Correlation Vocabulary: Spearman's vs Pearson's Coefficients

Ranks, or a straight line

When two sets of figures rise and fall together, statisticians want to say how strongly, and they have more than one way of doing it. Your specification names two coefficients, and the single most useful thing to know is that they are not two methods for the same job. They measure different things. Spearman's rank correlation coefficient works on the ORDER things come in. It asks whether the item ranked first on one measure tends to be ranked near the top on the other, and it does not care how far apart the values are. Pearson's product moment correlation coefficient works on the LINEAR relationship. It asks how closely the points lie to a straight line, using the actual values rather than their positions. That difference has a consequence worth carrying into the exam. The two can disagree about the same data, and neither is wrong when they do: they are answering different questions. A relationship that is perfectly ordered but curved will give a high Spearman's and a lower Pearson's. So when a question names one of them, it is telling you something about what is being asked. The first half of this module is the vocabulary you need before any coefficient means anything: what positive, negative and zero correlation look like, the difference between association and causation, and what interpolation and extrapolation let you claim. That vocabulary is not a preliminary. It is what the numbers are for.

Words for how two things move

Six terms from the vocabulary half of this topic, which is Foundation content and therefore on every paper. The last two are about what you are allowed to claim.

Match each term to its meaning here

Inside the data, or beyond it

A study collects data on people aged 20 to 60. Someone uses the line of best fit to estimate a value for a person aged 75. What is that, and why does it matter?

• Extrapolation, and it is unreliable because there is no evidence the relationship continues outside the range that was actually measured
• Interpolation, because the value is read from the line of best fit
• Extrapolation, and it is perfectly reliable provided the correlation is strong
• Causation, because the estimate assumes one variable affects the other

Two bands, and no middle one

Your specification gives exact thresholds for describing the strength of a correlation from its coefficient, and they are worth learning precisely because they are stated as numbers rather than left to judgement. If the size of the coefficient, ignoring its sign, is 0.6 or more, the correlation is described as STRONG. If it is at least 0.2 but less than 0.6, the correlation is described as WEAK. If it lies between -0.2 and 0.2, there is NO CORRELATION. ⚠️ Notice what is not there. Your specification does not use the word "moderate", so do not write it. Many textbooks and websites offer a middle band with that name, and using it here means describing a correlation in terms the mark scheme does not recognise. A coefficient of 0.45 is weak, and 0.72 is strong; there is nothing in between. Two more things about reading a coefficient. The sign tells you the direction: positive means the variables move the same way, negative means opposite ways. The size tells you the strength, and the sign has nothing to do with it, so -0.71 is exactly as strong as +0.71. ⚠️ And a word about tiers, because the wording catches people out. Describing strength by inspection is Foundation content, so it can appear on either paper. Interpreting a given Spearman's coefficient is marked [Higher, both tiers], which means Higher content that still appears on both papers rather than Higher-only. Interpreting a given Pearson's coefficient is Higher only, with no Foundation exposure at all.

Which two are banded correctly

Select the TWO statements that describe a coefficient correctly, using your specification's thresholds.

• A coefficient of -0.71 shows a strong negative correlation
• A coefficient of 0.45 shows a weak positive correlation
• A coefficient of 0.45 shows a moderate positive correlation
• A coefficient of -0.71 is weaker than one of +0.45 because it is negative

The thresholds in words

A correlation coefficient carries two pieces of information: its _____ gives the direction, and its size gives the strength. If the size, ignoring the sign, is _____ or more, the correlation is described as strong. If it is at least 0.2 but less than 0.6, it is described as _____. Between -0.2 and 0.2 there is _____ correlation. Estimating beyond the range of the data collected is called _____, and is unreliable.

One measures order, one measures line

The distinction your specification asks for, with the tier of each part marked. Read the last row of each column carefully.

Banding drill

Band each coefficient using your specification's thresholds. Remember there is no middle band.

Why the third factor matters

Correlation does not imply causation is the most quoted sentence in statistics and the least often explained, so here is what your specification actually wants, which is Higher content. The claim: two variables move together in the data. What that does not establish: that either one brings the other about. And the reason, which the specification names as multiple interacting factors, is where the marks are. When two things correlate, there is more than one account available. The first could cause the second. The second could cause the first, and correlation cannot tell you which direction it runs. Or a third factor could be driving both, producing an association between them with no direct link at all. Here is a worked example, invented for illustration. Suppose a town's data shows a strong positive correlation between ice-cream sales and the number of people swimming. A weak answer: "This shows ice cream makes people want to swim." A strong answer: "The correlation is strong, but it does not establish causation. It is more plausible that a third factor, hot weather, increases both independently: people buy more ice cream when it is hot and also swim more when it is hot. Because a third variable can produce an association between two others, a correlation alone cannot tell us that either one causes the other." Notice what the strong answer does. It accepts the correlation, offers a specific alternative account rather than a general warning, and names the mechanism. ⚠️ "Correlation is not causation" on its own earns very little. Naming a plausible third factor is what turns it into an explanation.

The claim that adds a cause

Four statements about a strong positive correlation between two variables. Select the ONE that claims causation the data cannot support.

• There is a strong positive correlation between the two variables in this sample.
• The strong correlation shows that the first variable causes the increase in the second.
• The two variables are associated, but a third factor could be affecting both.
• The correlation is strong, though that alone does not tell us which variable came first.

Say it without asserting a cause

Assemble a statement that reports the correlation and stops where the evidence stops.

Which coefficient, and what it shows

You are given data and asked to interpret it. Work through the decisions.

• Two judges each place the same eight competitors in order. You want to know how far they agree. Which coefficient suits it?
• Spearman’s gives a high value but Pearson’s is noticeably lower for the same data. What does that suggest?
• A coefficient of 0.45 is reported. How should it be described?
• Someone concludes from a strong correlation that one variable causes the other. What is the best response?

Explain correlation and the coefficients

Explain the vocabulary of correlation and how a correlation coefficient is interpreted, and explain the difference between the two coefficients your specification names.

• Define positive, negative and zero correlation
• Explain the difference between association and causation
• Explain interpolation and extrapolation, and say which is unreliable and why
• Give the thresholds for describing a correlation as strong, weak, or none
• Explain what the sign of a coefficient tells you and what the size tells you
• Explain what Spearman’s measures and what Pearson’s measures
• Explain why the two can disagree about the same data
• Finish by explaining why a correlation cannot establish causation, naming a possible third factor