Exploring Scales of Measurement in Non-Parametric Correlation

Every time a psychologist scores a personality test, ranks therapy outcomes, or records a patient’s diagnosis, they are working with data – but not all data is created equal. Before you can run any correlation analysis, you need to ask a more fundamental question: what kind of data do you actually have? The answer lies in understanding the four scales of measurement – nominal, ordinal, interval, and ratio – a framework that has guided statistical thinking since psychologist Stanley Smith Stevens introduced it in a landmark 1946 paper. These scales do far more than categorize data. They determine which statistical tests are valid, which correlation techniques are appropriate, and ultimately, how much you can trust your conclusions.

What are scales of measurement? 🔗

Scales of measurement refer to the nature and characteristics of data being collected or analyzed. In practical terms, they tell you how precisely a variable has been recorded and what mathematical operations are meaningful to apply. Think of them as a hierarchy – each level builds on the one before it, adding new properties and opening up more powerful forms of analysis. There are four levels, ranked from low to high precision: nominal, ordinal, interval, and ratio. The higher you go, the richer the information and the broader your statistical options.

Getting this right matters enormously in psychology. Use the wrong scale assumption, and you risk running tests that are simply not valid for your data – like calculating the average of a set of category labels, or treating ranked satisfaction scores as though the gaps between them are mathematically equal.

The nominal scale: naming without ordering 🔗

The nominal scale is the most basic level of measurement. It is a naming scale where variables are simply labeled or categorized with no specific order and no quantitative value attached. The word “nominal” itself comes from the Latin nomen, meaning name – and that captures exactly what this scale does. It names things.

Classic examples in psychology include a participant’s diagnostic category (e.g., anxiety disorder, depression, ADHD), their gender, ethnicity, or political affiliation. You can assign numbers to these categories – for instance, coding male as 1 and female as 2 – but those numbers carry no mathematical meaning. If you did calculate the mean or standard deviation of a group of nominal codes, the result would be meaningless.

What you can and cannot do with nominal data 🔗

With nominal data, you can only identify the most common category – the mode – as the appropriate measure of central tendency. You can count how many people fall into each group, display that with pie charts or bar charts, and run proportion tests. What you cannot do is rank the categories, measure the distance between them, or calculate an average. A sub-type worth knowing: a nominal scale with only two categories – like yes/no, or sick/healthy – is called a dichotomous scale.

The ordinal scale: when order matters 🔗

The ordinal scale takes things one step further. It allows for rank ordering among data points, but does not assume equal intervals between those ranks. You know that one value is higher or lower than another, but you cannot say by exactly how much.

Psychologists encounter ordinal data constantly. Likert scale responses (“strongly agree,” “agree,” “neutral,” “disagree,” “strongly disagree”) are ordinal. So are socioeconomic status categories, education levels, or a therapist’s ranking of a client’s symptom severity. Classic examples include income brackets (“low income,” “middle income,” “high income”) and satisfaction ratings (“extremely dislike” through “extremely like”).

The key limitation: unequal gaps 🔗

The critical point about ordinal data is that the intervals between ranks are not necessarily equal or even measurable. With ordinal scales, the order of values is significant, but the differences between each one are not truly quantifiable. Is the psychological distance between “very happy” and “happy” the same as the distance between “happy” and “neutral”? Almost certainly not – but the ordinal scale gives you no way to determine that.

This has direct consequences for statistical analysis. For ordinal variables, the median is a valid measure of central tendency because it refers to the middle-ranked value, while many statisticians consider the mean to be inappropriate since the differences between ranks may not be constant. For correlation purposes, this is precisely why ordinal data calls for non-parametric methods rather than Pearson’s r. Notably, IQ scores are a well-known example of ordinal measurement – a 10-point difference may carry different meanings at different points of the scale, and there is no zero point representing the complete absence of intelligence.

The interval scale: equal distances, no true zero 🔗

The interval scale adds a crucial property to the ordering of ordinal data: equal and measurable distances between values. An interval scale is a numerical scale where the variables’ order is known and the exact differences between variables are consistent and computable.

Temperature in Celsius or Fahrenheit is the textbook example. The difference between 20°C and 30°C is exactly the same as the difference between 60°C and 70°C – 10 degrees in each case. Time on a clock works similarly. In psychology, standardized test scores such as IQ (when treated as interval-level) or scores on validated psychometric instruments often approximate interval measurement.

Why the absence of a true zero matters 🔗

The interval level of measurement involves numerical scales where the intervals between values are equal, but there is no true zero point. On the Celsius scale, zero degrees does not mean “no temperature” – it is simply an arbitrary reference point. This means you cannot form meaningful ratios. Saying that 40°C is “twice as hot” as 20°C is mathematically incorrect, because the zero point is not absolute.

