# Level of measurement in statistics

## Why are there level of measurement?

The level of measurement is important because it determines what kind of statistics and analysis can be done. The higher the scale level, the more analysis possibilities there are.

Why are there level of measurements?

Why are there level of measurements for example, only frequency distributions can be calculated for nominal variables. For ordinal variables, medians and rank correlations can also be calculated. For interval-scaled variables, mean values and standard deviations can be calculated additionally. For ratio-scaled variables, geometric means and variances can be calculated additionally.

It is important to know the level of measurement of a variable, because some statistics and analyses only make sense at certain scale levels. For example, it would be pointless to calculate a median for a ratio-scaled variable because the median is only meaningful for ordinal or smaller scaled variables. Similarly, it would be meaningless to calculate a variance for a nominal-scaled variable, since variance is only meaningful for interval- or ratio-scaled variables.

## What are level of measurements?

Scale of measurements refers to the way the values of a variable are measured and coded. There are four main scale levels: nominal, ordinal, interval, and ratio.

• Nominal: At the nominal scale level, the values of a variable are divided into categories that have no natural order. Examples of nominal variables are gender, eye color, or country. The nominal scale is not metric.
• Ordinal: At the ordinal scale level, the values of a variable are divided into categories that have a natural order, but the distances between the values are not known. Examples of ordinal variables are educational level (elementary school, middle school, high school) or pain intensity (no pain, mild pain, moderate pain, severe pain). The nominal scale is not metric.
• Interval: at the interval scale level, the values of a variable are divided into categories where the distances between the values are known, but there is no absolute zero. Examples of interval-scaled variables are seasons (spring, summer, fall, winter) or temperature in degrees Celsius. The interval scale is metric.
• Ratio: In ratio-scale (also called ratio-scale), the values of a variable are divided into categories where the distances between the values are known and there is an absolute zero. Examples of ratio-scaled variables are age, weight, or distance. The ratio scale is metric.

## Examples of scale of measurements

### Scale Level: Nominal Variables

An example of a nominal variable would be a person’s eye color. A person’s eye color can be divided into categories such as brown, blue, green, gray. These categories have no natural order and one cannot quantitatively compare eye colors. For example, one could not say that green eyes are “better” or “worse” than brown eyes. So it is a nominal variable.

### Scale level: ordinal variables

An example of an ordinal variable would be a person’s education level. A person’s education level could be divided into categories such as elementary school, middle school, high school. These categories have a natural order, but the distances between the categories are not known. For example, one could say that a person with a college degree is “more” qualified than a person with an elementary school degree, but one could not say that a person with a college degree is twice as qualified as a person with an elementary school degree. So it is an ordinal variable.

### Scale Level: Interval Scaled Variables

An example of an interval-scaled variable would be temperature in degrees Celsius. Temperature can be divided into categories from, say, -20°C to 50°C. These categories have intervals between values that are known (e.g., the interval between 20°C and 30°C is 10°C), but there is no absolute zero. For example, one could say that 10°C is colder than 20°C, but one could not say that 10°C is twice as cold as 5°C because the absolute zero is missing. So it is an interval-scaled variable.

### Scale level: Ratio or ratio scaled variables

An example of a ratio-scaled variable would be a person’s age. A person’s age can be divided into categories from, say, 0 years to 120 years. These categories have distances between values that are known (e.g., the distance between 20 years and 30 years is 10 years), and there is an absolute zero or zero point (0 years). For example, one could say that someone who is 30 years old is twice as old as someone who is 15 years old, since there is an absolute zero. So it is a ratio-scaled variable.

An example of a ratio-scaled variable would be a person’s age. The age of a person can be divided into categories from, say, 0 years to 120 years. These categories have distances between values that are known (e.g., the distance between 20 years and 30 years is 10 years), and there is an absolute zero or zero point (0 years). For example, one could say that someone who is 30 years old is twice as old as someone who is 15 years old, since there is an absolute zero. So it is a ratio-scaled variable.

What is the natural zero point or the absolute zero?

A natural zero point is a value on a scale that serves as the
starting point for measuring change. A natural
zero point is important because it allows changes to be measured with respect to a fixed point.
fixed point. For example, when measuring temperatures, the zero point is defined as the
freezing point of water. This zero point serves as a starting point for
measurement of temperatures and makes it possible to measure changes in relation to this point.
to be measured in relation to this point.

A natural zero point is an important property of data at the
ratio scale level, since it allows changes to be measured with respect to a fixed
point and to measure and perform calculations such as division. Data in the
Interval scale level also have clear distances between values but
no natural zero point. Data in the nominal and ordinal scale levels have
neither a natural ranking nor clear distances between the values, and therefore
therefore no natural zero point. It is important to note that the presence of a natural
zero point should be taken into account when selecting appropriate analysis methods and when
interpretation of results.

## Mathematical operations and scale levels

Arithmetic operations are mathematical operations applied to data to produce specific results. Some examples of arithmetic operations are addition, subtraction, multiplication and division. The way in which arithmetic operations can be applied to data depends on the scale levels of the data. The scale levels determine whether certain arithmetic operations make sense and what kind of results can be achieved.

For example, arithmetic operations such as addition and subtraction can only be meaningfully applied to data on the interval and ratio scale level. This is because these scale levels have a natural ranking and clear spacing between values. In contrast, arithmetic operations such as addition and subtraction do not make sense on data at the nominal and ordinal scale levels because these scale levels have no natural ranking.

It is important to note that the scale of measurement levels of the data should be considered when choosing the appropriate arithmetic operations to produce meaningful results.

