Continuous and Discrete Levels of Measurement

When carrying out any kind of data collection or analysis, it's essential to understand the nature of the data you're dealing with. Within your dataset, you'll have different variables—and these variables can be recorded to varying degrees of precision. This is what's known as the level of measurement.

There are four main levels of measurement: nominal, ordinal, interval, and ratio. In this guide, we'll explain exactly what is meant by levels of measurement within the realm of data and statistics—and why it matters. We'll then explore the four levels of measurement in detail, providing some examples of each.

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1. What are levels of measurement in data and statistics?
2. Why are levels of measurement important?
3. What are the four levels of measurement?
4. Nominal level
5. Ordinal level
6. Interval level
7. Ratio level
8. Key takeaways

Let's get started!

Here's Dr. Humera Noor Minhas giving a general overview of the field of data, which helps to understand levels of measurement!

1. What are levels of measurement in data and statistics?

When gathering data, you collect different types of information, depending on what you hope to investigate or find out. For example, if you wanted to analyze the spending habits of people living in Tokyo, you might send out a survey to 500 people asking questions about their income, their exact location, their age, and how much they spend on various products and services. These are your variables: data that can be measured and recorded, and whose values will differ from one individual to the next.

When we talk about levels of measurement, we're talking about how each variable is measured, and the mathematical nature of the values assigned to each variable. This, in turn, determines what type of analysis can be carried out.

Let's imagine you want to gather data relating to people's income. There are various levels of measurement you could use for this variable. You could ask people to provide an exact figure, or you could ask them to select their answer from a variety of ranges—for example: (a) 10-19k, (b) 20-29k, (c) 30-39k, and so on. You could ask them to simply categorize their income as "high," "medium," or "low."

Can you see how these levels vary in their precision? If you ask participants for an exact figure, you can calculate just how much the incomes vary across your entire dataset (for example). However, if you only have classifications of "high," "medium," and "low," you can't see exactly how much one participant earns compared to another. You also have no concept of what salary counts as "high" and what counts as "low"—these classifications have no numerical value. Thus, the latter is a less precise level of measurement.

So, in a nutshell: Level of measurement refers to how precisely a variable has been measured.

2. Why are levels of measurement important?

Level of measurement is important as it determines the type of statistical analysis you can carry out. As a result, it affects both the nature and the depth of insights you're able to glean from your data. Certain statistical tests can only be performed where more precise levels of measurement have been used, so it's essential to plan in advance how you'll gather and measure your data.

3. What are the four levels of measurement? Nominal, ordinal, interval, and ratio scales explained

There are four levels of measurement (or scales) to be aware of: nominal, ordinal, interval, and ratio. Each scale builds upon the last, meaning that each scale not only "ticks the same boxes" as the previous scale, but also adds another level of precision.

So:

• The nominal scale simply categorizes variables according to qualitative labels (or names). These labels and groupings don't have any order or hierarchy to them, nor do they convey any numerical value. For example, the variable "hair color" could be measured on a nominal scale according to the following categories: blonde hair, brown hair, gray hair, and so on. Learn more in this complete guide to nominal data.
• The ordinal scale also categorizes variables into labeled groups, and these categories have an order or hierarchy to them. For example, you could measure the variable "income" on an ordinal scale as follows: low income, medium income, high income. Another example could be level of education, classified as follows: high school, master's degree, doctorate. These are still qualitative labels (as with the nominal scale), but you can see that they follow a hierarchical order. Learn more in this guide to ordinal data.
• The interval scale is a numerical scale which labels and orders variables, with a known, evenly spaced interval between each of the values. An oft-cited example of interval data is temperature in Fahrenheit, where the difference between 10 and 20 degrees Fahrenheit is exactly the same as the difference between, say, 50 and 60 degrees Fahrenheit. Learn more about interval data in this guide.
• The ratio scale is exactly the same as the interval scale, with one key difference: The ratio scale has what's known as a "true zero." A good example of ratio data is weight in kilograms. If something weighs zero kilograms, it truly weighs nothing—compared to temperature (interval data), where a value of zero degrees doesn't mean there is "no temperature," it simply means it's extremely cold! You'll find a full guide to ratio data here.

Another way to think about levels of measurement is in terms of the relationship between the values assigned to a given variable. With the nominal scale, there is no relationship between the values; there is no relationship between the categories "blonde hair" and "black hair" when looking at hair color, for example. The ratio scale, on the other hand, is very telling about the relationship between variable values.

