Errors and mean values

In most experimental studies, there are two sources of variation or error. The first arises from the intrinsic variability in individual measurements or observations that are all obtained in the same way. The second arises from errors in the apparatus or procedures which affect all of the measurements in a similar way. This type of variation, which leads to a consistent deviation from the correct or true measurement is termed systematic error. It is important that these two sources of error are clearly distinguished. The first (intrinsic variability) can be analysed statistically, and replicated measurements will give an estimate of the precision of the measurement or procedure. These errors are termed statistical errors. This does not mean, however, that the result obtained is necessarily an accurate or true one. Pipettes may have become uncalibrated, or solutions have deteriorated there are many ways in which systematic errors can lead to false values, and attention to calibration and independent verification of apparatus or procedures is essential to minimise the risk of this happening. Systematic errors are potentially much more problematic in biochemical work, since these can easily occur without the experimenter being aware of them. 

The magnitude of statistical errors are revealed by the use of the correct statistical procedures. 

A third source of uncertainty is the occurrence of rare or unique events in the measurement, such as an incorrect reading by the observer, or a chance disturbance in the equipment. Such errors can often produce large deviations from the other readings, and are hence termed outliers . There are statistical tests for recognising such data points, but the occurrence of outliers can be a real problem in statistical data analysis. 

If a number of observations is made of a variable, the results will be clustered around a value that we seek to define as an average. There are many sorts of average, and it is not always obvious which one should be used. 

In this equation, all of the readings have the same importance or weight, and this is therefore the unweighted arithmetic mean. This is a special case of a more generally defined weighted mean given by the equation  

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