n), T-score (unknown σ, small n) — Choose the critical value method. Auto selects based on sample size.

## Outputs

- **Lower Bound** — The lower limit of the 95% confidence interval.
- **Upper Bound** — The upper limit of the 95% confidence interval.
- **Margin of Error** — Half the width of the confidence interval.
- **Standard Error** — Standard error of the mean (s / sqrt(n)).
- **Critical Value Used** — The z* or t* value used for the 95% interval.
- **Method Used** — formatted as text — Whether z-score or t-score was used.
- **Interpretation** — formatted as text — How to interpret the 95% confidence interval.

## Details

A 95% confidence interval is the most commonly used interval in statistical analysis. It gives a range where you can be 95% confident the true population mean falls.

**The Formula:**

CI = x̄ +/- critical value x (s / sqrt(n))

For 95% confidence:
- z* = 1.96 (when population SD is known or n >= 30)
- t* varies by degrees of freedom (when population SD is unknown and n = 30 | Z (approximate) | 1.96 |
| Unknown sigma, n < 30 | T | Depends on df |

**How to Make the Interval Narrower:**

1. Increase sample size (most effective)
2. Reduce variability in measurements
3. Accept a lower confidence level (e.g., 90%)

**Example:** A sample of 40 test scores has mean 72.5 and SD 8.3. The 95% CI is 72.5 +/- 1.96 x (8.3 / sqrt(40)) = 72.5 +/- 2.57, giving the interval [69.93, 75.07].

## Frequently Asked Questions

**Q: Why is 95% the most common confidence level?**

A: The 95% level is a convention that balances precision and confidence. It corresponds to roughly 2 standard deviations, which is easy to remember and calculate. It is also the default in most statistical software and is widely accepted across scientific disciplines. However, there is nothing mathematically special about 95%, and other levels (90%, 99%) are appropriate in different contexts.

**Q: Does a 95% CI mean there is a 95% probability the mean is in this interval?**

A: This is a common misconception. The true mean is a fixed (but unknown) number. It is either in the interval or not. The 95% refers to the procedure: if you repeated the sampling and interval calculation many times, about 95% of the resulting intervals would contain the true mean.

**Q: When should I use the t-distribution instead of z?**

A: Use the t-distribution when the population standard deviation is unknown (you are estimating it from the sample) and the sample size is small (typically n < 30). The t-distribution has heavier tails than the normal, producing wider intervals that account for the extra uncertainty. As n grows, the t-distribution approaches the normal.

**Q: How does sample size affect the confidence interval?**

A: The margin of error is proportional to 1/sqrt(n). Doubling the sample size cuts the margin by a factor of sqrt(2), about 29%. To halve the margin of error, you need to quadruple the sample size. This is why there are diminishing returns as sample size grows.

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Source: https://vastcalc.com/calculators/statistics/95-confidence-interval
Category: Statistics
Last updated: 2026-04-08