Understanding Confidence Intervals | Easy Examples & Formulas

Published on August 7, 2020 by Rebecca Bevans. Revised on July 9, 2022.

When you make an estimate in statistics, whether it is a summary statistic or a test statistic, there is always uncertainty around that estimate because the number is based on a sample of the population you are studying.

The confidence interval is the range of values that you expect your estimate to fall between a certain percentage of the time if you run your experiment again or re-sample the population in the same way.

The confidence level is the percentage of times you expect to reproduce an estimate between the upper and lower bounds of the confidence interval, and is set by the alpha value.

What exactly is a confidence interval?
Calculating a confidence interval: what you need to know
Confidence interval for the mean of normally-distributed data
Confidence interval for proportions
Confidence interval for non-normally distributed data
Reporting confidence intervals
Caution when using confidence intervals
Frequently asked questions about confidence intervals

What exactly is a confidence interval?

A confidence interval is the mean of your estimate plus and minus the variation in that estimate. This is the range of values you expect your estimate to fall between if you redo your test, within a certain level of confidence.

Confidence, in statistics, is another way to describe probability. For example, if you construct a confidence interval with a 95% confidence level, you are confident that 95 out of 100 times the estimate will fall between the upper and lower values specified by the confidence interval.

Your desired confidence level is usually one minus the alpha ( a ) value you used in your statistical test:

Confidence level = 1 − a

So if you use an alpha value of p < 0.05 for statistical significance, then your confidence level would be 1 − 0.05 = 0.95, or 95%.

When do you use confidence intervals?

You can calculate confidence intervals for many kinds of statistical estimates, including:

Proportions
Population means
Differences between population means or proportions
Estimates of variation among groups

These are all point estimates, and don’t give any information about the variation around the number. Confidence intervals are useful for communicating the variation around a point estimate.

Calculating a confidence interval: what you need to know

Most statistical programs will include the confidence interval of the estimate when you run a statistical test.

If you want to calculate a confidence interval on your own, you need to know:

The point estimate you are constructing the confidence interval for
The critical values for the test statistic
The standard deviation of the sample
The sample size

Once you know each of these components, you can calculate the confidence interval for your estimate by plugging them into the confidence interval formula that corresponds to your data.

Point estimate

The point estimate of your confidence interval will be whatever statistical estimate you are making (e.g. population mean, the difference between population means, proportions, variation among groups).

Example: Point estimate

In the TV-watching example, the point estimate is the mean number of hours watched: 35.

Finding the critical value

Critical values tell you how many standard deviations away from the mean you need to go in order to reach the desired confidence level for your confidence interval.

There are three steps to find the critical value.

Choose your alpha ( a ) value.

The alpha value is the probability threshold for statistical significance. The most common alpha value is p = 0.05, but 0.1, 0.01, and even 0.001 are sometimes used. It’s best to look at the papers published in your field to decide which alpha value to use.

Decide if you need a one-tailed interval or a two-tailed interval.

You will most likely use a two-tailed interval unless you are doing a one-tailed t-test.

For a two-tailed interval, divide your alpha by two to get the alpha value for the upper and lower tails.

Look up the critical value that corresponds with the alpha value.

If your data follows a normal distribution, or if you have a large sample size (n > 30) that is approximately normally distributed, you can use the z-distribution to find your critical values.

For a z-statistic, some of the most common values are shown in this table:

Confidence level	90%	95%	99%
alpha for one-tailed CI	0.1	0.05	0.01
alpha for two-tailed CI	0.05	0.025	0.005
z-statistic	1.64	1.96	2.57

If you are using a small dataset (n ≤ 30) that is approximately normally distributed, use the t-distribution instead.

The t-distribution follows the same shape as the z-distribution, but corrects for small sample sizes. For the t-distribution, you need to know your degrees of freedom (sample size minus 1).

Check out this set of t tables to find your t-statistic. The author has included the confidence level and p-values for both one-tailed and two-tailed tests to help you find the t-value you need.

For normal distributions, like the t-distribution and z-distribution, the critical value is the same on either side of the mean.

