Confidence Interval Calculator

Enter the confidence level, sample size, sample mean, and standard deviation to find the confidence interval using the calculator below.

Results:

Confidence Interval:
 
Lower Bound:
 
Upper Bound:
 
Margin of Error:
 

Steps to Solve

Confidence Interval Formula

CI = x̄ ± z × s√n

Step One: Find the Z-Score

z-Score = ?

Step Two: Find the Margin of Error

MOE = z × s√n

Step Three: Find the Confidence Interval

CI = x̄ ± MOE



How to Find the Confidence Interval

In statistics, a confidence interval is a range of values from a sample that are likely to contain an unknown population parameter value.

Because surveys are often done on a sample of a larger population, there will inevitably be some amount of error or difference between the sample and the full population, which is known as the margin of error.

The confidence interval is used to measure the uncertainty in a sample variable. It’s essentially the range of the sample data that is expected to reflect a variable within the full population.

graph illustrating the confidence interval on a normal distribution with the upper and lower bound

For instance, if you have a sample and know the mean of a variable in the sample, you can say with some level of certainty that the mean of the population will fall within a range. The confidence interval is that range.

Because nothing is ever certain, statisticians also use a confidence level, which is the probability that a variable in the full population falls within the confidence interval range.

So if you’re using a 95% confidence level, then you might say that there is a 95% chance that the population mean will fall within a confidence interval range. The higher the confidence level, the wider the confidence interval will be, and vice versa.

The confidence interval is a range between a lower and upper bound. These bounds are equal to the sample mean minus the margin of error and the sample mean plus the margin of error, respectively.

You can use our margin of error calculator to find the margin of error for your sample.

Confidence Interval Formula

To find the confidence interval, you can use the following formula:

CI = x̄ ± z × σ√n

Thus, the confidence interval is equal to the sample mean plus or minus the z-score for the confidence level z times the sample standard deviation σ divided by the square root of the sample size n.

You can find the z-score for your desired confidence level in the table below. You can learn more about z-scores on our z-score calculator.

For example, let’s find the confidence interval for a sample with a mean of 14, a standard deviation of 2.5, and a size of 1500. Using a confidence level of 95%, we can use a z-score of 1.96 (see the table below).

Start by assigning values to the variables needed for the formula.

CI = 14 ± 1.96 × 2.5√1500
CI = 14 ± 1.96 × 2.538.73
CI = 14 ± 1.96 × 0.065
CI = 14 ± 0.1274

Thus, the confidence interval for this sample is equal to 14 plus or minus 0.1274.

You can find the lower and upper bounds to fully define the range for the confidence interval.

lower bound = 14 – 0.1274 = 13.8726
upper bound = 14 + 0.1274 = 14.1274

So, the confidence interval for this sample is between 13.8726 and 14.1274.

You can use our mean calculator and standard deviation calculator to find these values for your sample.

Z-Scores

The table below shows the z-scores for various confidence levels.

Table showing the z-scores for various confidence levels.
Confidence Level Z-Score
70% 1.04
75% 1.15
80% 1.28
85% 1.44
90% 1.645
91% 1.7
92% 1.75
93% 1.81
94% 1.88
95% 1.96
96% 2.05
97% 2.17
98% 2.33
99% 2.576
99.9% 3.29