# Confidence intervals explained: what 95% really means and how sample size shrinks it

> A poll says 52% support a plan, give or take 3 points. A confidence interval is that 'give or take', done properly. This guide shows the formula, a worked example, how level and sample size change the width, and the common misreading of '95% confident'.

Published 2026-10-09 by Toolcloak. 3 minute read. Original: https://toolcloak.pages.dev/articles/confidence-intervals-explained-what-95-percent-really-means-and-how-sample-size-shrinks-it
Related free tool: [Confidence Interval Calculator (Mean)](https://toolcloak.pages.dev/tools/confidence-interval-calculator)

You measured a sample and got one number: an average, or a percentage. The true value for the whole population is almost certainly a little different. A **confidence interval** is a range built around your number that says how far off it could reasonably be. It is the honest way to report an estimate.

> **In short**
> - For a mean: **interval = mean ± (critical value × standard error)**, where standard error = standard deviation ÷ √n.
> - At **95%** the critical value is about **1.96** (z) for large samples. At 90% it is 1.645, at 99% it is 2.576.
> - To **halve** the width you need **four times** the sample. Width shrinks with the square root of n.
> - '95% confident' describes the **method**: 95% of intervals built this way would capture the true value. It is not a 95% chance for one interval.

## A worked example

A company samples 36 delivery times. The average is 72 minutes and the standard deviation is 12 minutes. What range is the true average likely to sit in?

**95% interval for the average delivery time**

1. **Standard error.** 12 ÷ √36 = 12 ÷ 6 = **2 minutes**.
2. **Margin of error.** 1.96 × 2 = **3.92 minutes**.
3. **Interval.** 72 − 3.92 to 72 + 3.92 = **68.08 to 75.92 minutes**.
4. **Say it.** We are 95% confident the true average is between about 68 and 76 minutes.

*The sample mean sits in the middle. The margin of error is how far the interval reaches on each side.*

Try it: [Confidence Interval Calculator (Mean)](https://toolcloak.pages.dev/tools/confidence-interval-calculator). Enter mean, standard deviation and sample size to get the interval at your chosen level.

## Using the t value for small samples

The 1.96 above assumes you know the population standard deviation, or have a big sample. With a sample of 36 and an estimated spread, textbooks use the **t distribution**, which is a little wider to allow for that extra uncertainty. For 35 degrees of freedom the 95% critical value is about 2.03.

| Method | Critical value | Margin | Interval |
| --- | --- | --- | --- |
| z (1.96) | 1.960 | 3.92 | 68.08 to 75.92 |
| t, 35 df (2.030) | 2.030 | 4.06 | 67.94 to 76.06 |

*The difference is small at n = 36 and tiny by n = 100. For n under about 30 it matters more.*

**95% critical value by sample size**

|  | Critical value |
| --- | --- |
| n = 5 (t) | 2.776 |
| n = 10 (t) | 2.262 |
| n = 30 (t) | 2.045 |
| n = 100 (t) | 1.984 |
| Large n (z) | 1.960 |

*The t value falls toward 1.96 as the sample grows, because a bigger sample estimates the spread better.*

## What changes the width

Three things control how wide the interval is: how variable the data is, how many values you have, and how sure you want to be.

**Margin of error at 95% for different sample sizes (SD = 12, minutes)**

|  | Value |
| --- | --- |
| n = 9 | 7.84 |
| n = 36 | 3.92 |
| n = 100 | 2.35 |
| n = 400 | 1.18 |

*Going from 9 to 36 values (four times as many) cuts the margin in half. From 100 to 400 does the same again.*

| Confidence level | Critical value (z) | Margin at n = 36 | Interval around 72 |
| --- | --- | --- | --- |
| 90% | 1.645 | 3.29 | 68.71 to 75.29 |
| 95% | 1.960 | 3.92 | 68.08 to 75.92 |
| 99% | 2.576 | 5.15 | 66.85 to 77.15 |

*More confidence means a wider net. You cannot be more certain without being less precise.*

## Polls and percentages

For a proportion the formula changes slightly. The standard error is √(p × (1 − p) ÷ n). A poll of 1,000 people finds 52% in favour.