Example: Critical value

In the TV-watching survey, there are more than 30 observations and the data follow an approximately normal distribution (bell curve), so we can use the z-distribution for our test statistics.

For a two-tailed 95% confidence interval, the alpha value is 0.025, and the corresponding critical value is 1.96.

This means that to calculate the upper and lower bounds of the confidence interval, we can take the mean ±1.96 standard deviations from the mean.

Finding the standard deviation

Most statistical software will have a built-in function to calculate your standard deviation, but to find it by hand you can first find your sample variance, then take the square root to get the standard deviation.

Find the sample variance

Sample variance is defined as the sum of squared differences from the mean, also known as the mean-squared-error (MSE):

$s^2 = {\sum}^n _{i=1} {\frac {(Xi - \bar{X})^2}{n-1}}$

To find the MSE, subtract your sample mean from each value in the dataset, square the resulting number, and divide that number by n − 1 (sample size minus 1).

Then add up all of these numbers to get your total sample variance (s²). For larger sample sets, it’s easiest to do this in Excel.

Find the standard deviation.

The standard deviation of your estimate (s) is equal to the square root of the sample variance/sample error (s²):

$s = \sqrt{(s^2)}$

Example: Standard deviation

In the television-watching survey, the variance in the GB estimate is 100, while the variance in the USA estimate is 25. Taking the square root of the variance gives us a sample standard deviation (s) of:

10 for the GB estimate.
5 for the USA estimate.

Sample size

The sample size is the number of observations in your data set.

Example: Sample size

In our survey of Americans and Brits, the sample size is 100 for each group.

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Confidence interval for the mean of normally-distributed data

Normally-distributed data forms a bell shape when plotted on a graph, with the sample mean in the middle and the rest of the data distributed fairly evenly on either side of the mean.

The confidence interval for data which follows a standard normal distribution is:

$CI = \bar{X} \pm Z^* \frac {\sigma}{\sqrt{n}}$

Where:

CI = the confidence interval
X̄ = the population mean
Z* = the critical value of the z-distribution
σ = the population standard deviation
√n = the square root of the population size

The confidence interval for the t-distribution follows the same formula, but replaces the Z* with the t*.

In real life, you never know the true values for the population (unless you can do a complete census). Instead, we replace the population values with the values from our sample data, so the formula becomes:

$CI = \hat{x} \pm Z^* \frac {s}{\sqrt{n}}$

Where:

ˆx = the sample mean
s = the sample standard deviation

\begin{align*} CI &= 35 \pm 1.96 \dfrac{5}{\sqrt{100}} \\ &= 35 \pm 1.96(0.5) \\ &= 35 \pm 0.98 \end{align*} — Example: Calculating the confidence interval

\begin{align*} CI &= 35 \pm 1.96 \dfrac{10}{\sqrt{100}} \\ &= 35 \pm 1.96(1) \\ &= 35 \pm 1.96 \end{align*} — Example: Calculating the confidence interval

Confidence interval for proportions

The confidence interval for a proportion follows the same pattern as the confidence interval for means, but place of the standard deviation you use the sample proportion times one minus the proportion:

$CI = \hat{p} \pm Z^* \sqrt{\dfrac{{\hat{p}(1-\hat{p})}}{n}}$

Where:

ˆp = the proportion in your sample (e.g. the proportion of respondents who said they watched any television at all)
Z*= the critical value of the z-distribution
n = the sample size

Confidence interval for non-normally distributed data

To calculate a confidence interval around the mean of data that is not normally distributed, you have two choices:

You can find a distribution that matches the shape of your data and use that distribution to calculate the confidence interval.
You can perform a transformation on your data to make it fit a normal distribution, and then find the confidence interval for the transformed data.

Performing data transformations is very common in statistics, for example, when data follows a logarithmic curve but we want to use it alongside linear data. You just have to remember to do the reverse transformation on your data when you calculate the upper and lower bounds of the confidence interval.

Reporting confidence intervals

Confidence intervals are sometimes reported in papers, though researchers more often report the standard deviation of their estimate.

If you are asked to report the confidence interval, you should include the upper and lower bounds of the confidence interval.

