# What is the mean and the standard deviation

## Definition of standard deviation

The standard deviation is a measure of the spread of the values of a characteristic around its mean value (arithmetic mean). Put simply, the standard deviation is the average distance of all measured values of a characteristic from the average.

Example: 1,000 people were asked what their monthly mobile phone bill was. The mean value is 40 euros and the standard deviation is 27. This means that the average distance of all answers to the mean value is 27 euros.

The standard deviation is calculated using the square root of the variance. The symbol of the standard deviation for a random variable is indicated with “σ”, that for a sample with “s”. The standard deviation always has the same unit of measurement as the characteristic to be examined. This makes it easier to interpret compared to the variance. A smaller standard deviation usually indicates that the measured values of a characteristic are closer to the mean value, a larger standard deviation indicates a greater scatter. For normally distributed characteristics, the rule of thumb applies that around 68 percent of all response values lie within the distance of one standard deviation up and down from the mean. Around 95 percent of all values are within two standard deviations. Larger deviations are referred to as outliers.

Example: 1,000 people were asked how much money they spend on average when they have lunch. The mean value is 4.50 euros, the standard deviation is s = 0.60. This means that the average distance of all answers to the mean is 0.60 euros. The characteristic has a bell-like distribution - it is normally distributed. Based on the rule of thumb described, it can be deduced that around 68 percent of all respondents in the sample spend between 3.90 euros and 5.10 euros at lunchtime (4.50 +/- 0.60 euros). Around 95 percent spend between 3.30 euros and 5.70 euros (4.50 +/- 2 times 0.60 euros).

Please note that the individual definitions in our statistics lexicon are simplified explanations. The aim here is to bring the individual terms closer to the broadest possible user group. In this respect, it is possible that individual definitions do not fully correspond to scientific standards.

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