Mean Value and Mean Deviation

The most commonly used analysis of a series of measurements $x_i = x_1, \dots, x_n$ is calculating the mean value, the minimum, and the maximum as well as the mean deviation:

• The mean value provides a rough measure of the magnitude of the values in a series of measurements:

  $\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i$

• The minimum/maximum can be determined when calculating the mean value from $x_i$:

  $x_{min} = \min_{i=1}^n x_i$

  and

  $x_{max} = \max_{i=1}^n x_i$

  for a sorted series of measurements, these can be easily determined:

  $x_{min} = x_1$

  and

  $x_{max} = x_n$

• The mean deviation gives an idea how much the individual values deviate from the mean value:

  $\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^n (x_i-\bar{x})^2}$

Unfortunately it does not seem to be common knowledge that there are a number of issues that will probably lead to misinterpreted results without further invstigation:

1. Sensitivity wrt outliers:

The mean value is highly sensitive with respect to high or low outliers. If your series of measurements contains a few extremely high or low outliers or it contains some moderately high or low outliers, the mean value may not represent that which you anticipated. Although there is only a fraction of outliers in the entire series of measurements they are able to greatly affect the mean value. To prevent this you should always check outliers using 0.1 and 0.9 quantiles. Another useful method for visually identifying outliers is building a histogram from the series of measurements.

Please note that this also applied to the mean deviation because it is, in fact, the mean value of the deviation which is a series of measurements as well, although it was calculated from the original series.

1. Confidence:

It is often important to know how good the mean value is, i.e. how close would it be to the mean value if a large or even infinite number of measurements were taken. This is called the confidence. Keep in mind that this does not nullify the effect of outliers.