## Location parameters, mean values ​​and level of measurements

Means are statistical quantities used to describe the central tendency of data. There are different types of mean such as arithmetic mean, median, and mode. The way that means are calculated and which mean is best for a given data set depends on the scale levels of the data.

•
• For data at the nominal scale level, the mode is the best mean because this scale level does not have a natural ranking.

• For interval and ratio scale level data, both the arithmetic mean and the median can be used, depending on the distribution of the data. The mean is better suited to symmetric distributions, while the median is better suited to skewed distributions.

It is important to note that choosing the appropriate mean for a given data set is important to obtain meaningful results.

## Correlation coefficients with scale levels

Correlation coefficients are statistical measures that measure the strength and direction of a linear relationship between two variables. There are different types of correlation coefficients, such as the Pearson correlation coefficient and the Spearman correlation coefficient.

The way correlation coefficients are calculated and which correlation coefficient is most appropriate for a given data set depends on the scale levels of the data.

For ordinal scale level data, the Spearman correlation coefficient is most appropriate because this correlation coefficient takes into account the rank order of the values.

For interval and ratio scale level data, both the Pearson correlation coefficient and the Spearman correlation coefficient are appropriate, depending on the distribution of the data. The Pearson correlation coefficient is more appropriate for symmetric distributions, while the Spearman correlation coefficient is more appropriate for skewed distributions.

It is important to note that choosing the appropriate correlation coefficient for a given data set is important for obtaining meaningful results.

## Weitere Eigenschaften

Scale levels refer to the way data is collected and measured on a scale. There are four main scale levels: nominal, ordinal, interval, and ratio. Each scale level has certain characteristics and is appropriate for certain types of analyses.

Continuous variables are variables that can take an infinite number of values within a specified range.

• An example of a continuous variable is height, which can be measured in centimeters and can take on any number of values within a specified range (e.g., 1.56 cm, 1.57 cm, etc.).

• Discrete variables are variables that can only take on certain predefined values. An example of a discrete variable is the number of children a person has, which can only take on certain values (e.g. 0, 1, 2, 3, etc.).

• Qualitative variables are variables that describe characteristics or traits that cannot be measured numerically. An example of a qualitative variable is gender (male or female).

• Quantitative variables are variables that can be measured numerically. There are two types of quantitative variables: continuous variables and discrete variables. An example of a quantitative variable is height, which can be measured in centimeters.

### Was ist eine Skala?

A scale is a measurement system used to record and code the values of a variable. There are several types of scales that differ in their characteristics and scale level.
The most common scales are:

– Nominal scale: A nominal scale is a scale used to encode values of variables that have no natural order. Examples of nominal scales are gender (male, female), eye color (brown, blue, green, gray), or country (Germany, France, USA).

– Ordinal scale: An ordinal scale is a scale used to code values of variables that have a natural order, but no known distances between values. Examples of ordinal scales include educational level (elementary school, middle school, high school), pain intensity (no pain, mild pain, moderate pain, severe pain), or satisfaction (very dissatisfied, dissatisfied, satisfied, very satisfied).

– Interval Scale: An interval scale is a scale used to code values of variables where the intervals between values are known, but there is no absolute zero. Examples of interval-scaled variables are seasons (spring, summer, fall, winter) or temperature in degrees Celsius.

– Ratio Scale: A ratio scale is a scale used to encode values of variables where the distances between values are known and there is an absolute zero. Examples of ratio-scaled variables are age, weight, or distance.

### When ordinal scale?

An example would be to record the intensity of a person’s pain. One could use categories such as “no pain,” “mild pain,” “moderate pain,” and “severe pain” to classify the intensity of the pain.

It is important to note that when using an ordinal scale, the distances between categories are unknown. For example, one cannot say that the difference between “mild pain” and “moderate pain” is the same as the difference between “moderate pain” and “severe pain.”

The order of the categories is important, but the actual distances between the categories are not known.

### Are age groups ordinal?

It depends on how the age classes are defined. If the age classes have a natural order and have no known distances between the values, then they are ordinal. Examples of age classes that could be considered ordinal are “children,” “adolescents,” “adults,” and “seniors.”
If age is queried as an integer or if the year of birth is queried, then an absolute zero exists. Here we can assume that the age variable is ratio-scaled.
It is important to note that the scale used to capture age will in many cases depend on the specific needs and goals of the study.

### Are notes metric or ordinal?

Grades are usually ordinal. A grade is an assignment of a number or letter to a performance arranged on a scale from, for example, 0 to 100 or from F to A+. Grades have a natural order because a higher grade is usually considered a better performance than a lower grade.

However, there is no known spacing between grades. For example, one cannot say that the difference between a grade of 80 and a grade of 90 is the same as the difference between a grade of 90 and a grade of 100. The ordering of grades is important, but the actual spacing between grades is not known.

### What are external resources we scale levels?

Wikipedia: hier
Methodenportal der Universität Leipzig: hier
Informationen über Transformationen bei matheguru.com: hier

### Is the age steady or discrete?

Age is usually continuous.

A continuous variable is a variable that can take on an infinite number of values between two given values. Examples of continuous variables are age, weight, and height.

A discrete variable, on the other hand, is a variable that can only take on certain, given values. Examples of discrete variables are number of children, number of pets, and number of cars.

It is important to note that the fact that age is usually considered a continuous variable does not mean that it cannot be considered a discrete variable in certain contexts. For example, age could be categorized into certain age groups (e.g., “under 10 years,” “10-20 years,” “20-30 years”), which would make it considered a discrete variable.