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Confidence Limits for the Mean

NIST/SEMATECH Section 1.3.5.2 Confidence Limits for the Mean

What It Is

Confidence limits define an interval estimate around a sample statistic that contains the true population parameter with a specified level of confidence. For the mean, this interval is constructed using the t-distribution to account for the uncertainty from estimating the population standard deviation.

When to Use It

Use confidence limits to quantify the precision of a sample mean estimate. They provide a range of plausible values for the population mean, making them essential in quality control, process characterization, and scientific reporting. The width of the interval reflects both the sample variability and the sample size, guiding decisions about whether more data are needed.

How to Interpret

A 95% confidence interval means that if the sampling procedure were repeated many times, approximately 95% of the resulting intervals would contain the true population mean. A wider interval indicates greater uncertainty about the parameter, typically due to high variability or small sample size. If the confidence interval for the mean does not contain a specified target value, the hypothesis that the population mean equals that target can be rejected at that confidence level. The interval width is inversely proportional to the square root of the sample size, so quadrupling the sample size halves the interval width. Reporting confidence intervals alongside point estimates is considered best practice because it conveys both the estimate and its precision.

Assumptions and Limitations

The confidence interval for the mean assumes the data are independently and identically distributed from a population with a finite variance. For small samples, the underlying population should be approximately normal; for large samples (n > 30), the Central Limit Theorem ensures approximate validity regardless of the parent distribution.

Reference: NIST/SEMATECH e-Handbook, Section 1.3.5.2

Formulas

Confidence Interval for the Mean

xˉ±t1−α/2, ν⋅sn\bar{x} \pm t_{1-\alpha/2,\,\nu} \cdot \frac{s}{\sqrt{n}}

The interval is centered on the sample mean and extends by the critical t-value multiplied by the standard error. Here s is the sample standard deviation and nu = n - 1 degrees of freedom.

Standard Error of the Mean

SE=sn\text{SE} = \frac{s}{\sqrt{n}}

The standard error quantifies the variability of the sample mean across repeated sampling. It decreases as the sample size increases.

Degrees of Freedom

ν=n−1\nu = n - 1

The degrees of freedom for the t-distribution equal the sample size minus one, reflecting the loss of one degree of freedom from estimating the mean.