**The poll of 1,000 people**

|  | Value | Note |
| --- | --- | --- |
| Sample result | 52% | 520 of 1,000 |
| Margin of error | ±3.1 | 1.96 × √(0.52 × 0.48 ÷ 1000) |
| Lower end | 48.9% | Plausible low |
| Upper end | 55.1% | Plausible high |

*The interval includes 50%, so this poll cannot tell you whether most people are for or against. A 52% 'lead' is within the noise.*

Headline margins use the worst case, p = 50%, which gives the widest interval. That is where the familiar 'plus or minus 3 points for 1,000 people' comes from.

**Poll margin of error as the sample grows (percentage points)**

| People surveyed | Margin of error at 95% |
| --- | --- |
| 100 | 9.8 |
| 400 | 4.9 |
| 1,000 | 3.1 |
| 2,500 | 2 |
| 10,000 | 1 |

*Surveying ten times as many people shrinks the margin to only about a third of its size, and each extra person helps less. That is why most polls stop around 1,000.*

## What '95% confident' really means

**Two readings of the same interval**

| The correct reading | The common misreading |
| --- | --- |
| About the method | A 95% chance for this one |
| If we repeated the study many times | Treats the true value as a moving target |
| About 95% of the intervals would contain the true value | The true value is fixed, the interval varies |
| Any single interval either does or does not | A subtle but real difference |

*In everyday writing, 'we are 95% confident the true value lies between 68 and 76' is fine. Just do not read it as a probability about the true value itself.*

**100 intervals from 100 repeated samples at 95% confidence**

| Group | Count |
| --- | --- |
| Contain the true value | 95 |
| Miss it | 5 |

*Five in a hundred intervals will miss, and you cannot tell which ones from a single sample.*

## Assumptions behind the formula

- **Random sampling.** If the sample is biased, the interval is centred on the wrong value no matter how narrow it is.
- **Independent values.** Ten measurements on the same person are not ten independent people.
- **Roughly bell-shaped means.** With a large sample this holds even for skewed data. For tiny, skewed samples, be careful.
- **Spread is estimated.** The standard deviation comes from your sample, so it carries its own uncertainty.

> **Narrow does not mean right:** A very large biased sample gives a very narrow interval around the wrong answer. The interval measures random error only, not mistakes in how the data was collected.

## Choosing a sample size

You can work backwards. For a proportion, the sample needed for margin E at 95% is about 0.9604 ÷ E² (using p = 50%). For a margin of 3 points, that is 0.9604 ÷ 0.0009 ≈ **1,067 people**. For 5 points it is about 385.

1. **Decide the margin you can live with.**
2. **Pick a confidence level**, usually 95%.
3. **Estimate the spread**, from a pilot study or by using the worst case for proportions.
4. **Calculate n**, then add a margin for non-responses.
5. **Report the interval, not just the mean.**

## Frequently asked questions

### What is a confidence interval?

It is a range around a sample estimate, such as an average or percentage, that is likely to contain the true population value. It is calculated from the sample's size and variability.

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

Compute the standard error as the standard deviation divided by the square root of n, multiply by 1.96 (or the t value for small samples) and add and subtract the result from the mean.

### What does 95% confidence mean?

It means that if the study were repeated many times, about 95% of the intervals built this way would contain the true value. It does not mean there is a 95% chance this one does.

### How does sample size affect the interval?

A larger sample gives a narrower interval. The width shrinks with the square root of n, so you need four times the sample to halve the margin.

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

Use t when the population standard deviation is unknown and estimated from the sample, especially for samples under about 30. For large samples the two give nearly the same result.

### What is margin of error?

It is the amount added to and subtracted from the estimate to form the interval. In a poll of 1,000 people the margin is about 3 percentage points.