Properties and assumptions of Pearson's r

What should be checked first when deciding whether a stated value could be Pearson's r?

Predict the direction when the change per x unit is negative, zero and positive. Try −8, 0 and +8. The axes stay fixed and the points are not joined

What happens at zero change?

Calculate a coefficient and judge what it summarises

Pearson’s r measures linear association, not every possible relationship. Distinguish a defined numerical calculation from a useful description of the plotted pattern

1. 1Defined values For paired numerical observations with nonzero variation in each variable, −1≤r≤1. Pearson’s r is unitless and measures linear association. A constant variable makes its ordinary denominator zero, so r is undefined
2. 2Interpret the display Inspect the scatterplot before summarising. A coefficient can be calculated for a nonconstant curved dataset, but it can miss that relationship. Near-zero r does not rule out a strong non-linear pattern. Investigate unusual cases and separate groups without automatically deleting them
3. 3Separate sign, strength and cause Use sign for linear direction and magnitude for linear strength. Steepness depends on units. Correlation is unchanged by swapping the two lists, but the explanatory/response question still matters. Neither a large coefficient nor a precise calculation establishes causation
4. 4Declared practice convention For these exercises only, use |r|<0.25: the bank label “No association”; 0.25≤|r|<0.50: weak; 0.50≤|r|<0.75: moderate; 0.75≤|r|≤1: strong. The sign supplies direction outside the first bin. These are reporting conventions, not universal boundaries. The first label means little or no linear association under this convention; it does not prove no relationship or independence. At |r|=1 also state perfect linear association

Given values

\(\displaystyle r=-0.72\)

Find the magnitude of the stated coefficient

Using the declared exercise bands (moderate: 0.50≤|r|<0.75), classify r=−0.72

Check the sign. One of these lines is right. Which one?

Two plots use identical axes. In both, all points lie exactly along an increasing straight pattern, but one is steeper. Which conclusion is justified?

Which description fits this plotted pattern?

In this plot, U lies well above the main increasing pattern. What is the appropriate first response?

A cloud appears to have two separate groups. What should an investigation check?

Changing the vertical unit from grams to kilograms makes a graph look flatter if the same numerical tick spacing is retained. Does this alone weaken the association?

For these five pairs, the positive and negative centred products cancel and r=0. Does that establish no relationship?

All response values are 7 while x varies. What is ordinary Pearson correlation?

Construct examples and counterexamples for correlation claims

Use the invented pairs x=(−2,−1,0,1,2), y=(4,1,0,1,4). Draw a labelled scatterplot and calculate both means, the three centred sums and r. Compare it with y=3 x+7 on the same five x values, and with the constant response y=7.

1. 1Test changes systematically For the straight-line data, replace x by 60 x, then by x+10, then by −x, changing one thing at a time. Predict and verify r. Swap x/y while retaining pairs. Explain what positive unit changes, a sign reversal and swapping roles do and do not establish
2. 2Choose and verify technology Choose a correlation function and XY plot tool; record the paired ranges, axis units and windows. Use x=−3..3,y=−1..5 for the curved plot and a window containing every straight-line point. Independently verify the centred products. Record the tool’s zero-variance error for the constant column without converting it to r=0.
3. 3Complete before opening the model Save your labelled plots, calculations, written interpretation and technology record. Complete independent production, tool selection and external-tool execution are not automatically marked; use self or teacher review

Model response: zero is different from undefined

For the curved data, x̄=0 and ȳ=2. The centred sums are Sxx=10,Syy=14,Sxy=0, giving r=0 despite the exact relation y=x². For y=3 x+7, Sxx=10,Syy=90,Sxy=30 and r=1. For y=7, Syy=0 and ordinary r is undefined

