Confidence Interval One Sample T Test

Sample Mean

The average value observed in your sample.

Sample Standard Deviation

How spread out your sample's values are from the mean.

Sample Size

Total number of people or items in your sample.

Confidence Level (%)

Common choices are 90, 95, or 99.

About This Test

What Is a One-Sample T Confidence Interval?

When you don't know the true population standard deviation — which is almost always — you estimate the population mean from a sample and use the t-distribution instead of the normal distribution to build your confidence interval. It accounts for the extra uncertainty that comes from estimating variability alongside the mean itself.

When to Use It

Estimating the true average score, time, or measurement from a sample when the population standard deviation is unknown
Small to moderate sample sizes, where the t-distribution's wider tails matter
Any continuous outcome where you want a range, not just a point estimate

The Formula

\[ \bar{X} \pm t_{\alpha/2,\,df}\,\frac{s}{\sqrt{n}} \]

Worked Example

A sample of 25 students has a mean exam score of 78 with a standard deviation of 6. What's the 95% confidence interval for the true average score?

\( \bar{X} = 78, \ s = 6, \ n = 25 \)
Degrees of freedom: \( df = n - 1 = 24 \)
Critical value: \( t_{0.025,\,24} \approx 2.064 \)
Margin of error: \( 2.064 \times 6/\sqrt{25} \approx 2.48 \)
95% confidence interval: (75.52, 80.48)

Your Results

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Related Resources:

Understanding the Empirical Rule

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Perfect for students, researchers, and professionals looking to build real statistical skills.

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