confidence interval equation
Sample mean (x̅)
Standard deviation (s)
Sample size (n)
Confidence level
or Z-score (Z)
confidence interval figure
95% of samples contain the population mean (μ) within the confidence interval x̅ ± E.
Confidence interval
Lower bound
Upper bound
Margin of error (E)

Confidence Interval Calculator

By Bogna Szyk, Jasmine J Mah and Tibor Pal, PhD candidate
Last updated: Oct 22, 2020

This confidence interval calculator is a tool that will help you find the confidence interval for a sample, provided you give the mean, standard deviation and sample size. You can use it with any arbitrary confidence level. If you want to know what exactly the confidence interval is and how to calculate it, or are looking for the 95 confidence interval formula with no margin of error, this article is bound to help you.

What is the confidence interval?

The definition says that "a confidence interval is the range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter." But what does that mean in reality?

Imagine that a brick maker is concerned whether the mass of bricks he manufactures is in line with specifications. He has measured the average mass of a sample of 100 bricks to be equal to 3 kg. He has also found the 95% confidence interval to be between 2.85 kg and 3.15 kg. It means that he can be 95% sure that the average mass of all the bricks he manufactures will lie between 2.85 kg and 3.15 kg.

Of course, you don't always want to be exactly 95% sure. You might want to be 99% certain, or maybe it is enough for you that the confidence interval is correct in 90% of cases. This percentage is called the confidence level.

95 confidence interval formula

Calculating the confidence interval requires you to know three parameters of your sample: the mean (average) value, μ, the standard deviation, σ, and the sample size, n (number of measurements taken). Then you can calculate the margin of error according to the following formulas:

standard error = σ/√n

margin of error = standard error * Z(0.95)

where Z(0.95) is the z-score corresponding to the confidence level of 95%. If you are using a different confidence level, you need to calculate the appropriate z-score instead of this value. But don't fret, our z-score calculator will make this easy for you!

How to find the Z(0.95) value? It is the value of z-score where the two-tailed confidence level is equal to 95%. It means that if you draw a normal distribution curve, the area between the two z-scores will be equal to 0.95 (out of 1).

If you want to calculate this value using a z-score table, this is what you need to do:

  1. Decide on your confidence level. Let's assume it is 95%.
  2. Calculate what is the probability that your result won't be in the confidence interval. This value is equal to 100% - 95% = 5%.
  3. Take a look at the normal distribution curve. 95% is the area in the middle. That means that the area to the left of the opposite of your z-score is equal to 0.025 (2.5%) and the area to the right to your z-score is also equal to 0.025 (2.5%).
  4. The area to the right to your z-score is exactly the same as the p-value of your z-score. You can use the z-score tables to find the z-score that corresponds to 0.025 p-value. In this case, it is 1.959.

Once you have calculated the Z(0.95) value, you can simply input this value into the equation above to get the margin of error. Now, the only thing left to do is to find the lower and upper bound of the confidence interval:

lower bound = mean - margin of error

upper bound = mean + margin of error

How to calculate confidence interval: an example

Luckily, our confidence level calculator can perform all of these calculations on its own. All you need to do is follow these steps to find the confidence interval.

  1. Find the mean value of your sample. Let's assume that we are solving the brick example and the mean mass of a brick is 3 kg.
  2. Determine the standard deviation of the sample. Let's say it is equal to 0.5 kg.
  3. Write down the sample size. Let's say your calculations were based on a sample of 100 bricks.
  4. Determine your confidence level. You can leave it at a default value of 95%.
  5. Our confidence interval calculator automatically finds the Z(0.95) score equal to 1.959.
  6. Find the standard error equal to σ/√n = 0.5/√100 = 0.05.
  7. Multiply this value by the Z(0.95) score to obtain the margin of error: 0.05 * 1.959 = 0.098.
  8. Now all you have to do is to add or subtract the margin of error from the mean value to obtain the confidence interval. In this case, the confidence interval is between 2.902 kg and 3.098 kg.

Confidence interval application in time series analysis

One peculiar way of making use of confidence interval is the time series analysis, where the sample data set represents a sequence of observations in a specific time frame.

A frequent subject of such a study is whether a change in one variable affects another variable in question.

To be more specific, let's consider the following general question that often raises economists' interest: "How does a change in the interest rate affect the price level?"

There are several ways to approach this issue, which involves complex theoretical and empirical analysis, that is far beyond the scope of this text. Besides, there are multiple techniques to estimate and apply confident intervals, but still, through this example, we can represent the functionality of confidence interval in a more complicated problem.

Confidence interval - visual representation

The horizontal axis represents the number of months after one unit change in the interest rate, the vertical axis shows the response of price level. Note that this example with the figure is hypothetical and displayed here only for illustrative purposes.

The above graph is a visual representation of an estimation output of an econometric model, a so-called Impulse Response Function, that shows a reaction of a variable at the event of a change in the other variable. The red dashed lines below and above the blue line represent a 95% confidence interval, or in another name, confidence band, which defines a region of most probable results. More specifically, it shows that after a change in interest rate, it is only the second month when a significant response occurs at the price level.

To sum, we hope that with the above examples and short description, you get more insight into the purpose of the confidence interval, and you gain the confidence to use our calculator.

Bogna Szyk, Jasmine J Mah and Tibor Pal, PhD candidate
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