For example, if your variable is "number of clients" (which constitutes ratio data), you know that a value of four clients is double the value of two clients. As such, you can get a much more accurate and precise understanding of the relationship between the values in mathematical terms.

In that sense, there is an implied hierarchy to the four levels of measurement. Analysis of nominal and ordinal data tends to be less sensitive, while interval and ratio scales lend themselves to more complex statistical analysis. With that in mind, it's generally preferable to work with interval and ratio data.

Now we've introduced the four levels of measurement, let's take a look at each level in more detail.

4. Nominal level

What is the nominal level?

The nominal level is the first level of measurement, and the simplest. It classifies and labels variables qualitatively. In other words, it divides them into named groups without any quantitative meaning. It's important to note that, even where numbers are used to label different categories, these numbers don't have any numerical value.

For example: If you collected data on hair color, when entering your data into a spreadsheet, you might use the number 1 to represent blonde hair, the number 2 to represent gray hair, and so on. These numbers are just labels; they don't convey any mathematical meaning.

When using the nominal scale, bear in mind that there is no order to the groups you use to classify your variable. One category is not higher than, better than, or greater than another.

Examples of nominal data

Some examples of nominal data include:

• Eye color (e.g. blue, brown, green)
• Nationality (e.g. German, Cameroonian, Lebanese)
• Personality type (e.g. introvert, extrovert, ambivert)
• Employment status (e.g. unemployed, part-time, retired)
• Political party voted for in the last election (e.g. party X, party Y, party Z)
• Type of smartphone owned (e.g. iPhone, Samsung, Google Pixel)

As you can see, nominal data describes certain attributes or characteristics.

How to analyze nominal data

1. Descriptive statistics for nominal data

Descriptive statistics describe or summarize the characteristics of your dataset. Two useful descriptive statistics for nominal data are:

• Frequency distribution
• Mode

A frequency distribution table (e.g. a pivot table) summarizes how many responses there were for each category—for example, how many people selected "brown hair," how many selected "blonde," and so on. You can also use percentages rather than count, in which case your table will show you what percentage of the overall sample has what color hair.

Here's what a pivot table might look like for our hair color example, with both count and percentages:

The mode is a measure of central tendency, and it's the value that appears most frequently in your dataset. So, if 38 out of 129 questionnaire respondents have gray hair, and that's the highest count, that's your mode.

2. Statistical tests for analyzing nominal data

You can analyze nominal data using certain non-parametric statistical tests, namely:

• The Chi-square goodness of fit test if you're looking at just one variable. This allows you to assess whether the sample data you've collected is representative of the whole population. It does this by evaluating the extent to which your observations deviate from what you expected or hypothesized.
• The Chi-square test of independence is used to explore the relationship between two nominal variables. It does this by comparing the frequency of each category of one nominal variable across the categories of the second nominal variable, allowing you to see if there's some kind of correlation.

5. Ordinal level

What is the ordinal level?

The ordinal level of measurement groups variables into categories, just like the nominal scale, but also conveys the order of the variables. For example, rating how much pain you're in on a scale of 1-5, or categorizing your income as high, medium, or low.

As you can see from these examples, there is a natural hierarchy to the categories—but we don't know what the quantitative difference or distance is between each of the categories. We don't know how much respondent A earns in the "high income" category compared to respondent B in the "medium income" category; nor is it possible to tell how much more painful a rating of 3 is compared to a rating of 1.

So, although the ordinal level of measurement is more precise than the nominal scale, it's still a qualitative measure and thus not as precise or informative as the interval and ratio scales.

Examples of ordinal data

Some examples of ordinal data include:

• Academic grades (A, B, C, and so on)
• Happiness on a scale of 1-10 (this is what's known as a Likert scale)
• Satisfaction (extremely satisfied, quite satisfied, slightly dissatisfied, extremely dissatisfied)
• Income (high, medium, or low). Note that income is not an ordinal variable by default; it depends on how you choose to measure it.
• Level of education completed (high school, bachelor's degree, master's degree)
• Seniority level at work (junior, mid-level, senior)

As is clear from our examples, the ordinal scale naturally ranks variables into a meaningful order or hierarchy.

How to analyze ordinal data

1. Descriptive statistics for ordinal data

The following descriptive statistics can be used to summarize your ordinal data:

• Frequency distribution
• The mode and/or the median
• The range

Frequency distribution describes, usually in table format, how your ordinal data are distributed, with values expressed as either a count or a percentage. Let's imagine you've conducted a survey asking people how painful they found the experience of getting a tattoo (on a scale of 1-5). Here's how your frequency distribution table might look:

The mode and the median are measures of central tendency (the other possible measure of central tendency is the mean, but this doesn't apply to ordinal data). The mode is, quite simply, the value that appears most frequently in your dataset. In our pivot tables, we can see that the pain rating "5" received the highest count, so that's the mode.

