Correlation Coefficient

Correlation Coefficient ($r$)

Linear correlation is a measure of how close a set of points lie to a straight line. Correlation is measured using a value called the Product Moment Correlation Coefficient, or $r$, for short.

How to Calculate the $r$ Value on Different Calculators

Casio ClassWiz (fx-991EX / fx-570EX / 83/85/991 CW series)

Goal: Enter paired data (xi,yi)(x_i,y_i)(xi,yi) and read off r from the regression results.

1. Enter STAT mode Press [MENU] (or [MODE]) → choose STAT.

2. Choose linear regression Select A+Bx (that's the straight-line model).

3. Type your data You'll see two columns (usually List1 and List2).

• Type an x value, press = to place it.
• Move right, type the matching y value, press =.
• Move down and continue until all pairs are entered.

4. Get the regression/correlation Press [OPTN] → choose Reg (Regression) → choose r.

The display shows the correlation coefficient.

Tip: To clear a whole list: while in the table, press [OPTN] → Data → Delete All (wording can vary slightly by model).

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Casio fx-83/85GTX and fx-991ES PLUS (ES family)

1. STAT mode Press [MODE] → choose STAT.

2. Pick A+Bx Select the A+Bx regression type.

3. Enter pairs

• Type x, press =.
• Move right, type y, press =.
• Repeat for each pair.

4. Show r Press [SHIFT] [1] (STAT) → choose Reg → choose r → =.

The calculator prints the value of r.

Notes:

• If you hit AC you do not lose data; it just exits the table. Use SHIFT 1 → Reg → r from the main screen any time after entry.
• If "Reg" is missing, you're not in a regression (A+Bx) setup—return to STAT and re-select A+Bx.

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Texas Instruments TI-84 Plus CE / TI-84 Plus / TI-83

1. Enter data in lists Press [STAT] → 1
   …

Put xxx-values in L1, yyy-values in L2 (use the arrow keys to move).

2. (One-time only) Make sure 'r' will display If your regression output doesn't show r/r², turn diagnostics on:

[2nd] [0] (CATALOG) → press D to jump to D-items → choose DiagnosticOn → ENTER, ENTER. (This setting is remembered.)

3. Run the linear regression [STAT] → CALC → 8
   (a+bx)

If needed, specify lists: type L1 , L2 (use [2nd][1] for L1, [2nd][2] for L2).

Press ENTER. You'll see a, b, r², and r.

Optional (nice for graphs): Before pressing ENTER, you can store the regression into Y1:

Type , then [VARS] → Y-VARS → 1

→ 1

so the line plots automatically.

Common fixes: If you get DOMAIN ERROR, some lists are different lengths or contain blanks—clean them in STAT → Edit.

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Casio fx-CG50 / fx-CG20 (graphing)

1. Open STAT [MENU] → STAT → [1] Edit.

2. Enter data Put x in List1, y in List2.

3. Compute regression Press [F2] CALC → [F1] SET and choose Linear Reg (A+Bx) if asked.

Then [F1] CALC (or [F3] depending on OS) to show results.

4. Read r The results screen lists a, b, r (and often r²). Use arrows to scroll if needed.

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Sharp (e.g., EL-W531/531XH/506 family)

(Menu labels differ slightly by model, but this sequence fits most modern Sharp scientifics.)

5. STAT mode Press [MODE] until you see STAT, or select STAT from the menu.

6. Select 2-variable linear regression Choose LinReg a+bx (wording may be "a+bx" or "LR").

7. Enter paired data Many Sharps accept an x, then , (comma), then y, then = to store the pair.

Repeat for all pairs. (On models with a data table, enter xxx and yyy in side-by-side columns like Casio.)

8. Show r Open the STAT calculation menu (often 2ndF → STAT or a dedicated STAT key), choose Reg results, then select r to display it.

If the exact key names differ: look for STAT, a+bx, and a results page listing a, b, and r—that's the right place.

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In this case, $r = 1$. This means perfect linear positive correlation between $x$ and $y$.

Types of Correlation

There are different types of correlation that can be observed on the graph or through calculating the $r$ value.

Strong Positive Correlation

• The points lie close to a straight line with a positive slope.
• $r$ will be close to $1$.

Weak Positive Correlation

• The points show some positive correlation but are more spread out.
• $0 < r < 0.5$.

No Correlation

• The points do not show any trend or pattern.
• $r = 0$.

Weak Negative Correlation

• The points show some negative correlation but are more spread out.
• $-0.5 < r < 0$.

Strong Negative Correlation

• The points lie close to a straight line with a negative slope.
• $r$ is close to $-1$.

Perfect Negative Correlation