Example: Reporting a confidence interval

“We found that both the US and Great Britain averaged 35 hours of television watched per week, although there was more variation in the estimate for Great Britain (95% CI = 33.04, 36.96) than for the US (95% CI = 34.02, 35.98).”

One place that confidence intervals are frequently used is in graphs. When showing the differences between groups, or plotting a linear regression, researchers will often include the confidence interval to give a visual representation of the variation around the estimate.

Caution when using confidence intervals

Confidence intervals are sometimes interpreted as saying that the ‘true value’ of your estimate lies within the bounds of the confidence interval.

This is not the case. The confidence interval cannot tell you how likely it is that you found the true value of your statistical estimate because it is based on a sample, not on the whole population.

The confidence interval only tells you what range of values you can expect to find if you re-do your sampling or run your experiment again in the exact same way.

The more accurate your sampling plan, or the more realistic your experiment, the greater the chance that your confidence interval includes the true value of your estimate. But this accuracy is determined by your research methods, not by the statistics you do after you have collected the data!

Frequently asked questions about confidence intervals

What is the difference between a confidence interval and a confidence level?: The confidence level is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way.

The confidence interval consists of the upper and lower bounds of the estimate you expect to find at a given level of confidence.

For example, if you are estimating a 95% confidence interval around the mean proportion of female babies born every year based on a random sample of babies, you might find an upper bound of 0.56 and a lower bound of 0.48. These are the upper and lower bounds of the confidence interval. The confidence level is 95%.
How do you calculate a confidence interval?: To calculate the confidence interval, you need to know:

The point estimate you are constructing the confidence interval for

The critical values for the test statistic

The standard deviation of the sample

The sample size

Then you can plug these components into the confidence interval formula that corresponds to your data. The formula depends on the type of estimate (e.g. a mean or a proportion) and on the distribution of your data.
What is a standard normal distribution?: The standard normal distribution, also called the z-distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.

Any normal distribution can be converted into the standard normal distribution by turning the individual values into z-scores. In a z-distribution, z-scores tell you how many standard deviations away from the mean each value lies.
What are z-scores and t-scores?: The z-score and t-score (aka z-value and t-value) show how many standard deviations away from the mean of the distribution you are, assuming your data follow a z-distribution or a t-distribution.

These scores are used in statistical tests to show how far from the mean of the predicted distribution your statistical estimate is. If your test produces a z-score of 2.5, this means that your estimate is 2.5 standard deviations from the predicted mean.

The predicted mean and distribution of your estimate are generated by the null hypothesis of the statistical test you are using. The more standard deviations away from the predicted mean your estimate is, the less likely it is that the estimate could have occurred under the null hypothesis.
What is a critical value?: A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval, or which defines the threshold of statistical significance in a statistical test. It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data (i.e. 90%, 95%, 99%).

If you are constructing a 95% confidence interval and are using a threshold of statistical significance of p = 0.05, then your critical value will be identical in both cases.
What does it mean if my confidence interval includes zero?: If your confidence interval for a difference between groups includes zero, that means that if you run your experiment again you have a good chance of finding no difference between groups.

If your confidence interval for a correlation or regression includes zero, that means that if you run your experiment again there is a good chance of finding no correlation in your data.

In both of these cases, you will also find a high p-value when you run your statistical test, meaning that your results could have occurred under the null hypothesis of no relationship between variables or no difference between groups.
How do I calculate a confidence interval if my data are not normally distributed?: If you want to calculate a confidence interval around the mean of data that is not normally distributed, you have two choices:

Find a distribution that matches the shape of your data and use that distribution to calculate the confidence interval.

Perform a transformation on your data to make it fit a normal distribution, and then find the confidence interval for the transformed data.

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Bevans, R. (July 9, 2022). Understanding Confidence Intervals | Easy Examples & Formulas. Scribbr. Retrieved October 19, 2022, from https://www.scribbr.com/statistics/confidence-interval/

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Rebecca Bevans

Rebecca is working on her PhD in soil ecology and spends her free time writing. She's very happy to be able to nerd out about statistics with all of you.

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Understanding Confidence Intervals | Easy Examples & Formulas

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