Despite this limitation, interval data unlocks significantly more statistical power. With interval data, the mode, median, and mean can all be used as measures of central tendency, and standard deviation can also be calculated. This makes interval-level data well suited for Pearson’s correlation, which requires that variables be measured at the interval or ratio level.

The ratio scale: the most informative level 🔗

The ratio scale is the highest and most informative level of measurement. It has all the properties of an interval scale – meaningful order and equal intervals – but adds one defining feature: an absolute zero point that represents the true absence of the quantity being measured.

Ratio data maintains order and equal intervals and also has a true zero point, indicating the complete absence of the attribute being measured, as observed in measurements like height or weight. Reaction time, the number of therapy sessions attended, the duration of a depressive episode, and cortisol levels in milligrams – all of these are ratio-scale measurements in psychology research.

What ratio data makes possible 🔗

Because zero is meaningful, ratio data allows for the full range of mathematical operations. Ratio scale variables can be meaningfully added, subtracted, multiplied, and divided, allowing for the calculation of ratios. You can legitimately say that a patient who attended 10 therapy sessions attended twice as many as one who attended 5. When measuring the duration of depressive episodes, for instance, a 6-month episode is meaningfully twice as long as a 3-month episode – a statement you could not make with interval data. All standard descriptive statistics – mode, median, mean, standard deviation, and the coefficient of variation – are appropriate for ratio data.

How scales of measurement determine which correlation to use 🔗

This is where scales of measurement move from being a theoretical taxonomy to a practical decision-making tool. The scale at which you measure your variables directly dictates which correlation technique is statistically appropriate – and using the wrong one can produce misleading results.

Pearson’s r: for interval and ratio data 🔗

Pearson’s product-moment correlation is one of the most commonly used correlations in statistics and measures the strength and direction of a linear relationship between two variables – but it requires your data to be at the interval or ratio level. It also assumes the data is approximately normally distributed and that the relationship between variables is linear. When those conditions are met, Pearson’s r is the most powerful and precise tool available. When they are not – particularly when your data is ordinal – it is inappropriate to apply it.

Spearman’s rho: for ordinal data and non-normal distributions 🔗

Spearman correlation can be used with either continuous or ordinal data, and it is relatively robust to outliers. Rather than working with raw scores, Spearman’s rho converts data into ranks and then measures the strength and direction of the monotonic relationship between those ranks. While Pearson’s correlation assesses linear relationships, Spearman’s correlation assesses monotonic relationships – whether linear or not. This makes it a natural fit when you are working with Likert scale data, ranked outcomes, or any data that violates the normality assumption. Unlike Pearson’s r, Spearman’s rho does not require the assumption that the relationship between variables is linear, nor does it require variables to be measured on interval scales.

Kendall’s tau: for ordinal data with small samples 🔗

Kendall’s Tau is a non-parametric measure that makes no assumptions about data distribution and can be used to explore correlation between two ordinal variables. Where Spearman’s rho works with rank deviations, Kendall’s tau is based on concordant and discordant pairs – comparing whether pairs of observations move in the same or opposite direction in ranking. It is particularly valued for small sample sizes and is considered by many statisticians to be a more robust measure when data is sparse or has tied ranks. Spearman’s rho and Kendall’s tau are invaluable when analyzing data sets that do not adhere to the assumptions required for Pearson’s correlation, allowing researchers to uncover relationships that might otherwise be obscured.

Putting it together: a practical guide 🔗

Choosing the right correlation coefficient is not guesswork – it follows directly from understanding your scale of measurement. If you are correlating two interval or ratio variables that meet assumptions of normality and linearity, Pearson’s r is your tool. If your variables are ordinal, or your continuous data is non-normal or contains outliers, Spearman’s rho is appropriate. For small ordinal datasets or when you need a particularly robust non-parametric measure, Kendall’s tau is often the better choice.

There is a hierarchy in the complexity and precision of measurement levels, from low (nominal) to high (ratio) – and the higher the level of measurement, the more precise your data and the broader your analytical options. Whenever you have a choice between scales, opting for the higher level gives you more flexibility. A researcher who records exact income figures (ratio) has more analytical options than one who groups participants into income brackets (ordinal).

Ultimately, scales of measurement are not just a textbook concept. They are a practical map that guides every step of quantitative research in psychology – from how you design your variables to which statistical test you trust with your conclusions. Getting this foundation right is what separates a well-designed study from one built on shaky ground.

What do you think? When you look at commonly used psychological measurements – like self-report mood scales or standardized personality inventories – which scale of measurement do you think best describes them, and does that change which correlation method researchers should be using? If most psychological data is ordinal in nature, does that mean Pearson’s correlation is being over-applied in psychology research?

References

1. https://en.wikipedia.org/wiki/Level_of_measurement
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11. https://en.wikipedia.org/wiki/Spearman's_rank_correlation_coefficient
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