1. 1Transformation evidence For the straight-line data, multiplying x by 60 or adding 10 preserves r=1. Replacing x by −x changes r to −1. Swapping the numerical lists preserves r=1, but does not establish that either variable causes the other or resolve which variable a research question uses to explain the other
2. 2One reproducible workflow Enter X in A2:A6 and Y in B2:B6. Use an XY plot and =CORREL(A2:A6,B2:B6), retaining all paired rows. Independently compute each mean and the sums of centred products/squares. The curved products are −4,1,0,−1,4 and sum to 0. Keep separate columns for each transformation; verify transformed units/window labels and save the constant-column error as evidence that the statistic is undefined

Write a decision about whether to investigate a linear model

Four original fictional measurement profiles are supplied. Each row pairs a setting x with a reading y for the same test: L=(1,4),(2,5),(3,5),(4,7),(5,9); C=(1,1),(2,4),(3,9),(4,16),(5,25); H=(1,6),(2,6),(3,6),(4,6),(5,6); V=(3,2),(3,4),(3,5),(3,6),(3,8). The question uses setting to predict reading. Separately, a proposed fifth investigation uses storage-zone labels 1–5 to predict a measured reading

1. 1Produce the evidence State the observational unit, roles and variable types. Construct four labelled XY plots, keeping each pair together. Use x=0–6 throughout; use y=0–10 for L,H,V and y=0–27 for C. Explain why C needs a different vertical range. Decide which displays support investigating an approximately straight trend, which reveal curvature, and which have a constant variable
2. 2Separate three questions For each profile distinguish whether ordinary Pearson correlation is defined, whether a least-squares line can be fitted with a unique slope, and whether one straight trend adequately describes the displayed pattern. Do not decide adequacy from the magnitude of r alone. Explain why numerical category codes do not establish measurements or equal spacing
3. 3Choose and verify technology Choose an XY plotting and correlation tool. Record paired ranges, units, windows and the treatment of zero-variation inputs. For C, independently calculate the centred sums and r, then compare that result with successive changes in y. Reorder whole rows and verify that the result is unchanged. State what additional evidence a real investigation would need before making predictions or causal claims
4. 4Complete before continuing Save your plots, decision table, qualified explanation and tool record before opening the model response. Complete drawing, writing and external-tool execution are not automatically marked; use self or teacher review

Model response: a calculation and a useful model are different decisions

Each observational unit is one fictional test. Setting x is explanatory and reading y is response; both are numerical measurements in L,C,H,V. L shows a positive, approximately straight trend with scatter, so a linear model is worth investigating. That preliminary decision is not a completed fit assessment. C follows an increasing curve, so its large r does not make one straight line an adequate description

1. 1Independent calculation for C For C, x̄=3 and ȳ=11. The centred sums are Sxx=10, Syy=374 and Sxy=60. Thus the squared correlation is 180 divided by 187, and the correlation is approximately 0.981 to three decimal places. The successive changes in y are 3,5,7,9 for equal x steps, which expose the systematic curve. The coefficient describes a linear component; it does not erase curvature
2. 2Constant response and constant explanatory value For H, x varies but y is always 6. Ordinary r is undefined because the response variation is zero. The horizontal fitted line ŷ=6 nevertheless has a unique slope 0 and zero residuals. For V, all x=3, so the slope of a line cannot be identified uniquely: different slopes can pass through the same mean point. The ordinary r is also undefined. Do not turn either undefined coefficient into r=0.
3. 3Codes, workflow and limits Storage-zone labels are categories; their digits do not establish a quantitative x scale for an ordinary measurement regression. Enter each valid profile in separate paired columns, use point-only XY plots and the stated windows, and verify every row. A reproducible C check uses X in A2:A6, Y in B2:B6 and =CORREL(A2:A6,B2:B6); independently sum centred products and squares. Retain any zero-variance message for H or V. In a real study, investigate unusual observations, measurement quality, grouped cases and the supported x range. Inspect residual patterns after fitting; that is developed later. Neither an explanatory label nor a strong association proves a causal effect

C has r≈0.981. What does the displayed evidence support?

Which statement correctly distinguishes the two calculations for H?

Why do the V observations fail to determine a unique regression slope?

What is a justified next step for L?

Storage-zone labels are numbered 1–5. Which conclusion is supported?