Standard deviation

Those Standard deviation <math>\sigma_x</math> is a measure for those Dispersion the values of one Random variable. It is defined as the positive square root from that Variance. The variance is second Moment a distribution, that Expectancy value the first moment.

If a quantity of comparable objects lies forwards (here {<math>x_1, x_2, x_3, \dots, x_N</math>}), for instance, and one wants to make a statement about their uniformity for notes of a classroom test, then the most important yardsticks the number of notes are <math>N</math>, that <math>\bar{x}</math> and the standard deviation.

Becomes in that an evaluation over a quantity of values necessarily, indicates the standard deviation a meaningful measure for the dispersion around the average value.

She is called also middle error or RMS error (root mean square). As indications are ?, s, m.F. or English rms usually. One calls the middle error also Plus/minus (±) and writes it directly behind the central and/or . The latter becomes z. B. with MW or Ø shortened.

Table of contents

An example (with range)

Middle age (for example in one Dance school) = (17.5 ± 1,2) years.
Both values together result in the middle Range, MW ± s = 16.3 to 18.7 years.
It applies in the case more normaldistributed (see Bath tub curve) with one of approx.. 68 % (those of 2? with approx.. 95 %). Therefore above range suggests that

  • 16 % the dance pupil than 16.3 years are younger (and 2 - 3 % under 15.1 years) and
  • 16 % than 18.7 years (and 2 - 3 % over 19.9 years) are older.

Our example has however hardly Normal distribution, because there is supposed from the class participants more than 2.5 % older than 20 years.

  • Values outside of that two to three-way One calls standard deviation . Outliers can a reference up gross errors that its. It can be appropriate for the data in addition, a strongly inclined distribution to reason. On the other hand must approx.. are appropriate for each 20ste measured value outside of the double standard deviation.

Middle error, dispersion and variance

The standard deviation (m. F.) the square root of another dispersion yardstick is, that Variance. The standard deviation has the advantage in relation to the variance that it has the same unit as the original measured values.

If the number of the children in a household is examined, then the unit of the variance is a square child, the unit of the standard deviation however again a child.

Mathematical definition of the standard deviation

<math> 

\sigma_x:= {\frac{1}{N-1 \sqrt} \sum_{i=1}^N{(x_i-\bar{x})^2}}

</math>

Is

Web on the left of

 

  > German to English > de.wikipedia.org (Machine translated into English)