Confidence Interval Calculator (2024)

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 for z-score, 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. More precisely: if the brick maker took lots of samples of 100 bricks and used each sample to compute the confidence interval, then 95% of these intervals would cointain the true average mass of a brick.

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 value, μ, the standard deviation, σ, and the sample size, n (number of measurements taken). Then you can calculate the standard error and then 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 of your z-score is also equal to 0.025 (2.5%).
  4. The area to the right of 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?

To calculate a confidence interval (two-sided), you need to follow these steps:

  1. Let's say the sample size is 100.
  2. Find the mean value of your sample. Assume it's 3.
  3. Determine the standard deviation of the sample. Let's say it's 0.5.
  4. Choose the confidence level. The most common confidence level is 95%.
  5. In the statistical table find the Z(0.95)-score, i.e., the 97.5th quantile of N(0,1) – in our case, it's 1.959.
  6. Compute the standard error as σ/√n = 0.5/√100 = 0.05.
  7. Multiply this value by the z-score to obtain the margin of error: 0.05 × 1.959 = 0.098.
  8. Add and subtract the margin of error from the mean value to obtain the confidence interval. In our case, the confidence interval is between 2.902 and 3.098.

That's it! That was quite of a lot of computations, wasn't it? Luckily, our confidence level calculator can perform all of these calculations on its own.

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 Calculator (1)

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 up, 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 confidence interval calculator.

FAQ

How to interpret confidence intervals?

If you repeatedly draw samples and use each of them to find a bunch of 95% confidence intervals for the population mean, then the true population mean will be contained in about 95% of these confidence intervals. The remaining 5% of intervals will not contain the true population mean.

What is the z-score for 95% confidence interval?

The z-score for a two-sided 95% confidence interval is 1.959, which is the 97.5-th quantile of the standard normal distribution N(0,1).

What is the z-score for 99% confidence interval?

The z-score for a two-sided 99% confidence interval is 2.807, which is the 99.5-th quantile of the standard normal distribution N(0,1).

What will increase the width of a confidence interval?

The width of a confidence interval increases when the margin of error increases, which happens when the:

  • Significance level increases;
  • Sample size decreases; or
  • Sample variance increases.

What will decrease the width of a confidence interval?

The width of a confidence interval decreases when the margin of error decreases, which happens when the:

  • Significance level decreases;
  • Sample size increases; or
  • Sample variance decreases.

The sample mean has no impact on the width of a confidence interval!

Confidence Interval Calculator (2024)

FAQs

How to calculate a 95% confidence interval? ›

Since 95% of values fall within two standard deviations of the mean according to the 68-95-99.7 Rule, simply add and subtract two standard deviations from the mean in order to obtain the 95% confidence interval. Notice that with higher confidence levels the confidence interval gets large so there is less precision.

Is a 90% confidence interval OK? ›

Choosing a confidence interval range is a subjective decision. You could choose literally any confidence interval: 50%, 90%, 99,999%... etc. It is about how much confidence do you want to have. Probably the most commonly used are 95% CI.

How many samples do I need for 95 confidence? ›

To be 95% confident that the true value of the estimate will be within 5 percentage points of 0.5, (that is, between the values of 0.45 and 0.55), the required sample size is 385. This is the number of actual responses needed to achieve the stated level of accuracy.

How to calculate 95% confidence interval with standard error? ›

We use the standard error to calculate the 95% confidence interval. This interval equals, for large samples: [ˉy−1.96×se(y);ˉy+1.96×se(y)], i.e. the mean +/- 1.96 times the standard error.

What is a confidence interval for dummies? ›

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.

What is the 95 confidence interval example? ›

Analysts often use confidence intervals that contain either 95% or 99% of expected observations. Thus, if a point estimate is generated from a statistical model of 10.00 with a 95% confidence interval of 9.50 to 10.50, it means one is 95% confident that the true value falls within that range.

Why do we calculate 95 confidence interval? ›

It's this callous nature that makes 95% confidence intervals so useful. It's a strict gatekeeper that passes statistical signal while filtering a lot of noise out. It dampens false positives in a very measured and unbiased manner. It protects us against experiment owners who are biased judges of their own work.

Is it better to use 95 or 99 confidence interval? ›

A 99% confidence interval will allow you to be more confident that the true value in the population is represented in the interval. However, it gives a wider interval than a 95% confidence interval. For most analyses, it is acceptable to use a 95% confidence interval to extend your results to the general population.

Should I choose 90 or 95 confidence interval? ›

Traditionally 95% confidence interval use is widespread, but in social sciences, 90% confidence interval can also be used, especially in small sample sizes. Obviously, for a used estimation method, the confidence interval will decrease as well as the level of confidence.

What is considered a bad confidence interval? ›

Intervals that are very wide (e.g. 0.50 to 1.10) indicate that we have little knowledge about the effect, and that further information is needed. A 95% confidence interval is often interpreted as indicating a range within which we can be 95% certain that the true effect lies.

What is a good value for 95 confidence interval? ›

The constant for 95 percent confidence intervals is 1.96.

What is the difference between confidence level and confidence interval? ›

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.

What is the 95 confidence rule? ›

The confidence interval can be expressed in terms of probability with respect to a single theoretical (yet to be realized) sample: "There is a 95% probability that the 95% confidence interval calculated from a given future sample will cover the true value of the population parameter." This essentially reframes the " ...

What is the 95% confidence interval for the mean score? ›

The critical value for a 95% confidence interval is 1.96, where (1-0.95)/2 = 0.025. A 95% confidence interval for the unknown mean is ((101.82 - (1.96*0.49)), (101.82 + (1.96*0.49))) = (101.82 - 0.96, 101.82 + 0.96) = (100.86, 102.78).

What is the correct formula for the 95 percent confidence interval for a population mean? ›

The formula for calculating a 95% confidence interval for a population mean is: Confidence Interval for Population Mean: sample mean – E < population mean < sample mean + E Error “E” = (1.96)*(s) / sqrt(n) “s” is the standard deviation and “n” is the sample size.

How do you find the confidence interval for the population mean? ›

ci = μ ± Zα/2*(s/√n)*√FPC, where: FPC = (N-n)/(N-1), Zα/2 is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96), μ is the sample mean, s is the sample standard deviation, n is the sample size and N is the population size.

What is mean and 95% confidence intervals? ›

The means and their standard errors can be treated in a similar fashion. If a series of samples are drawn and the mean of each calculated, 95% of the means would be expected to fall within the range of two standard errors above and two below the mean of these means.

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