Why is a standard deviation required

Standard deviation and variance

Description:

The variance (s2) and standard deviation (s) are by far the most frequently used measures to characterize the spread of a distribution and are a measure of the spread of individual values ​​of a random variable around its mean. The variance is the sum of the squared deviations of all measured values ​​from the arithmetic mean, divided by the number of all measured values. The square root of the variance is the standard deviation, usually referred to as the variance for short. The standard deviation is therefore the positive square root of the variance.

The variance or the standard deviation are mathematically simple and therefore a useful measure for mapping the variability of a distribution:

Strictly speaking, the variance is nothing more than the arithmetic mean of the deviations from that arithmetic mean. The squaring of the deviations is necessary because otherwise the positive and negative deviations would cancel each other out, since the sum of all deviations from the arithmetic mean is always zero.

Tips & other notes:

  • The standard deviation is very large when a distribution is very bipolar. This means that half of the individual values ​​have very low scores, while the other half has very high scores. The other extreme case is when all individual values ​​have exactly the same expression. Then the standard deviation is zero.
  • One should always consider the possibilities of calculating and interpreting the variance or standard deviation. Provided there are a sufficient number of measured values ​​and at least interval-scaled data, the variance or standard deviation can always be calculated as a numerical value. On the contrary, the interpretation of the variance or standard deviation as a measure of the scatter can only be used meaningfully if the type of distribution is known.
  • The standard deviation is easy to interpret if the individual values ​​correspond to a normal distribution: Approx. 68% of all observations are in the "mean minus standard deviation" and "mean plus standard deviation" range, while the "mean minus twice the standard deviation" and "mean plus twice the standard deviation" range Standard deviation "are approx. 95% of the observations.