D. Statistical Analysis · Two-Variable Data Analysis and Validity

Using Models: Predictions and Comparing Relationships

D2.4 D2.5About 70 minutes

Learning goals

I can use a regression model to make predictions and judge whether each prediction is reasonable.
I can tell interpolation from extrapolation and explain the risks of extrapolating.
I can compare two related two-variable data sets using r, r², slopes and graphs.
I can write a complete, justified description of a two-variable relationship, and spot summaries that misrepresent data.

Words to know

interpolation
Predicting y for an x-value inside the range of the data used to build the model.

extrapolation
Predicting y for an x-value outside the range of the data. Risky, because the pattern may change.

model
A mathematical description (here, an equation) that approximates the pattern in the data.

lurking variable
A variable that is not in the analysis but affects the variables being studied.

Is math more like science or like English?

The Ontario curriculum suggests a classic investigation: compare the relationship between Grade 12 math and English marks with the relationship between science and math marks. A guidance counsellor collected the final marks of 10 students who took all three courses. Is a student's math mark a good predictor of their science mark? Of their English mark?

Student	A	B	C	D	E	F	G	H	I	J
Math	55	62	66	70	74	78	81	85	90	95
Science	58	60	68	67	75	74	83	82	88	93
English	72	65	80	70	68	84	75	71	86	78

Final marks (%) for 10 students (made-up data).

Same students, same math marks. Science marks follow math marks closely; English marks drift up only slightly, with a lot of scatter.

Worked example

Example 1: Comparing the two relationships

Use technology or the formulas to find r, r² and the least-squares line for science vs math and for English vs math. Compare the relationships.

Science vs math: r ≈ 0.981, r² ≈ 0.963, ŷ ≈ 0.908x + 6.18.
English vs math: r ≈ 0.492, r² ≈ 0.242, ŷ ≈ 0.270x + 54.48.
Strength: science is very strongly related to math (about 96% of the variation in science marks is explained by the linear relationship with math); English is only moderately related (about 24%).
Slope: each extra percentage point in math goes with about 0.91 points in science but only about 0.27 points in English.
Graphs: both are roughly linear with no clear outliers; the English plot has much more scatter.

Math marks are a much better predictor of science marks than of English marks for these students. A reasonable explanation: science courses use many of the same skills (algebra, graphs, problem solving) as math. Remember the limits: only 10 students, all from one school, and correlation does not show that math skill causes science success.

Making predictions

A model is useful for predicting y from x, but every prediction needs a reality check.

Interpolation: predicting for an x inside the range of the data. Usually reasonable, if the model fits well.
Extrapolation: predicting outside the range. The pattern might change, so be cautious and say so.

Also consider the precision: a model with a small s_e (or r close to ±1) gives tighter predictions than one with lots of scatter.

The model was built from days between −16 °C and 4 °C (green band). Predicting inside the band is interpolation. The dashed parts are extrapolation: at 20 °C the outdoor rink would have no ice, and at −35 °C the rink might close.

Worked example

Example 2: Interpolate or extrapolate?

The rink model is ŷ = −2.37x + 56.99 (data from −16 °C to 4 °C). Predict sales at −7 °C, at −30 °C and at 15 °C, and comment on each.

−7 °C: ŷ = −2.37(−7) + 56.99 ≈ 74 cups. Inside the data range: interpolation, reasonable.
−30 °C: ŷ = −2.37(−30) + 56.99 ≈ 128 cups. Extrapolation. At −30 °C, fewer people might come to skate at all, so the true number could be much lower.
15 °C: ŷ = −2.37(15) + 56.99 ≈ 21 cups. Extrapolation, and the outdoor rink would have melted: the situation itself changes.

About 74 cups at −7 °C is a sensible prediction. The other two are extrapolations; the model may not apply, so report them with a clear warning (or not at all).

Worked example

Example 3: Predicting a science mark

Predict the science mark of a student with 72% in math, using the science model, and decide how much to trust it.

ŷ = 0.908(72) + 6.18 ≈ 71.5%.
72 is inside the math range (55 to 95): interpolation.
r ≈ 0.98 and the residuals are small (typically 2 to 3 points), so the prediction is fairly precise.

About 72%, and we can expect the actual mark to be within a few percentage points. A prediction from the English model would be much less precise.

How summaries can mislead

Two-variable summaries are powerful, but they can misrepresent data when used carelessly:

r alone says nothing about the slope. Two data sets can have the same r but very different slopes.
A steep slope does not mean a strong relationship; the points might be widely scattered.
Tiny samples can produce impressive r values by chance.
Extrapolating a trend ("at this rate, by 2100...") assumes the pattern never changes.
Hiding the graph hides curves, outliers and clusters.
Combining groups can create or hide relationships (the lesson on hidden variables explores this).

