Compute and interpret the correlation coefficient of a linear fit, and distinguish correlation from causation

Learning goal: Compute the correlation coefficient of a linear fit using technology, interpret its sign and size, know what it does not measure, and distinguish correlation from causation by naming the other explanations an association can have.

The data. A hypothetical market stall sold hot cocoa on ten Saturdays and recorded the midday temperature in degrees Celsius and the cups sold. The pairs are invented for teaching: (2, 44), (4, 47), (6, 36), (8, 41), (10, 30), (12, 33), (14, 24), (16, 27), (18, 18), (20, 16). A scatter plot shows a clear negative linear association: warmer days sold fewer cups. A fitted line describes the trend; the correlation coefficient measures how tightly the points hold to a line.

What the correlation coefficient is. The correlation coefficient r turns each x and y into a z-score, multiplies the two z-scores for each point, adds the products and divides by n − 1: r = Σ(z_x · z_y) ÷ (n − 1), where the z-scores use the sample standard deviations s_x and s_y, which also divide by n − 1. When large x-values pair with large y-values, both z-scores are positive and the product is positive; when large x pairs with small y, the product is negative. So r comes out positive for a rising pattern and negative for a falling one, and its size shows how consistent the pattern is. Here the mean temperature is 11 °C with standard deviation 6.06 and the mean sales are 31.6 cups with standard deviation 10.60. The first point gives z_x = (2 − 11) ÷ 6.06 = −1.486 and z_y = (44 − 31.6) ÷ 10.60 = 1.170, product −1.739. Adding all ten products gives −8.541, and dividing by 9 gives r = −0.949.

Computing with technology. For a real data set r is almost always computed with technology, and the standard asks for technology. In a spreadsheet, put the temperatures in one column and the sales in another and enter a correlation function such as CORREL(A2:A11, B2:B11); it returns −0.949. On a graphing calculator, enter the two lists, run the linear regression command, and with diagnostics turned on the output shows the slope a = −1.66, the intercept b = 49.87, r = −0.949 and r² = 0.901. The fitted line is therefore cups ≈ −1.66 × temperature + 49.87. Always check the sign of r against the sign of the slope: they must agree, because both describe the same direction.

Interpreting r. r is defined only when neither variable is constant (a standard deviation of 0 would leave the z-scores undefined), and then it is always between −1 and 1. Values near +1 mean the points lie close to a rising line; values near −1 mean close to a falling line; values near 0 mean no linear pattern. A common informal scale calls |r| above 0.8 strong, between 0.5 and 0.8 moderate and below 0.5 weak, though those cutoffs are conventions, not laws. The cocoa data give r = −0.949, a strong negative linear association: in the sentence a statistician would write, "for these ten Saturdays, higher temperatures were strongly associated with lower cocoa sales, roughly linearly." Three properties matter for interpretation. First, r has no units: converting the temperatures to degrees Fahrenheit or the sales to dollars at a fixed price per cup changes every z-score not at all, because each is a shift and a multiplication by a positive number, so r stays −0.949. Second, r is symmetric: the correlation of sales with temperature equals the correlation of temperature with sales. Third, r² = 0.90 says that 90% of the variation in sales is accounted for by the linear relationship with temperature, leaving 10% to everything else.

What r does not measure. r measures only linear association. The seven hypothetical points (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9) lie exactly on the parabola y = x², so y is completely determined by x, yet r = 0 exactly: the positive products on the right cancel the negative products on the left. A correlation near 0 means "no linear pattern", never "no relationship", which is why the scatter plot is looked at before r is trusted. r is also sensitive to outliers. Change the last cocoa point from (20, 16) to (20, 44), as if a festival crowd came on the warmest day, and r moves from −0.949 to −0.561 and the slope from −1.66 to −0.90. One point out of ten changed the description from strong to moderate. An outlier is checked and explained, not silently deleted.

Correlation is not causation. A strong r says that two variables move together in the data. It does not say that changing one would change the other. The misconception is natural because a fitted line reads like a recipe: "each degree warmer, 1.66 fewer cups." But the line only describes what was observed together. Four common reasons an association can exist without x causing y are these. A lurking variable (also called a confounding variable) drives both: in a hypothetical town, monthly ice-cream sales and monthly rescues at the public pool are strongly positively correlated, not because ice cream causes drownings but because hot weather raises both. Reverse causation: the arrow runs the other way. In a hypothetical survey, people who take pain relievers more often report more headaches; the headaches cause the pain-reliever use, not the reverse. Coincidence: among many quantities that happen to trend upward over the same years, some pairs correlate strongly by chance alone. Selection: if only students who already enjoy mathematics sign up for a club, a correlation between club membership and grades says little about the club.

