# 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 gender (male and female), we
also speak of **dichotomous** or **binary** variables.

- Only relations "equal", "unequal" 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 |
---|

1 = single |

2 = married |

3 = divorced |

4 = widowed |

You read the newspaper: |
---|

1 = The Washington Post |

2 = The New York Times |

2 = USA Today |

... |

## Ordinal variables

The **ordinal level of measurement** is the next higher level, it contains
nominal information, only with the difference that a ranking can be formed,
therefore the term ranking scale is often used. In these cases, however, 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" or "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 = declines |

## 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 can have a ranking, but do not have to.

## 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", "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 scalar 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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