Topic 4.10 · teacher page · Standard Level

One outlier, two very different answers

Why Spearman exists, what ties do to the ranks, and the justification that carries the mark.

The one thing to do with the animation

Drag the outlier and read both coefficients at once.

Pearson collapses from 0.9789 to 0.7924 while Spearman does not move off 1. Say the number out loud: one point out of eight has cost Pearson nearly a fifth of its value.

That is the entire argument for Spearman, and it is far more convincing as a live number than as a sentence in a textbook. Keep dragging back and forth; the stillness of the Spearman readout is the thing they remember.

Ties, which is where the method is actually tested

Everyone can rank 3, 7, 9. The examined case is 7, 7, 9, where both sevens take the mean of the ranks they occupy, so 1.5 and 1.5, and the nine takes 3 rather than 2.

Get this wrong and every subsequent difference is wrong. Make them write the ranks under the data in a row, not in their heads.

The answers

1. rs on y = x²As x goes up, y goes up every single time, so the ranks match perfectly and Spearman is exactly 1.
2. Rank for each 7They occupy ranks 1 and 2, so both get 1.5, and the 9 gets rank 3.
3. Which to quoteB, Spearman. It uses only the order, so one stray value barely shifts it. The justification is where the mark is.

Where the marks go

1 markRanking both variables correctly, including shared ranks for ties.

1 markThe coefficient itself, usually straight from the calculator.

1 markA reason for preferring one coefficient, in the context of the data. “Because of the outlier” alone is thin; name what the outlier does.

What each wrong answer tells you

They giveWhat it means
0.9789 (Q1)Gave Pearson. On y = x² the relationship is perfect but not linear, which is exactly the case that separates the two coefficients.
1 (Q2)Gave the first tied rank to both and then used 2 for the nine. The ranks no longer sum correctly, which is the check to show them.
2 (Q2)Averaged the wrong pair, or ranked from the top. Either is worth distinguishing before correcting.
"A, Pearson" (Q3)They default to the familiar one. Send them back to the widget and make them read the two numbers again.
"Neither, remove the outlier" (Q3)Tempting and sometimes right, but it needs justifying as a data decision, not used as an escape. If they say this, ask what they would write in an IA to defend it.

Other things they will say

"Which one do I use?" Linear and clean: Pearson. Monotonic but curved, ordinal data, or an outlier present: Spearman. Say which and why, because the why is the mark.

"Can Spearman be 1 when Pearson is not?" Yes, and the widget is sitting on that case. Any strictly increasing relationship gives Spearman 1, however curved.

"Does Spearman mean causation?" No, no more than Pearson does. Worth saying, because the unfamiliar name makes it sound stronger than it is.

A possible order

 What is happening
1Drag the outlier. Read both numbers. Say the fifth-of-its-value line.
2Ranking by hand, including the tie case, written under the data.
3The three questions.
4A real data set with a defensible outlier, and a written justification each.
5Flag that the IA rewards exactly this judgement, and that quoting both coefficients with a reason is a strong move.

Two things not to say

Do not say Spearman is “the one for small data sets”. Size is not the criterion; shape and outliers are.

Do not accept “because it is better” as the justification. It earns nothing and it is not true in general.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

  1. Compute from ranksSeven items ranked 1 to 7 by judge A are ranked 2, 1, 4, 3, 6, 5, 7 by judge B. Find Σd² and rs.
    Σd² = 6, so rs = 1 − 6(6) / (7 × 48) = 0.893.
  2. State the meaningWhat does rs = 0.893 say about the two judges?
    They agree strongly on the order, though not perfectly. It says nothing about whether their actual scores were close.

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

  1. Choose the measureA set of marks has one student who scored far above everyone else. Would you quote Pearson or Spearman, and why?
    Spearman. Ranks cap the influence of that one student, while Pearson uses the actual distances and a single extreme point drags the whole line.
  2. Monotonic but not linearTwo quantities are related by y = x³ exactly, for positive x. State rs and explain.
    rs = 1 exactly. Cubing preserves order, and Spearman only uses order. Pearson would be below 1 because the relationship is not a straight line.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

  1. The point of the pageOne outlier takes Pearson from near 1 down to 0.79 while Spearman stays at 1. Explain how both can be describing the same data honestly.
    They answer different questions. Pearson asks how close to a straight line the values lie, and the outlier is far from it. Spearman asks whether the order is preserved, and it is. Quoting one without saying which is the real error.
  2. Limits of ranksGive a situation where Spearman would hide something important.
    When the sizes matter. Ranking salaries 1 to 10 loses the fact that the top one might be twenty times the second. Perfect rank agreement can sit on top of wildly different magnitudes.

Practicalities

Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.