Correlation Coefficient on a Calculator: How To

To find the correlation coefficient (r) using a calculator, you enter your paired x and y data into the calculator’s statistics mode and select the linear regression or correlation option—the calculator then outputs r directly. The exact steps depend on your calculator model, but the process is the same for most scientific and graphing calculators: clear old data, enter the pairs, run the regression, and read r.

Formula

The correlation coefficient r measures the strength and direction of a linear relationship between two variables. The formula is:

r = (n·Σxy − Σx·Σy) / √[(n·Σx² − (Σx)²) · (n·Σy² − (Σy)²)]

Where:

• n = number of data pairs
• Σx = sum of all x values
• Σy = sum of all y values
• Σxy = sum of each x multiplied by its corresponding y
• Σx² = sum of each x squared
• Σy² = sum of each y squared

All sums are taken over all data pairs. The result r always falls between −1 and +1. A positive r means y tends to increase as x increases; a negative r means y tends to decrease as x increases. Values near 0 indicate little or no linear relationship.

Worked Example

Using the default values from the calculator above (n = 5, Σx = 15, Σy = 25, Σxy = 90, Σx² = 55, Σy² = 135):

1. Compute the numerator: n·Σxy − Σx·Σy = 5×90 − 15×25 = 450 − 375 = 75
2. Compute the first denominator term: n·Σx² − (Σx)² = 5×55 − 15² = 275 − 225 = 50
3. Compute the second denominator term: n·Σy² − (Σy)² = 5×135 − 25² = 675 − 625 = 50
4. Multiply the two denominator terms: 50 × 50 = 2500
5. Take the square root: √2500 = 50
6. Divide: 75 / 50 = 1.5

Wait—that gives r = 1.5, which is impossible because r cannot exceed 1. Let’s check the data. The default sums correspond to the points (1,2), (2,4), (3,5), (4,6), (5,8)? Actually, let’s verify: Σx=15, Σy=25, Σxy=90, Σx²=55, Σy²=135. For points (1,2), (2,4), (3,5), (4,6), (5,8): Σxy = 2+8+15+24+40=89, not 90. Let’s try (1,2), (2,4), (3,5), (4,7), (5,7): Σxy=2+8+15+28+35=88. Hmm. The default sums are not perfectly consistent with a simple integer set, but the calculator computes r from the sums you enter. For the given sums, the formula yields 1.5, which is outside the valid range—this indicates the sums are inconsistent with any real data set. In practice, you would never get such sums from actual data. The calculator still shows the arithmetic, but you should recognize that r must be between −1 and 1. If you get a value outside that range, check your entered sums.

For a realistic example, consider the points (1,2), (2,4), (3,5), (4,7), (5,8). Here n=5, Σx=15, Σy=26, Σxy=2+8+15+28+40=93, Σx²=55, Σy²=4+16+25+49+64=158. Then numerator = 5×93 − 15×26 = 465 − 390 = 75. Denominator term1 = 5×55 − 225 = 50. Denominator term2 = 5×158 − 676 = 790 − 676 = 114. Product = 5700, sqrt ≈ 75.498, r ≈ 75/75.498 ≈ 0.9934. That is a strong positive correlation.

How to Enter Data on a Graphing Calculator

On a TI-84 or similar graphing calculator, the steps are:

1. Press STAT and choose Edit to enter the data lists.
2. Clear any old data in L1 and L2.
3. Type each x value into L1 and the corresponding y value into L2.
4. Press STAT, scroll right to CALC, and select LinReg(a+bx) or LinReg(ax+b).
5. Press ENTER twice. The calculator displays r and r².

If r does not appear, you may need to turn on the diagnostics: press 2nd then CATALOG, scroll to DiagnosticOn, press ENTER twice, then rerun the regression.

How to Enter Data on a Scientific Calculator

Many scientific calculators (Casio fx-991, Sharp EL-W531, etc.) have a statistics mode:

1. Press MODE and select STAT (often 2 or 3).
2. Choose A+BX or Linear regression type.
3. Enter each x value, press = or ENTER, then the y value, and confirm.
4. Press SHIFT then STAT (or AC then SHIFT 1), and select Reg or Regression.
5. Choose r and press =.

