# Level of measurement

One of the most important properties of variables is the
**level of measurement**, also called scales of measurement. The measurement scale is
important because it determines the permissible arithmetic operations and thus specifies
the possible statistical tests. The higher the level of measurement, the more
comparative statements and arithmetic operations are possible.

The **level of measurement** of a variable can be either **nominal**,
**ordinal** or **metric**. In a nutshell: For nominal variables the values can be
differentiated, for ordinal variables the values can be sorted and for metric scale
level the distances between the values can be calculated. Nominal and ordinal variables
are also called categorical variables.

## Nominal variables

The nominal measurement scale is the **lowest level of measurement** in statistics
and therefore has the lowest information content. Possible values of the variables can
be distinguished, but a meaningful order is not possible. If there are only two
characteristics, such as in the case of gender (male and female), we also speak of
**dichotomous** or **binary** variables.

- Only relations "equal" and "unequal" are possible.
- No logical ranking of categories.
- The order of the answer categories is interchangeable.
- Nominal characteristics with only two expressions are also called "binary" or "dichotomous"

##### Examples:

Gender |
---|

1 = male |

2 = female |

Marital status |
---|

single |

married |

divorced |

widowed |

Preferred newspaper: |
---|

The Washington Post |

The New York Times |

USA Today |

... |

## Ordinal variables

The **ordinal level of measurement** is the next higher level, it also contains
nominal information, only with the difference that a ranking can be formed. The term
ranking scale is hence often used. In these cases, the distances between the values are
not interpretable, so it is not possible to make a statement about the absolute distance
between two values. A classic representative of the ordinal scale are
**school grades**, here a ranking can be formed, but it cannot be said that the
distance between *A* and *B* is the same as the distance between *B* and
*C*.

- Next higher scale of measurement.
- "Equal" and "unequal", as well as "greater" and "smaller", can be determined.
- There is a logical hierarchy of categories.
- The distances between the numerical values are not equal, i.e. cannot be interpreted.

##### Examples:

Frequency of television: |
---|

1 = daily |

2 = several times a week |

3 = less frequently |

4 = never |

The government is doing a good job: |
---|

1 = agree with |

2 = undecided |

3 = disagree with |

## Categorical variables

Variables that have a nominal scale or an ordinal scale are also called categorical
variables. In other words, *categorical* is an umbrella term for variables scaled
nominally and ordinally.

Categorical variables can have a limited and usually fixed number of expressions, e.g.
*country* with "Germany", "Austria", ... or *gender* with "female" and "male".
It is important, however, that it must be a finite number of categories or groups. The
different categories may have a ranking, but must not.

## Metric variables

Metric variables have the **highest possible level of measurement**. With a metric
level of measurement, the characteristic values can be compared and sorted and distances
between the values can be calculated. Examples would be the *weight* and
*age* of subjects.

- Highest level of measurement.
- Creation of rankings possible.
- "Equal" and "unequal", as well as "greater" and "smaller", can also be determined
- Differences and sums can be formed meaningfully.

##### Examples:

Income |
---|

1820 $ |

3200 $ |

800 $ |

... |

Weight |
---|

81 kg |

70 kg |

68 kg |

... |

Age |
---|

18 years |

27 years |

64 years |

... |

Electricity consumption |
---|

520 kWh |

470 kWh |

340 kWh |

... |

### Ratio scale and interval scale

The metric level of measurement can be further subdivided into interval scale and ratio scale. As the name suggests, the values of the ratio scale can be put into a ratio. Thus, a statement like the following can be made: "One value is twice as large as another". For this, an absolute zero must be available as a reference.

##### Example ratio scale:

The time of marathon runners is measured. Here the statement can be made that the fastest runner is twice as fast as the last runner. This is possible because there is an absolute zero point at the beginning of the marathon where all runners start from zero.

##### Example interval scale:

If, however, the stopwatch is forgotten to start at the start of the marathon and only
the differences are measured starting from the fastest runner, the runners cannot be
put in proportion. In this case it can be said how big the interval between the
runners is (e.g. runner *A* is 22 minutes faster than runner *B*), but it
cannot be said that runner *A* ran 20 percent faster than runner *B*.

The classic example is the temperature indication in degrees Celsius and Kelvin. The zero point of the Kelvin temperature scale is absolute zero, therefore it is a ratio scale. At degrees Celsius the absolute zero point is -273.15 °C, therefore the value zero on the degree Celsius scale cannot be assumed as natural zero and therefore it is an interval scale.

## Scale level examples

Scale level | ||
---|---|---|

1 | States of the USA | nominal |

2 | Product rating on a scale from 1 to 5 | ordinal |

3 | religious confession | nominal |

4 | CO2 emissions in the year | metric, ratio scale |

5 | IQ-Score of students | metric, interval scale |

6 | examination grades from 1 to 5 | ordinal |

7 | telephone numbers of respondents | nominal |

8 | care level of a patient | ordinal |

9 | Living space in m2 | metric, ratio scale |

10 | job satisfaction on a scale from 1 to 4 | ordinal |

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