The median is the middle value in your dataset, and it's useful as it gives you an insight into the average answer or value provided. If you arranged all survey respondents' answers (i.e. their pain rating) in ascending order, you could work out the median (middle) value. In the following example, we've highlighted the median in red:

In a dataset where you have an odd number of responses (as with ours, where we've imagined a small, hypothetical sample of thirty), the median is the middle number. In a dataset with an even number of responses, the median is the mean of the two middle numbers. However, bear in mind that, with ordinal data, it may not always be possible or sensical to calculate the median. For example, if your two middle values were "agree" and "strongly agree," it would not be possible to calculate the mean; so, in this case, you would have no median value.

The final descriptive you can use for ordinal data is variability. Variability identifies the highest and lowest values within your dataset, and tells you the range—i.e. How much the highest and lowest values differ from each other. When looking at variability, it's important to make sure that your variables are numerically coded (i.e. represented by number labels). In our tattoo pain rating example, this is already the case, with respondents rating their pain on a scale of 1-5. However, if you'd asked participants to select from a range of categories such as "painless," "slightly painful," "painful," "very painful," and "excruciating," you would need to convert these ratings into numbers (e.g. 1 = painless, 2 = slightly painful, and so on).

Once the data are numerically coded, you simply look for the highest and lowest values that appear in your dataset. If the highest pain rating given was "very painful," your maximum value would be 4. But, if at least one respondent answered with "excruciating," your maximum value would be 5.

Once you've identified the highest and lowest values, simply subtract the lowest from the highest to get the range. So, for example: 5 − 1 = 4, meaning 4 is your range. This is useful as it tells you, at a glance, that at least one respondent gave a pain rating at either end of the scale.

2. Statistical tests for analyzing ordinal data

How you analyze ordinal data depends on both your goals (what do you hope to investigate or achieve?) and the number and type of data samples you're working with. Just like nominal data, ordinal data is analyzed using non-parametric tests. Some possible options include:

• Mood's median test which enables you to compare the medians of two or more samples of data.
• The Mann-Whitney U-test for comparing the sum of rankings of scores across two independent data samples. For example: How do happiness scores of people living in Berlin compare to happiness scores of people living in New York?
• Wilcoxon matched pairs signed-rank test. This looks at the distribution of scores in two dependent data samples, comparing how they differ (the direction of difference) and to what extent (the magnitude of difference). For example: How do happiness scores among New York residents differ for those living in the city center versus those in the suburbs?
• The Kruskal-Wallis H test which compares the mean (average) ranking of scores across three or more data samples. For example: How do happiness scores differ between full-time employed, part-time employed, and unemployed people in their thirties?
• Spearman's rho (rank correlation efficient). This explores whether there's a relationship (or correlation) between two ordinal variables. For example: Is there a relationship between happiness score (1-5) and income level (high, medium, and low)?

6. Interval level

What is the interval level?

The interval level is a numerical level of measurement which, like the ordinal scale, places variables in order. Unlike the ordinal scale, however, the interval scale has a known and equal distance between each value on the scale (imagine the points on a thermometer).

Unlike the ratio scale (the fourth level of measurement), interval data has no true zero; in other words, a value of zero on an interval scale does not mean the variable is absent. This is best explained using temperature as an example. A temperature of zero degrees Fahrenheit doesn't mean there is "no temperature" to be measured—rather, it signifies a very low or cold temperature.

Examples of interval data

Some examples of interval data include:

• Temperature in degrees Fahrenheit or Celsius (but not Kelvin)
• IQ score
• Income categorized as ranges ($30-39k, $40-49k, $50-59k, and so on)

How to analyze interval data

1. Descriptive statistics for interval data

One of the first steps in the data analysis process is to summarize your data. For interval data, you can obtain the following descriptive statistics:

• Frequency distribution
• The mode, median,and mean
• Range, standard deviation, and variance

As we saw previously with nominal and ordinal data, frequency distribution presents a summary of the data in a table, allowing you to see how frequently each value occurs (either as a count or a percentage).

The mode, median, and mean are all measures of central tendency. The mode is the most frequently occurring value; the median is the middle value (refer back to the section on ordinal data for more information), and the mean is an average of all values. So, to calculate the mean, add all values together and then divide by the total number of values.

