Building an Interval from Summary Statistics

From Summary Statistics to an Interval

A sample mean gives a point estimate of a population mean, but it does not show how much uncertainty comes from sampling. A confidence interval adds a margin of error around that estimate. When the population standard deviation is unknown and the sample standard deviation is used instead, the standard interval for a mean uses the t distribution.

In many reports, the individual observations are not provided. Instead, the report gives a sample mean, a sample standard deviation, a sample size, and a desired confidence level. These summaries are enough to calculate the one-sample t interval. The calculation is compact, but each input has a particular role: the sample mean centers the interval, the standard deviation and sample size determine its standard error, and the confidence level and degrees of freedom determine the t critical value.

Inputs, Assumptions, and the Calculation

The interval estimates the population mean \(\mu\) from a quantitative sample. Its usual model assumes that the observations are independent and come from a population with a common mean and variance. For an exact t procedure, the observations are normally distributed. In practice, a t interval is also often reasonable for a sufficiently large sample when the data do not have severe skewness or extreme outliers. The sample should be obtained in a way that supports inference to the population of interest; a calculation cannot fix a biased sample.

The summary statistics do not reveal every feature of the data. In particular, \(\bar{x}\), \(s\), and \(n\) alone cannot show whether there are extreme outliers or serious departures from the model. Those conditions need to be considered using information about how the data were collected and, when available, the observations themselves. Here, the examples explicitly state that the hypothetical sampling and data shape are adequate for using the t interval.

The calculation follows a fixed sequence. First, calculate \(df=n-1\). Then use the confidence level to determine the two-sided t critical value for those degrees of freedom. Next, calculate \(SE=s/\sqrt{n}\), multiply by \(t^*\) to obtain the margin of error, and subtract and add that margin to \(\bar{x}\). A t table or statistical software can supply \(t^*\); the confidence level alone is not enough, because the critical value also depends on the degrees of freedom.

Why the Formula Uses a t Critical Value

The reason for the t critical value can be seen by starting with the standardized sample mean. If \(X_1,\ldots,X_n\) are independent observations from a normal population with mean \(\mu\) and standard deviation \(\sigma\), then standardizing \(\bar{x}\) using the unknown \(\sigma\) gives a standard normal quantity. In practice, \(\sigma\) is usually unknown, so it is replaced by \(s\). This additional estimation of spread leads to a t statistic with \(n-1\) degrees of freedom:

Choose \(t^*\) so that the central probability under this t distribution is \(C\). Equivalently, the probability of being between \(-t^*\) and \(t^*\) is \(C\). Rearranging this event gives the interval:

The rearrangement multiplies by \(S/\sqrt{n}\), which is positive, and then isolates \(\mu\). In repeated samples under the model, the method produces intervals that contain the fixed population mean a proportion \(C\) of the time. It does not mean that after one interval has been calculated, there is probability \(C\) that this fixed \(\mu\) lies in that particular interval.

Worked Examples with Different Inputs

Reading and Checking a Constructed Interval

A useful check is to verify the interval’s center and half-width. The midpoint of the lower and upper bounds should equal \(\bar{x}\), apart from rounding, and the distance from \(\bar{x}\) to either endpoint should equal \(ME\). If the interval is symmetric around the sample mean but its endpoints do not agree with the reported margin of error, recheck the arithmetic or rounding.

The units of the standard error, margin of error, and interval endpoints are the same as the units of the original measurements. The sample size and degrees of freedom are counts and have no measurement units. Include the measurement units when reporting the interval so the estimated quantity is clear.

Common Mistakes

• Using \(s\) instead of the standard error. The interval is not \(\bar{x}\pm t^*s\). The standard deviation describes variation among observations; the standard error \(s/\sqrt{n}\) describes estimated variation in the sample mean.
• Using the wrong degrees of freedom. For this one-sample t interval, \(df=n-1\), not \(n\). The degrees of freedom are needed to choose the correct critical value.
• Using the wrong tail probability. For a two-sided confidence level \(C\), each tail has probability \((1-C)/2\). A 95% interval therefore uses 0.025 in each tail.
• Changing the method without a reason. When the population standard deviation is unknown, the one-sample t interval uses \(s\) and a t critical value. Do not substitute a normal critical value merely because it is more familiar.
• Rounding intermediate calculations too early. Keep calculator precision through the standard error and margin of error, then round the endpoints for reporting. If rounded intermediate values are displayed, the stated arithmetic should still match those values.
• Treating the confidence level as a probability about this fixed interval. The confidence level describes the long-run success rate of the method under its assumptions, not the probability that a fixed population mean changes or falls inside a particular computed interval.