What would support a causal claim? A randomized experiment, in which the investigator assigns the values of x at random, so that lurking variables are balanced across the groups on average. That design is the subject of later lessons. For the cocoa stall, temperature cannot be assigned, so the honest statement stays "associated with". A plausible mechanism (people want hot drinks when cold) and consistent findings in other markets make the causal story more believable, but the correlation coefficient by itself cannot establish it.

The model sentence. A complete interpretation of a computed r names the direction, the strength, the linear form, the units-free number itself, the data it came from and the limit on causal language. For the cocoa data: "r = −0.95 for these ten Saturdays shows a strong negative linear association between temperature and cups sold; the fitted line predicts about 1.66 fewer cups per degree; the data cannot show that temperature caused the change in sales, though the mechanism is plausible."

Worked calculation by hand on a tiny set: Five hypothetical points: (1, 2), (2, 4), (3, 5), (4, 4), (5, 7). Means: x̄ = 3, ȳ = 4.4. Deviations in x: −2, −1, 0, 1, 2; in y: −2.4, −0.4, 0.6, −0.4, 2.6. Products: 4.8, 0.4, 0, −0.4, 5.2, total 10.0. Sum of squared x-deviations 10; of y-deviations 5.76 + 0.16 + 0.36 + 0.16 + 6.76 = 13.2. Then r = 10 ÷ √(10 × 13.2) = 10 ÷ 11.489 = 0.870. This equivalent form, Σ(dx·dy) ÷ √(Σdx² · Σdy²), is what the technology computes; the z-score form gives the same value. A spreadsheet CORREL on the two columns returns 0.870.

Worked interpretation in context: A calculator reports r = −0.949 for the cocoa data. Direction: negative, so higher temperatures go with lower sales. Strength: |r| = 0.95 is close to 1, so the points lie close to the fitted line; the scatter plot confirms a linear shape with no curve. Statement: "there is a strong negative linear association between midday temperature and cups sold on these ten Saturdays." The word "causes" does not appear.

Worked outlier effect: Replacing (20, 16) with (20, 44) changes r from −0.949 to −0.561. The single point now sits far above the line on the right, so its own product z_x·z_y changes from about −2.19 to about +1.47; the new value also raises the mean of sales from 31.6 to 34.4 and lowers their standard deviation from about 10.60 to about 9.67, which shifts every other product a little. The sum of the products moves from −8.54 to −5.05, and −5.05 ÷ 9 = −0.561. The correct response is to check the record: if the 44 was a festival day, report the analysis with and without that day and say why; if it was a typing error for 14, correct it. Deleting it merely because it weakens r is not allowed.

Worked causation judgment: A hypothetical report says towns with more ice-cream shops per resident have more pool rescues per resident, r = 0.7, and recommends limiting ice-cream shops. The association is real, but a lurking variable, summer heat and the number of people swimming, plausibly raises both quantities. No random assignment of shops occurred, so the causal recommendation is unsupported. The correct statement is that the two rates are associated, with heat and swimming as a likely common cause.

Represent and compute: Enter the ten cocoa pairs into a spreadsheet or calculator. Produce the scatter plot, the fitted line and r. Then, in a second copy, change (20, 16) to (20, 44) and recompute; record both values of r and both slopes and write two sentences on why one point moved r so much. Finally enter the seven parabola points and confirm that r = 0 while the plot shows a perfect curve.

Independent performance: Choose or collect a bivariate data set of at least twelve pairs (for example, hours of daylight and heating used per day in a hypothetical house, or two measurements from classmates). (1) Plot it and describe the form; (2) compute r with technology and state the fitted line; (3) interpret r in a full sentence naming direction, strength and the linear restriction; (4) name one lurking variable, one possible reverse-causation story and one design that could test causation; (5) write the conclusion using "associated with" and explain why "causes" was not used. Success means r is reported to two decimals with the tool named, the sign of r matches the slope, and every causal alternative is specific to your variables rather than generic.