confidence limits (about the mean)

Symmetric confidence limits ( $± C$ ) about the estimated mean, which cover the population mean with probability $1 − α$ . The quantity $C$ is calculated by the formula:

$C = t p , v S n$

Here $t p , v$ , is the critical value from the $t$ - (or Student) distribution function corresponding to the confidence level $1 − α$ and degrees of freedom $v$ . The symbol $p$ represents the percentile (or percentage point) of the $t$ -distribution. For 1-sided intervals, $p = 1 − α$ ; for 2-sided intervals, $p = 1 − α 2$ . In each case, the confidence level is $1 − α$ . The confidence interval is given as $x _ ± C$ .

Note:

If the population standard deviation $σ$ is known, confidence limits about a single result may be calculated with the formula:

$C = t p , ∞ σ$

The coefficient $t p , ∞$ , is the limiting value of the $t$ -distribution function for $v = ∞$ at confidence level $1 − α$ . This is identical to $z p$ , the $p$ th percentage point of the standard normal variate.

Source:

PAC, 1994, 66, 595 (Nomenclature for the presentation of results of chemical analysis (IUPAC Recommendations 1994)) on page 601