Range, standard deviation, and variance are all measures of variability within your dataset. You can calculate the range by subtracting the lowest value in your dataset from the highest. Standard deviation calculates, on average, how much each individual score deviates from the mean, allowing you to gauge how your data are distributed.

Variance looks at how far and wide the numbers in a given dataset are spread from their average value. These concepts can be confusing, so it's worth exploring the difference between variance and standard deviation further. For now, though, let's look at how you might analyze interval data.

2. Statistical tests for analyzing interval data

As long as your interval data are normally distributed, you have the option of running both parametric and non-parametric tests. However, parametric tests are more powerful, so we'll focus on those. Here are some of the most common parametric tests you might use:

• T-test to compare the mean values of two data samples. For example: What is the difference in the average IQ score of forty-fifty year olds living in London and Leeds?
• ANOVA test to compare the mean values across three or more samples of data. For example: What is the difference in the average IQ score of forty-fifty year olds living in London, Leeds, and Birmingham?
• Pearson's r to see if there is a correlation between two variables. For example: Is there a relationship between a person's income range and their IQ score?
• Simple linear regression to model or predict the relationship between two variables, or the impact of one variable on another. For example: Can a person's IQ score be used to predict their salary range?

7. Ratio level

What is the ratio level?

The fourth and final level of measurement is the ratio level. Just like the interval scale, the ratio scale is a quantitative level of measurement with equal intervals between each point. What sets the ratio scale apart is that it has a true zero. That is, a value of zero on a ratio scale means that the variable you're measuring is absent. Population is a good example of ratio data. If you have a population count of zero people, this means there are no people!

So what are the implications of a "true zero?" As the name suggests, having a true zero allows you to calculate ratios of your values. For example, if you have a population of fifty people, you can say that this is half the size of a country with a population of one hundred.

Examples of ratio data

Ratio variables can be discrete (i.e. expressed in finite, countable units) or continuous (potentially taking on infinite values). Here are some examples of ratio data:

• Weight in grams (continuous)
• Number of employees at a company (discrete)
• Speed in miles per hour (continuous)
• Length in centimeters (continuous)
• Age in years (continuous)
• Income in dollars (continuous)
• Sales made in one month (discrete)

How to analyze ratio data

The great thing about data measured on a ratio scale is that you can use almost all statistical tests to analyze it. So how do you analyze ratio data? Let's take a look.

1. Descriptive statistics for ratio data

You can use the same descriptive statistics to summarize ratio data as you would for interval data (with the addition of coefficient of variation). We'll recap briefly here, but for a full explanation, refer back to section five.

• Frequency distribution: This shows you how frequently each value occurs within your dataset, and is often presented as a table. The frequency can be expressed as either a count or a percentage.
• Mode, median, or mean: The mode is the value that occurs most frequently in your dataset, while the median is the middle value. The mean value is the average of all values within your dataset. The mode, median, and mean are all measures of central tendency which help you to gauge how your data are distributed.
• Range, standard deviation, variance, and coefficient of variation all show you the variability within your dataset. Coefficient of variation is unique to ratio data because it's a fraction, calculated by dividing the standard deviation by the mean.

2. Statistical tests for analyzing ratio data

As with interval data, you can use both parametric and non-parametric tests to analyze your data. Still, as we know, parametric tests are more powerful and therefore allow you to draw more meaningful conclusions from your analysis. Here are some common parametric tests you might use to analyze ratio data:

• T-test to compare the mean values of two data samples. For example: What is the difference in the average income of 40-50 year olds living in London and Leeds?
• ANOVA test to compare the mean values across three or more samples of data. For example: What is the difference in the average income of 40-50 year olds living in London, Leeds, and Birmingham?
• Pearson's rto see if there is a correlation between two variables. For example: Is there a relationship between a person's age in years and their income?
• Simple linear regression to model or predict the relationship between two variables, or the impact of one variable on another. For example: Can a person's age in years be used to predict their income?

8. Key takeaways

So there you have it: the four levels of data measurement and how they're analyzed. In this post, we've learned the difference between the various levels of measurement, and introduced some of the different descriptive statistics and analyses that can be applied to each. If you're looking to pursue a career in data analytics, this fundamental knowledge will set you in good stead.

If you enjoyed learning about the different levels of measurement, why not get a hands-on introduction to data analytics with this free, five-day short course? At the same time, keep building on your knowledge with these guides:

• What's the difference between descriptive and inferential statistics?
• An introduction to multivariate analysis
• What is data visualization and why is it important?

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Source: https://careerfoundry.com/en/blog/data-analytics/data-levels-of-measurement/

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