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Statistics & Data Science · pattern F, Dataset

A confidence interval for the mean, from your data.

Paste your sample and the tool returns the interval for the population mean, using the t distribution when n is small and reporting every intermediate quantity.

Inputs

Your text

Confidence level%90, 95, 98 or 99.Value of interestUsed by the z-score tools; ignored here.

Sample or population

Divide by n − 1 for a sample drawn from a wider population; by n only when the data is the entire population.

The formula used

x̄ ± t × s ÷ √n

With fewer than thirty observations the t distribution with n − 1 degrees of freedom is used; above that the normal approximation is applied.

Confidence interval

65.745 to 77.255

95% interval for the mean, n = 10, using the t distribution with 9 degrees of freedom (n < 30)

Mean: 71.500
Standard deviation: 8.045
Standard error: 2.5441
Critical value: 2.262
Margin of error: 5.755
n: 10

Your data — accent bars sit beyond one deviation

59.0085.00

What 95% confidence actually claims.

It is a statement about the procedure, not about this interval: if you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true mean. It does not say there is a 95% probability the mean lies inside the interval you happen to have. That distinction is worth stating correctly in a report, because markers look for it.

Questions about confidence interval

Why does a small sample need the t distribution?: Because the standard deviation is estimated from the same small sample, which adds uncertainty. The t distribution has heavier tails to absorb it, and converges on the normal as n grows.
How do I narrow an interval?: Collect more data — width falls with the square root of n, so quartering the width needs sixteen times the sample. Lowering the confidence level narrows it too, at the cost of being right less often.
Does the data need to be normal?: For small samples, approximately. Above about thirty observations the central limit theorem makes the interval for the mean fairly robust, though heavy skew still matters.

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Statistical output here is for coursework and learning. Reported p-values and intervals assume the conditions stated on the page; nothing here checks whether those conditions hold for your data.