Confidence Interval Calculator

Confidence interval calculator

The confidence interval calculator computes both the confidence interval of a mean and the confidence interval of the standard deviation. The calculation uses the normal distribution or the student's t distribution for the confidence interval of the mean, and the chi-squared distribution for the confidence interval of the standard deviation.

When working with sample data, we are familiar with the sample's statistics, but the true value of the population parameter remains unknown. To calculate the confidence interval, we treat the population parameter as a random variable and subsequently determine the confidence interval.

The confidence level is the required certainty level that the parameter's true value will be in the confidence interval. Researchers commonly use a confidence level of 0.95 hence the default is set as a 95 confidence interval calculator, but you may change the confidence level.
This confidence interval calculator reports the results in APA style.
The online confidence interval calculator displays the formulas and provides a step-by-step calculation.

How to use the confidence interval calculator?

1. Data is:
   Average, SD , n - enter the average, the standard deviation, and the sample size (n).
   Raw data - enter the delimited data, separated by comma, space or enter. In this case the tool will calculate the average, the standard deviation, and the sample size.
2. Outliers: - this option is relevant only when you enter raw data, using Tukey's fences method with k equal 1.5
   included - the calculator will calculate the outliers but will include them in the calculation.
   Excluded - The calculator will exclude the outliers before calculating the average and the standard deviation.
You should remove outliers only if you identify them as invalid observations!
3. Confidence Level (CL) - The certainty level that the true value of the estimated parameter will be in the confidence interval.
4. Do you know the population SD (σ)? - this option relevant for the mean confidence interval.
   Yes - when you know the population standard deviation, the calculation uses the normal distribution with the population standard deviation.
   No - when you don't know the population standard deviation, the calculation uses the t-distribution with the sample standard deviation.
   
5. Population standard deviation (σ) - from a preliminary knowledge, usually from other researches.
6. Rounding - when the number is bigger than one the calculator rounds to the required decimal places, but when the number is smaller than one, it rounds to the required significant figures For example, when you choose 2, it will format 88.1234 to 88.12 , and 0.001234 to 0.0012.
7. Step by step - show the calculation steps

What is a confidence interval?

The confidence interval is the range in which the population parameter is most likely to be found.
The degree of certainty for which it is likely to be within that range is called the confidence level.
When you collect sample data, you can not know the exact value of the parameter.

What is a confidence level?

The confidence level is the required degree of certainty that the population parameter will be in the confidence interval. This is the probability that the calculated confidence interval contains the population parameter.
Note: researchers commonly use a confidence level of 0.95.

What is a 95 confidence interval?

The 95% confidence interval is a proposition as follows: if one were to calculate the confidence interval for an infinite number of samples, then 95% of the calculated ranges will contain the population parameter.

Mean confidence interval calculator

When we know the population's standard deviation (σ), use the normal distribution. The average's (x̄) distribution is normal (Mean, σ/√n). Otherwise, use the sample size standard deviation with the t distribution with n-1 degrees of freedom. The (x̄-Mean)/(S/√n) distribution is T.

What is the mean confidence interval formula?

When we know the population standard deviation.

x̄ ± Zα/2 *	σ
	√n

When we don't know the population standard deviation and use the sample standard deviation.

x̄ ± Tα/2(df) *	S
	√n

Standard deviation confidence interval calculator

The statistic (n-1)S²/σ² distributes chi-squared with n-1 degrees of freedom.

What is the standard deviation confidence interval formula?

(n - 1)S2	≤ σ2 ≤	(n - 1)S2
χ1-α/2(df)		χα/2(df)

Where:

x̄ - the sample average.
σ - the population standard deviation, usually you don't know the population standard deviation, you may get it from other researches as a sample standard deviation with a larger sample size, in this case, you may assume it is the population standard deviation.
S - the sample standard deviation.
n - the sample size (the number of observations).
CL -confidence level
α = 1 - CL.
Z_α/2 - the z-score based on the standard normal distribution, p(z < Z_α/2) = α/2.
T_α/2 - the t-score based on the t distribution, p(t < T_α/2) = α/2.
df - degrees of freedom. df = n -1.