Definition Standard deviation

The standard deviation measures the dispersion or variation of the values of a variable around its mean value (arithmetic mean). Put simply, the standard deviation is the average distance from the mean value of all values in a set of data.

An example:

1,000 people were questioned about their monthly phone bill. The mean value is $40 and the standard deviation 27. Which means that the average distance of all answers (=values) to the mean is $27.

We calculate the standard deviation with the help of the square root of the variance. The symbol of the standard deviation of a random variable is "σ“, the symbol for a sample is "s". The standard deviation is always represented by the same unit of measurement as the variable in question. This makes its interpretation easier, compared to the variance.

A lower standard deviation generally indicates that the measured values of a variable are distributed closer to the mean; a higher standard deviation indicates that the data points a spread more widely.

For normally distributed variables, the rule of thumb is that about 68 percent of all data points are spread from the mean within the standard deviation. Within two standard deviations that would include around 95 percent of all data points. Deviations higher than this average are called outliers.

Another example:

We asked 1,000 people how much money they spend on average for their lunch. The mean result is $4.50, the standard deviation is s=$0.60. This means that the average distance of all data points to the mean is $0.60. The variable has a bell-shaped distribution – it is a normal distribution.

Due to the above-mentioned rule of thumb we can deduce that around 68% of all respondents in the sample spend $3.90-$5.10 on lunch ($4.50 +/- $0.60). Around 95% of all respondents spend $3.30-$5.70 on lunch ($4.50 +/- 2 times $0.60). 

Please note that the definitions in our statistics encyclopedia are simplified explanations of terms. Our goal is to make the definitions accessible for a broad audience; thus it is possible that some definitions do not adhere entirely to scientific standards.