A correlation coefficient summarizes how two numerical measurements vary together. The common Pearson coefficient, written r, describes the direction and strength of their linear association. Its value runs from −1 to +1 when it is defined. Read it alongside a scatterplot: a single number can conceal a curve, a separate cluster or one unusually influential observation.
For example, a positive correlation between the length and mass of paper samples means that longer samples tend to be heavier within those observations. It does not say how many grams each extra centimeter adds, explain why the pattern occurs or guarantee what a new sample will weigh.
What do the sign and size of r mean?
A positive r indicates an upward linear tendency: larger values of one measurement tend to accompany larger values of the other. A negative r indicates a downward tendency. The sign describes direction, not quality. Negative correlation is not automatically weaker or less useful than positive correlation.
Compare absolute values to compare linear strength. For example, −0.85 is farther from zero than +0.35. Values of +1 and −1 describe exact straight-line arrangements with positive and negative slopes, respectively, provided both variables vary. A value near zero indicates little linear association in the observed data, not necessarily little association of every kind.
There is no universal dividing line that makes a coefficient “good,” “strong enough” or useful for a decision. Context, the number and range of observations, measurement quality and the plotted shape all matter. OpenStax explains the coefficient’s range and its distinction from slope.
Start with pairs, then make a scatterplot
This invented practice dataset contains six paper samples with different dimensions and thicknesses. Each row belongs to one sample. The values are for arithmetic practice, not measurements inferred from the accompanying photographs.
| Sample | Length x (cm) | Mass y (g) |
|---|---|---|
| A | 8 | 3 |
| B | 10 | 5 |
| C | 12 | 4 |
| D | 14 | 7 |
| E | 16 | 6 |
| F | 18 | 9 |
Put length on the horizontal axis and mass on the vertical axis. Plot one point per row: the first point is (8, 3), the second is (10, 5), and so on. Use numerical spacing on both axes and label the units. Leave the points unconnected so a line does not imply an observation between samples.
The six points generally rise from left to right, but they do not form an exact straight line. Sample C is lighter than B despite being longer; E is lighter than D. Those local reversals can coexist with an overall positive association. NIST’s scatterplot guide describes using plots to inspect linear patterns, curves and outliers.
Never sort the length column and mass column independently. That creates new pairs and can manufacture a pattern. If you reorder samples, move entire rows together. If a measurement is missing, keep the original record and document which complete pairs enter the calculation rather than filling the gap with an invented zero.
How is Pearson correlation calculated?
One calculation uses each value’s distance from its column mean. For these six rows, the mean length is 13 cm and the mean mass is 17/3 g, approximately 5.6667 g. Keep the fraction or full calculator precision until the final result. If needed, review how to calculate a mean first.
Subtract the appropriate mean from every value. Multiply the two deviations within each row, then add those products. Separately, square each length deviation and add them; do the same for the mass deviations. The coefficient is the sum of paired products divided by the square root of the product of the two squared-deviation sums.
Here, the paired-product sum is 36. The squared-deviation sums are 70 for length and 70/3 for mass. Therefore r = 36 ÷ √(70 × 70/3) ≈ 0.891. This is a positive linear association close to +1 in this small invented dataset, with visible departures from an exact line.
In a spreadsheet or statistics tool, use its Pearson correlation function on the two matching numeric columns. Check that both ranges start and finish on the same rows and exclude headings. An independent calculation should reproduce approximately 0.890769. Do not interpret that decimal as a percentage of samples, a percentage increase or a probability.
Why is correlation different from slope?
Slope expresses a change in one variable per unit of another, such as grams per centimeter. Correlation has no units. Converting every length from centimeters to millimeters leaves r unchanged, although a slope expressed per millimeter changes. Swapping the two columns also leaves Pearson r unchanged.
If your question is the steepness between two coordinate points, use the separate two-point slope method. A correlation calculation describes the paired collection; its output does not supply a conversion rate.
What can make the number misleading or undefined?
A curved pattern can cancel out in Pearson’s calculation. For another example, pair x values −3, −1, 0, 1, 3 with y values 9, 1, 0, 1, 9. Every y equals x squared, yet r is exactly zero. Plotting the five points reveals a U shape that the linear summary misses.
An outlier can substantially change r. Inspect the original record for unit mistakes, transcription errors or a genuinely unusual sample. Do not delete a valid point merely to obtain a cleaner result. If comparing results with and without it helps explain sensitivity, identify both versions and the reason for the comparison.
If every x value is identical, or every y value is identical, one squared-deviation sum is zero. Pearson r is then undefined because its denominator is zero. A software error or blank result in that situation should not be rewritten as “no correlation.”
Finally, association alone does not establish cause and effect or reliable prediction beyond these observations. Report what was paired, how many complete pairs were used, the scatterplot’s shape and the coefficient together. For hand plotting, existing squared paper is enough; you can also browse graph-paper notebooks on Amazon. The useful habit is inspecting the pairs before trusting the summary.
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