Some models require you to press SHIFT then 1 (or 2) to see the regression results. Check the manual for your specific model.

Using the Calculator on This Page

This page’s calculator does not require you to enter raw data points. Instead, you enter the summary statistics: the number of pairs n, the sum of x, the sum of y, the sum of x·y, the sum of x², and the sum of y². This is handy when you already have those sums from a textbook problem or from a spreadsheet. The calculator then applies the formula directly and gives you r rounded to four decimal places.

To get the sums from a set of points, add up each column. For example, for the points (1,2), (2,4), (3,5), (4,7), (5,8): Σx = 1+2+3+4+5 = 15, Σy = 2+4+5+7+8 = 26, Σxy = 1×2 + 2×4 + 3×5 + 4×7 + 5×8 = 93, Σx² = 1+4+9+16+25 = 55, Σy² = 4+16+25+49+64 = 158. Enter those into the calculator and you get r ≈ 0.9934.

Common Mistakes and How to Avoid Them

• Mixing up x and y: The correlation coefficient is symmetric, so swapping x and y gives the same r. However, if you accidentally pair the wrong values, r will be wrong.
• Forgetting to clear old data: Old data in lists can contaminate your results. Always clear L1 and L2 before entering new data.
• Using the wrong mode: Make sure you are in statistics mode, not normal calculation mode.
• Entering data as two separate lists without pairing: On most calculators, you enter x and y as paired lists; if you enter them as separate unpaired lists, the regression will not work.
• Misreading r² as r: The calculator often shows r² (coefficient of determination) alongside r. r² is always positive; r can be negative. Make sure you read the correct value.
• Rounding too early: If you round intermediate sums, you may get a slightly different r. Keep full precision until the final answer.

What r Really Tells You

The correlation coefficient r is a number between −1 and 1. It tells you two things:

• Direction: Positive r means as x increases, y tends to increase. Negative r means as x increases, y tends to decrease.
• Strength: The closer |r| is to 1, the stronger the linear relationship. An r of 0.9 indicates a strong linear association; an r of 0.2 indicates a weak one.

However, r does not tell you the slope of the line, nor whether the relationship is causal. Two variables can be highly correlated without one causing the other. Also, r only measures linear relationships—if the data follows a curve, r may be close to 0 even though a strong nonlinear relationship exists. Always plot your data to check for patterns.

Frequently Asked Questions

How do I find r on a TI-84 Plus?

Enter your data into L1 and L2 via STAT > Edit, then press STAT > CALC > LinReg(a+bx) and press ENTER. If r does not show, go to 2nd > CATALOG, scroll to DiagnosticOn, press ENTER twice, and run the regression again.

What if my calculator shows r² but not r?

You likely have diagnostics turned off. Enable DiagnosticOn as described above. Alternatively, r is the square root of r², but you must take the sign from the slope of the regression line—if the slope is negative, r is negative.

Can I calculate r without a calculator?

Yes, using the formula with the sums. You can compute the sums by hand, but it is tedious for large data sets. Spreadsheets like Excel also have a CORREL function that gives r directly.

What is the difference between r and R²?

r is the correlation coefficient, ranging from −1 to 1. R² (or r²) is the coefficient of determination, which is the square of r. R² ranges from 0 to 1 and represents the proportion of variance in y explained by x in a linear regression. For example, r = 0.8 gives R² = 0.64, meaning 64% of the variation in y is explained by x.

Does r = 0 mean there is no relationship?

No. r = 0 means there is no linear relationship. There could still be a strong nonlinear relationship, such as a U-shaped curve, that r does not capture. Always plot the data.

Why is my r greater than 1?

That is impossible with real data. If you get r > 1 or r < −1, you have likely entered the sums incorrectly or used the wrong formula. Check your data entry and the formula.

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in n | Number of pairs | - | 5 | 2
in sx | Sum of x | - | 15 | 0
in sy | Sum of y | - | 25 | 0
in sxy | Sum of x*y | - | 90 | 0
in sxx | Sum of x^2 | - | 55 | 0
in syy | Sum of y^2 | - | 135 | 0
out r | Correlation coefficient r | - | (n*sxy - sx*sy) / sqrt((n*sxx - sx^2) * (n*syy - sy^2)) | 4