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Statistics

T Confidence Interval Calculator

Estimate a population mean interval from sample data when the population standard deviation is unknown.

Data stays on this deviceConvention explained below
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Check the dataset and sample convention before interpreting results.

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Review the method below for assumptions and conventions.

How to use this tool

  1. Enter sample a, confidence level.
  2. Select Calculate to view the result.
  3. Check the method and assumptions below before using the result.

Input reference and sample values

Use the units and choices shown for each field. Sample values are provided for practice, not as recommendations.

InputFormat and unitsSample
Sample AKeep the structure and separators shown in the sample. Line breaks are retained in this input.8, 10, 9, 12, 11
Confidence levelNumeric value. Unit: %. Minimum: 80. Maximum: 99.9.95

Result for these sample inputs

8.036756838522441 to 11.963243161477559
Sample mean
10
Margin of error
1.96324316
Critical t
2.77644511
Degrees of freedom
4

The confidence level describes the long-run coverage of this procedure, not a probability assigned to this one fixed population mean.

Select Try the example in the form to load these values and calculate them again. This replaces the current form inputs. Compare the result with the method and worked explanation below.

The method, explained

Interval = sample mean ± t critical × sample SD/√n, using df = n−1 and a two-sided confidence level.

A WORKED EXAMPLE

Using sample a = 8, 10, 9, 12, 11, confidence level = 95 %, the result is 8.036756838522441 to 11.963243161477559 probability. Change these example inputs to match your task; use the method above to check each step.

What to keep in mind

Independent sample from an approximately normal population, particularly for small samples. Outliers or selection bias can invalidate the interpretation. A descriptive or probability calculation under the stated assumptions. Check the sample design and model before interpreting results.

Reference: NIST: statistical methods handbook

Common questions

Why use t instead of z?

The sample standard deviation estimates an unknown population standard deviation. The t distribution accounts for this estimation under the model; it approaches the normal distribution as degrees of freedom increase.

Does 95% confidence mean 95% of observations fall inside?

No. This is an interval for the population mean, not a prediction interval or a range containing a chosen percentage of individual observations.

Methodology maintained by ClarityKit. How these tools are built and checked.