What is a ratio scale of measurement?

A ratio scale is a quantitative scale where there is a true zero and equal intervals between neighboring points. Unlike on an interval scale, a zero on a ratio scale means there is a total absence of the variable you are measuring.

Length, area, and population are examples of ratio scales.

Levels of measurement

The ratio level is the highest of four hierarchical levels of measurement. The levels, or scales, of measurement indicate how precisely data is recorded. The higher the level, the more complex the measurement is.

The ratio level contains all of the features of the other 3 levels. At the ratio level, values can be categorized, ordered, have equal intervals and take on a true zero.

The 4 levels of measurement: nominal, ordinal, interval, and ratio

While nominal and ordinal variables are categorical variables, interval and ratio variables are  quantitative variables. Many more statistical tests can be performed on quantitative than categorical data.

What is a true zero?

On a ratio scale, a zero means there’s a total absence of the variable of interest. For example, the number of children in a household or years of work experience are ratio variables: A respondent can have no children in their household or zero years of work experience.

With a true zero in your scale, you can calculate ratios of values. For example, you can say that 4 children is twice as many as 2 children in a household. Similarly, 8 years is double 4 years of experience.

Some variables, such as temperature, can be measured on different scales. While Celsius and Fahrenheit are interval scales, Kelvin is a ratio scale.

In all 3 scales, there are equal intervals between neighboring points. However, unlike the Celsius and Fahrenheit scales where zero is just another temperature value, the Kelvin scale has a true zero (0 K) where nothing can be colder.

That means that you can only calculate ratios of temperatures in the Kelvin scale. Although 40° is twice as many degrees as 20°, it isn’t twice as hot on the Celsius or Fahrenheit scales. However, in the Kelvin scale, 40 K is twice as hot as 20 K because there is a true zero at the starting point of this scale.

A true zero makes it possible to multiply, divide or square root values. Collecting data on a ratio level is always preferable to the other levels because it is the most precise.

Examples of ratio scales

Many variables in the natural and social sciences are measured using ratio scales.

Like interval variables, ratio variables can be discrete or continuous.

A discrete variable is expressed only in countable numbers (e.g., integers) while a continuous variable can potentially take on an infinite number of values.

Ratio variable Discrete or continuous?
Number of vehicles owned in the last 10 years Discrete
Number of people in a household Discrete
Number of students who identify as religious Discrete
Reaction time in a computer task Continuous
Years of work experience Continuous
Speed in miles per hour Continuous

Ratio data analysis

After you’ve collected ratio data, you can gather descriptive and inferential statistics. Almost all statistical tests can be performed on ratio data because all mathematical operations are permissible.

Ratio data example
You collect data on the commute duration of employees in a large city. The data is continuous and in minutes.

To summarize your data, you can collect the following descriptive statistics:

Distribution

You can get an overview of the frequency of different values in a table and visualize their distribution in a graph.

To organize your data, enter it into a grouped frequency distribution table. Create groups with equal intervals on the left hand column and enter the number of scores that fall within each interval into the right hand column.

Commute time (minutes) Frequency
1 – 10 3
11 – 20 1
21 – 30 9
31 – 40 19
41 – 50 13
51 – 60 3
61 – 70 2
71 – 80 1
To visualize your data, plot it on a frequency distribution polygon.  Plot the groupings on the x-axis and the frequencies on the y-axis, and join the midpoint of each grouping using lines.A frequency distribution polygon can be used to visualize the distribution of your data.

Central tendency

From your graph, you can see that your data is fairly normally distributed. Since there is no skew, to find where most of your values lie, you can use all 3 common measures of central tendency: the mode, median and mean.

The mode is the most frequently repeating value in your data set. For continuous variables, there’s often no mode because there is an infinite number of possible values that your data can take. In our example, there is no mode because each value only appears once.
The median is the value exactly in the middle of your data set. To find the middle position, take the value at position (n+1)/2 where n is the total number of values.

(n+1)/2 = (51+1)/2 = 26

The median is in the 26th position, which has a value of 36.4 minutes.

The mean (M) uses all values to give you a single number for the central tendency of your data. Use the formula of ⅀x/n. Sum up all values (⅀x) and divide the sum by n.

x = 1883.5
n = 52
M = ⅀x/n = 1883.5/52 = 36.9

 

The mean is usually considered the best measure of central tendency when you have normally distributed quantitative data. That’s because it uses every single value in your data set for the computation, unlike the mode or the median.

Variability

The range, standard deviation and variance describe how spread your data is. The range is the easiest to compute; the standard deviation and variance are more complicated, but also more informative.

The coefficient of variation is a measure of spread that only applies to ratio variables.

To find the range, subtract the lowest value from the highest value in your data set. Our maximum commute time is 72.5 minutes, and our minimum is 7 minutes.

Range =  72.5 – 7 = 65.5

The standard deviation (s) is the average amount of variability in your dataset. It tells you how far each score lies from the mean on average. Most computer programs will easily calculate the standard deviation for you. If you want to do it by hand, use these steps.

s = 13.34

The variance (s2) is the average squared deviation from the mean. A deviation from the mean is the difference between a value in your data set and the mean. To find the variance, square the standard deviation.

s2 = 178.04

Also known as relative standard deviation (RSD), the coefficient of variation (CV) is a standardized measure of dispersion. It tells you how variable your data is relative to the mean. It can only be used with ratio levels of measurement because it is a fraction. The coefficient of variation (CV) is the standard deviation divided by the mean.

CV = s/M

CV = 36.9/13.34 = 0.36

The coefficient of variation is 0.36 or 36%. This tells you that your data has a large spread because the standard deviation is 36% of the mean.

Statistical tests

Now that you have an overview of your data, you can select appropriate tests for statistical inferences. With a normal distribution of ratio data, parametric tests are best for testing hypotheses.

Parametric tests are more powerful than non-parametric tests and let you make stronger conclusions regarding your data. However, your data must meet several requirements for parametric tests to apply.

The following parametric tests are some of the most common ones applied to test hypotheses about ratio data.

Aim Samples or variables Test Example
Comparison of means 2 samples T-test Is there a difference in the average commute time of employees in Boston and Los Angeles?
Comparison of means 3 or more samples ANOVA Is there a difference in the average commute times of employees in North America, Asia and Europe?
Correlation 2 variables Pearson’s r How are commute times and income levels related?
Regression 2 variables Simple linear regression Does income predict average commute time?

Frequently asked questions about ratio data

What are the four levels of measurement?

Levels of measurement tell you how precisely variables are recorded. There are 4 levels of measurement, which can be ranked from low to high:

  • Nominal: the data can only be categorized.
  • Ordinal: the data can be categorized and ranked.
  • Interval: the data can be categorized and ranked, and evenly spaced.
  • Ratio: the data can be categorized, ranked, evenly spaced and has a natural zero.
What is the difference between interval and ratio data?

While interval and ratio data can both be categorized, ranked, and have equal spacing between adjacent values, only ratio scales have a true zero.

For example, temperature in Celsius or Fahrenheit is at an interval scale because zero is not the lowest possible temperature. In the Kelvin scale, a ratio scale, zero represents a total lack of thermal energy.

What is the difference between discrete and continuous variables?

Discrete and continuous variables are two types of quantitative variables:

  • Discrete variables represent counts (e.g. the number of objects in a collection).
  • Continuous variables represent measurable amounts (e.g. water volume or weight).
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Pritha Bhandari

Pritha has an academic background in English, psychology and cognitive neuroscience. As an interdisciplinary researcher, she enjoys writing articles explaining tricky research concepts for students and academics.

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