This page is a simple utility to calculate precision of measurements. This exercise is particularly important in laboratory research, where the 95% confidence interval for a measurement and the coefficient of variation provides quality assessment of the method of measurement.

A common method of assessment is to use duplicated measurements (2 measurements of each sample or subject) and this uses the paired difference as an estimate of precision.

The method presented on this page allows for 2 or more measurements being made on each sample or subject, and uses the analysis of variance to partition variances between and within samples. The residual (error or within sample) variance is used to estimate the Standard Deviation of precision.

In some methods, the 95% confidence interval is calculated by multiplying the Standard Deviation by 1.96, which is the z value for p=0.025 (2.5% each side of the normal distribution curve). This assumes the measurements are made on a population of infinite size.

The calculation on this page uses the two sided t value for p=0.05 and residual degrees of freedom. This produces a wider confidence interval than using 1.96, but is more realistic. t will approach 1.96 when sample size approaches infinity. If confidence interval based on population assumption is requred, the value can be divided by t and multiply by 1.96.

The model of analysis of variance assumes equal number of measurements are made on each sample or subject. Although minor variations in the number of measurements do not distort the result much, increasing variations in the number of measurements in each sample, particularly if the number of samples are small, may produce unreliable results.

Notes on default example data:

The default example data contains 5 samples, each with 3 measurements. Please note that this small size is intended to make visualization of the data easy. In a real exercise, the number of sample will need to be very much larger, and the number of measurements in each sample may vary

Please also note that the overall mean is not necessarily the same as the mean of the averaged sample values. They are only the same if all samples have the same number of measurements (as in the default example, 3 in every sample). When the number of measurements in the samples varies, the samples with more measurements weigh more towards the calculation of the overall mean

Reference for One Way Analysis of Variance

Armitage P. Statistical Methods in Medical Research (1971). Blackwell Scientific Publications. Oxford. P.189-207.

The data is a two column numerical data    - Column 1 is sample number    - Column 2 is the value of the measurement from that sample