In the case of a series of n measurements of the same quantity, the dispersion of the results obtained around the mean is characterised by the average quadratic deviation σ which is given by the following formula:
$$ \sigma = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^n {{\left( {{x_i} – \bar x} \right)}^2}}}{{n – 1}}} $$

In the case of a large numbers of measurement, the result of measurement number i (i = 1,2,3,…,n) is designated by $$ \bar x $$ and the average $$ \bar x = \frac{{\mathop \sum \nolimits_{i = 1}^n {x_i}}}{n} $$.

Increasing the number of measurements decreases the significance of chance errors and an average of the results can be accepted as the result of a series of measurements.

The “GUM” (Guide to Uncertainty in Measurements) however recommends weighting this standard deviation by a coefficient, known as the “Student’s”, symbol s, if the number of measurements is less than or equal to 5.

  • For 3 measurements, s=9.2
  • For 4 measurements, s=6.6
  • For 5 measurements, s=5.5

Thus, the standard deviation formula becomes:
$$ \sigma = \frac{s}{{3\sqrt n }}\sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^n {{\left( {{x_i} – \bar x} \right)}^2}}}{{n – 1}}} $$

Be careful not to confuse the standard deviation with the spread which is the difference between the minimum and maximum values of the quantities measured: $$ e = {x_{max}} – {x_{min}} $$.