Understanding Confidence Intervals  Easy Examples & Formulas
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 resample 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.
Table of contents
 What exactly is a confidence interval?
 Calculating a confidence interval: what you need to know
 Confidence interval for the mean of normallydistributed data
 Confidence interval for proportions
 Confidence interval for nonnormally 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).
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 onetailed interval or a twotailed interval.
You will most likely use a twotailed interval unless you are doing a onetailed ttest.
For a twotailed 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 zdistribution to find your critical values.
For a zstatistic, some of the most common values are shown in this table:
Confidence level  90%  95%  99% 

alpha for onetailed CI  0.1  0.05  0.01 
alpha for twotailed CI  0.05  0.025  0.005 
zstatistic  1.64  1.96  2.57 
If you are using a small dataset (n ≤ 30) that is approximately normally distributed, use the tdistribution instead.
The tdistribution follows the same shape as the zdistribution, but corrects for small sample sizes. For the tdistribution, you need to know your degrees of freedom (sample size minus 1).
Check out this set of t tables to find your tstatistic. The author has included the confidence level and pvalues for both onetailed and twotailed tests to help you find the tvalue you need.
For normal distributions, like the tdistribution and zdistribution, the critical value is the same on either side of the mean.
Finding the standard deviation
Most statistical software will have a builtin 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 meansquarederror (MSE):
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 (s^{2}). 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 (s^{2}):
Sample size
The sample size is the number of observations in your data set.
Confidence interval for the mean of normallydistributed data
Normallydistributed 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:
Where:
 CI = the confidence interval
 X̄ = the population mean
 Z* = the critical value of the zdistribution
 σ = the population standard deviation
 √n = the square root of the population size
The confidence interval for the tdistribution 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:
Where:
 ˆx = the sample mean
 s = the sample standard deviation
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:
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 zdistribution
 n = the sample size
Confidence interval for nonnormally 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.
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 redo 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 is the actual 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%.
This means that 95% of the time, you can expect your estimate to fall between 0.56 and 0.48.
 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 zdistribution, 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 zscores. In a zdistribution, zscores tell you how many standard deviations away from the mean each value lies.
 What are zscores and tscores?

The zscore and tscore (aka zvalue and tvalue) show how many standard deviations away from the mean of the distribution you are, assuming your data follow a zdistribution or a tdistribution.
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 zscore 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 pvalue 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.