A complete description of a relationship gives the form, direction and strength (with r), the model with its slope interpreted, how well it fits (r², residuals), any unusual points, the range where predictions are reasonable, and the limits of the data (sample size, how it was collected, correlation ≠ cause).

A template for writing about a relationship

"For [who / how many], there is a [strong/moderate/weak] [positive/negative] [linear/nonlinear] relationship between [x] and [y] (r = ___). The least-squares line is ŷ = ___; each additional [unit of x] is associated with about ___ [units of y]. The model explains about ___% of the variation in [y]. [Unusual points]. Predictions are reasonable for [x] between ___ and ___. Limitations: ___."

Just for fun

At the current rate, the length of the average Grade 12 essay grows by a page every week of the semester. Extrapolate to the end of university and every student will be writing a 200-page essay a week. This is why statisticians are nervous about extrapolation.

Hands-on activity

Marks or minutes: compare two relationships

You need: data you can find at home, a calculator, and a spreadsheet or Desmos if possible.

Choose one of these (or invent your own pair of related comparisons):

Family sports or games: for 8 or more games, the minutes played vs points scored for two different players.
Grocery receipts: for 10 or more items, the price vs the mass (from the label) for fruit and vegetables compared with snacks.
Your own marks: your test marks in two courses across the semester (only your own data).

Make two scatter plots on the same scales.
Find r, r² and the least-squares line for each.
Write a comparison using the template above.
Make one interpolation and one extrapolation, and comment on each.

What to look for: a good comparison uses numbers from both relationships, interprets both slopes in context, and names at least one limitation (small sample, other variables, extrapolation).

Your turn

A model ŷ = 1.5x + 20 was built from x-values between 10 and 40. Predict y for x = 30 and for x = 100. Which prediction is extrapolation?

Relationship P has r = 0.9 and slope 0.2. Relationship Q has r = 0.5 and slope 3. Which is stronger? Which has y changing faster with x?

Using the English model, predict the English mark for a student with 80% in math.

A model of a child's height vs age (ages 2 to 12) has slope 6 cm/year. Why would predicting height at age 40 be silly?

A company says, "r = 0.95 between our ad spending and sales over 4 months." Give two reasons to be cautious.

Write a complete description for: 12 cities, x = average January temperature (°C), y = winter heating cost ($), r = −0.88, ŷ = −35x + 900, x from −20 to 0.

More practice: warm-up

Is predicting inside the data range interpolation or extrapolation?

A model is built for x from 0 to 50. Is x = 75 interpolation?

r = 0.98 for model A and 0.60 for model B. Which gives more precise predictions (if all else is similar)?

ŷ = 0.5x + 4. Predict y at x = 12.

Does a steep slope always mean a strong correlation?

More practice: on your own

For the science model, find the residual for student F (math 78, science 74).

Two models: ŷ = 2x + 5 with r = 0.4, and ŷ = 2x + 5 with r = 0.95. Same line. How do their scatter plots differ?

Over 10 years, a town's population grew from 12 000 to 15 000 in a nearly straight line. A councillor predicts 45 000 in 100 more years. Comment.

Why should a complete description include the range of x-values in the data?

The math and science data have r ≈ 0.98. A student concludes "being good at math makes you good at science." Write a better conclusion.

Challenge

Using the summary statistics, verify the science slope: s_x (math) ≈ 12.66, s_y (science) ≈ 11.71, r ≈ 0.9815.

Explain why the English model's slope (about 0.27) is so much smaller than the science model's slope (about 0.91), even though both use the same math marks.

A model is built from Grade 9 to 12 students: y = hours of homework, x = grade. It predicts a university first-year student ("grade 13") does about 2 hours more homework than a Grade 12 student. Discuss whether this extrapolation is reasonable.

Check your understanding

Quiz

A model was built from data with x between 20 and 60. Predicting at x = 45 is:

Which situation makes extrapolation most risky?

Science vs math has r = 0.98; English vs math has r = 0.49. Which conclusion is best?

Two relationships have the same value of r. Which statement is true?

A model ŷ = −0.8x + 70 (x from 0 to 50) predicts y for x = 100. The prediction is:

Which is part of a complete description of a two-variable relationship?

For parents and teachers

Big idea (D2.4, D2.5): a model is only as good as the situation it came from. Predictions inside the data (interpolation) are usually fine; predictions outside (extrapolation) need a warning. When comparing two relationships, use r, r², slopes and the graphs together.

How to help: Ask your teen to make one sensible and one silly prediction from any model, and explain which is which. Discuss news claims like "at this rate, by 2050...".

Common difficulties:

Confusing a steep slope with a strong relationship.
Extrapolating without noticing the change in situation.
Writing conclusions that claim cause from correlation.
Leaving out the sample size and the range of the data when describing a model.