Mean Confidence Interval Calculator
Build a transparent z-based confidence interval around a mean using the standard error and a selected 90%, 95% or 99% confidence level.
Read the interval step by step, and know when to switch to t
Worked example with a known population standard deviation: mean 50, population standard deviation 10, n = 100 at 95%. Standard error = 10 ÷ √100 = 1.00; z* = 1.96; margin = 1.96 × 1.00 = 1.96; interval 48.04 to 51.96. With the same mean and standard deviation but n = 400, the standard error halves to 0.50 and the interval narrows to 49.02 to 50.98.
For mean 72, known population standard deviation 8 and n = 15, this z calculator gives about 67.95 to 76.05, provided the sampling assumptions are suitable. If 8 is instead the sample standard deviation and the population value is unknown, use an appropriate Student t interval with 14 degrees of freedom. This tool does not calculate that t interval.
A 95% confidence level describes the long-run performance of the interval-building method across repeated samples. It is not the same as assigning a 95% probability to the fixed population mean after this one interval has been calculated.
This tool always uses a z critical value. When the population standard deviation is unknown and you substitute a sample estimate, a Student t interval is the usual mean-interval method under its assumptions; a larger sample does not make this calculator a t calculator. Strongly skewed small samples and proportions also need a method suited to the data.
Source: NIST/SEMATECH e-Handbook of Statistical Methods, Confidence Limits for the Mean.
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Formula
interval = mean ± z* × standard deviation ÷ √n
Keep the normal approximation separate from a small-sample t interval
- Mean 50, standard deviation 12 and n = 36 give standard error 2.
- The 95% normal critical value is about 1.959964, giving a margin of 3.919928 and an interval about 46.08 to 53.92.
What the result tells you
This is the stated z-based method. With an unknown population standard deviation, a t procedure using the sample standard deviation is a different method, particularly for smaller samples.
Before you use the result
A 95% confidence procedure concerns repeated-sampling coverage, not a claim that 95% of individual observations lie inside this interval. Bias and dependent observations need separate assessment.
References for this method
Useful for
- Intro statistics homework
- Mean confidence intervals
- Sampling uncertainty checks
Questions
Why does this use a z critical value?
The calculator is explicitly a normal-approximation interval using 90%, 95% or 99% z critical values.
Should I use a t interval instead?
Often yes for smaller samples when population standard deviation is unknown. Follow the method required by your course or analysis.