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Pearson r from your data, exact as a radical, with its strength and the working behind it.
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The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
The correlation coefficient r is a single number between -1 and 1. Its sign is the direction of the trend: positive when y tends to rise as x rises, negative when it tends to fall. Its size is how tightly the points hug a straight line: 1 or -1 when every point sits on one line, 0 when there is no straight-line trend at all. It carries no units, so it does not change when you measure x in centimetres instead of metres.
The calculation is the one a statistics course asks you to show. Make a table with x, y, xy, x² and y², add each column, and put the sums into r = (nΣxy minus ΣxΣy) divided by the square root of (nΣx² minus (Σx)²) times (nΣy² minus (Σy)²). This page prints that table and the filled-in formula under the answer, so the number arrives with its working.
The answer is exact. The square of r is always a fraction on data you can type, so r itself is the square root of a fraction with a sign attached, and that is what this page prints: √15/5 for the textbook set rather than 0.77, which is two digits of it. The decimal sits beside the radical, and the strength label reads it against the thresholds stated below.
For every point write x, y, xy, x² and y², then add each column and count the points. For x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 5 that gives n = 5, Σx = 15, Σy = 20, Σxy = 66, Σx² = 55 and Σy² = 86.
nΣxy minus ΣxΣy = 5 × 66 minus 15 × 20 = 30. Its sign is the sign of r.
nΣx² minus (Σx)² = 275 minus 225 = 50, and nΣy² minus (Σy)² = 430 minus 400 = 30. Both are always positive unless every x or every y is the same.
r = 30 / √(50 × 30) = 30 / √1500 = 30 / (10√15) = √15/5. As a decimal that is about 0.7746, a strong positive correlation.
The labels are the ones this calculator prints. r² is r multiplied by itself, so the sign disappears.
| r | Label | r² | What the scatter plot looks like |
|---|---|---|---|
| 1 | perfect positive | 1 | Every point on one rising line |
| 0.9 | strong positive | 0.81 | A narrow band climbing to the right |
| 0.7 | strong positive | 0.49 | A clear upward band with some spread |
| 0.5 | moderate positive | 0.25 | An upward trend with a lot of scatter |
| 0.2 | weak positive | 0.04 | A tilt you would miss without the number |
| 0.05 | no linear correlation | 0.0025 | A shapeless cloud, or a curve |
| -0.8 | strong negative | 0.64 | A clear band falling to the right |
| -1 | perfect negative | 1 | Every point on one falling line |
x: 1, 2, 3, 4, 5; y: 2, 4, 5, 4, 5The top of the formula is 30 and the brackets underneath are 50 and 30, so r = 30 / √1500 = √15/5, about 0.7746. r² = 0.6 exactly. The label is strong positive, and the line of best fit is y = 0.6x + 2.2.
x: 1, 2, 3, 4, 5, 6; y: 10, 8, 9, 5, 4, 2The top is -168 and the brackets are 105 and 296, so r = -168 / √31080 = -2√7770/185, about -0.9529. r² = 168/185, about 0.9081. As x goes up, y reliably goes down; the one point that breaks the pattern, (3, 9), is not enough to weaken it much.
x: 1, 2, 3, 4, 5; y: 3, 1, 4, 1, 5r = √2/4, about 0.3536, and r² = 0.125, so a straight line accounts for an eighth of the variation in y. The label is weak positive: there is an upward tilt, but a scatter plot would show mostly noise.
x: -2, -1, 0, 1, 2; y: 4, 1, 0, 1, 4Every point lies on y = x², and yet r = 0 exactly. The top of the formula is 5 × 0 minus 0 × 10 = 0: the falling left half cancels the rising right half. r measures straight-line relationships only, which is why the scatter plot comes first.
x: 1, 2, 3, 4, 5, 10; y: 2, 4, 5, 4, 5, 0This is the textbook set with one extra point at (10, 0). r drops from about 0.7746 to -52√8845/8845, about -0.5529: the single point is far enough out to reverse the sign. The same thing works in the other direction: x = 1 to 5 with y = 3, 1, 4, 1, 5 has r ≈ 0.3536, and adding (12, 12) lifts it to about 0.9084.
