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 === Weighted Mean === A weighted mean is the best way to average data that have different precisions, if we know or can estimate those precisions. The weighted mean is calculated like an ordinary mean, except that we multiply each measurement by a weighting factor and divide the sum of these products not by N but by the sum of the weights, as follows:

$$\bar X = \sum w_ix_i / \sum w_i $$ where wi is the weighting factor of the ith measurement. If we use equal weights, then this equation reduces to the equation for the ordinary mean. Various techniques for weighting can be used. If each of the values to be averaged is itself a mean with an associated known variance, then the most theoretically satisfying procedure is to weight each value according to the inverse of the variance of the mean:$$w_i = 1/\sigma ^2 \bar x_i = N / \sigma^2_i$$. The weighted variance is: $$\sigma ^2 _\bar x = 1 / \sum (1/\sigma^2 _\bar x) = 1/\sum w_i$$

For example, suppose that three laboratories measure a variable Y and obtain the following:

Then $\overline{X}$ =(0.20•109 + 0.41•105 + 1.02•103)/(0.20+0.41+1.02) = 104.2. The variance of this weighted mean is $$ \sigma ^2 \bar x = 1/(0.20+0.41+1.02) = 0.613$$, and so the standard deviation of the weighted mean is σ $\overline{X}$ = 0.78. Note that the importance or weighting of the measurements from Lab 2 is twice as high as from Lab 1, entirely because Lab 2 was able to achieve a 30% lower standard deviation of measurements than Lab 1 could. Lab 3, which obtained the same standard deviation as Lab 2 but made 2.5 times as many measurements as Lab 2, has 2.5 times the importance or weighting of results.

95% Confidence Limits on Mean
Usually we want to use our measurements to make a quantitative estimate of the true mean M. One valuable way of doing so is to state the 95% confidence limits on the true mean, which for convenience we will call α95. Confidence limits for the true mean M can be calculated as follows:

95% confidence limits: α95=σ $\overline{X}$•t95 $\overline{X}$ -α95 < M < $\overline{X}$+α95

99% confidence limits: α99=σ $\overline{X}$•t99 $\overline{X}$ -α99 < M < $\overline{X}$+α99

Just multiply the standard error of the mean by the ‘t-factor’, finding the t-factor in the table below for the appropriate number of measurements.

By stating the mean (our best estimate of the true mean M) and its 95% confidence, we are saying that there is only a 5% chance that the true mean is outside the range $\overline{X}$±α95. One’s desire to state results with as high a confidence level as possible is countered by the constraint that higher confidence levels encompass much broader ranges of potential data values. For example, our random-number dataset (N=100, σ $\overline{X}$=0.095, $\overline{X}$=0.02) allows us to state with 95% confidence that the true mean lies within the interval -0.17 to 0.21 (i.e., $\overline{X}$ ± α95, or 0.02 ± 0.19). We can state with