# Dispersion parameter

Standard deviation, variance and range are among the measures of dispersion (measurement
of variability) in descriptive statistics. They are calculated to describe the
**scatter of values** of a sample around a location parameter. Put simply,
**dispersion parameters** are a measure of how much a sample fluctuates around a mean
value.

Measurements of Central Tendency give you the information about the centre of your data, dispersion measures give you the information how much your data is spread around this centre.

## Standard Deviation & Variance

The most common measures of dispersion for metric variables are
**standard deviation** and variance. These two measures relate each characteristic of
a variable to the mean value and thus indicate how far the individual characteristics
are scattered around the mean value.

## What is the Standard Deviation?

The **standard deviation** indicates the spread of a variable around its mean value.
Thus, the standard deviation is the mean deviation (root mean square) of all measured
values from the mean.

The standard deviation thus indicates how much the distribution of values scatters around the mean value. If the individual values scatter strongly around the mean value, a large standard deviation of the variable results. There are two slightly different equations for the calculation. On the one hand, the entire population can be used to calculate the standard deviation. On the other hand it can also be calculated if only one sample is available. If all values of the population are available, the following results are obtained

Often, however, the data of the entire population are not available. Therefore, a sample is usually used to estimate the standard deviation of the population. In this case, the calculation results in

The difference between the two formulas is that one is divided by *n* and the other
by *n-1*. It is customary to use *s* for the
**standard deviation of a sample** and σ for the
**standard deviation of the population**.

## What is the Variance?

Just like the standard deviation, the variance measures the deviation from the mean. For the calculation of the variance, the sum of the squared variances is divided by the number of values.

The variance thus describes the squared average distance from the mean. Because the values are squared, the result has a different unit (the unit squared) than the original values. Therefore, it is difficult to relate the results.

### Variance vs. Standard deviation

So the difference between the dispersion parameter variance and standard deviation is that the standard deviation measures the average distance from the mean and the variance measures the squared average distance from the mean. In other words, the variance is the squared standard deviation and the standard deviation is the root of the variance.

However, this squaring results in a key figure that is difficult to interpret, since the unit does not correspond to the original data. For this reason, it is advisable to always use the standard deviation to describe a sample, as this makes interpretation easier.

## Range

The **range**, also called **span**, is the distance between the minimum and
maximum of a distribution, i.e. the distance between the smallest and the largest value.
For example, if the height of 7 people is queried and the largest value is 1.90m and the
smallest is 1.50m, the span is calculated as 1.90m - 1.50m to 0.4m.

#### Definition Range:

The **range** indicates the distance between the highest and the lowest value in a
sample.

The span, often abbreviated with R, is therefore calculated by

## Quartile

Quartiles divide your data into four parts, as equal as possible. For the calculation quartiles, the data must be sorted from the smallest to the largest value.

- Quartile (Q1): The middle value between the smallest value (minimum) and the median.
- Quartile (Q2): The median of the data, i.e. 50% of the values are smaller and 50% of the values are larger.
- Quartile (Q3): The middle value between the median and the largest value (maximum).

Thus, 25% of all values are below the lower quartile (Q1) and 75% are below the upper quartile (Q3).

## Interquartile Range

In contrast to the range in which 100% of all values lie, one often wants to know the range in which the middle 50% of all values lie. This scattering parameter is called interquartile range (IQR). The upper and lower 25% of the values are therefore not taken into account for the interquartile range.

## Example dispersion parameter

The calculation of range, variance and standard deviation shall now be illustrated by an example. For this purpose, the results of students in a statistics exam (scores) will be used.

Student | Score |
---|---|

1 | 4 |

2 | 5 |

3 | 5 |

4 | 8 |

5 | 9 |

6 | 12 |

7 | 14 |

8 | 16 |

9 | 17 |

10 | 20 |

That's how it works with DATAtab: The calculator for descriptive statistics on DATAtab will give you the range, variance and standard deviation. Copy the above data into the Online Statistics Calculator, click on Descriptive Statistics and select the Score variable. The result will look like this:

Score | |
---|---|

Standard deviation | 5.637 |

Variance | 31.778 |

Range | 16 |

##### Calculate Variance:

Variance is the sum of the squared deviations from the mean value of all values divided by the number of values.

##### Calculate standard deviation:

The standard deviation is the square root of the variance.

##### Calculate range:

The range is obtained by subtracting the smallest value from the largest value.

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