Covariance on Casio Calculator: How to Do It

Covariance on a Casio calculator is found using the Statistics mode, not a single dedicated key. You enter paired data into two lists, run the 2-Var stats calculation, and read the value labeled Σxy, then apply the covariance formula yourself: covariance = (Σxy − n·x̄·ȳ) / (n − 1) for sample covariance, or divide by n for population covariance. This article explains the exact key presses for common Casio models (fx-991EX, fx-991CW, fx-82/350EX, and older fx-991ES Plus), how to interpret the result, and the common mistakes that lead to wrong answers.

Formula

The covariance measures how two variables move together. For a sample of paired observations (x�,y�), (x₂,y₂), …, (xₙ,yₙ), the sample covariance is:

s_xy = Σ(xᵢ − x̄)(yᵢ − ȳ) / (n − 1)

where:

• Σ(xᵢ − x̄)(yᵢ − ȳ) is the sum of the products of each deviation from the mean
• x̄ is the mean of the x-values
• ȳ is the mean of the y-values
• n is the number of paired observations

An algebraically equivalent form that is easier to compute from the calculator’s output is:

s_xy = (Σxy − n·x̄·ȳ) / (n − 1)

where Σxy is the sum of each x multiplied by its paired y. For a population (all data, not a sample), divide by n instead of n−1.

Worked Example

Using the default values from the calculator above: x = 2, y = 4, n = 3. That means the data pairs are (2,4), (2,4), and (2,4) – all identical. Let’s compute the sample covariance step by step.

1. Compute Σxy = 2×4 + 2×4 + 2×4 = 8 + 8 + 8 = 24
2. Compute x̄ = (2+2+2)/3 = 2
3. Compute ȳ = (4+4+4)/3 = 4
4. Compute n·x̄·ȳ = 3 × 2 × 4 = 24
5. Apply the formula: (Σxy − n·x̄·ȳ) / (n−1) = (24 − 24) / 2 = 0

The sample covariance is 0, which makes sense because there is no variation in the data – every point is identical. The calculator block gives the same result (0.00) for both sample and population covariance.

How to Enter Data in Statistics Mode

The exact key sequence depends on your Casio model, but the logic is the same across all scientific calculators with a Statistics mode.

For fx-991EX, fx-82EX, fx-350EX (ClassWiz series)

1. Press MENU, then 2 (or navigate to Statistics).
2. Choose 2-Var (2-variable statistics).
3. You will see two columns: X and Y. Enter each x-value, press =, then the y-value, press =. Continue for all pairs.
4. Press AC to leave the data editor.
5. Press OPTN, then select 3: Summary stats (or similar).
6. Scroll to find Σxy, x̄, ȳ, and n. Write them down.

For fx-991CW (newer ClassWiz)

1. Press MENU, then Statistics (usually option 6).
2. Select 2-Variable.
3. Enter data in the X and Y columns.
4. Press AC, then TOOLS (or the “Stats� soft key).
5. Choose Summary and scroll to Σxy, x̄, ȳ, n.

For fx-991ES Plus (older models)

1. Press MODE, then 3: STAT.
2. Press 2: A+BX (or any 2-Var option).
3. Enter x and y pairs, pressing = after each value.
4. Press AC, then SHIFT + 1 (STAT).
5. Select 5: Reg or 4: Sum to find Σxy, x̄, ȳ, n.

Once you have Σxy, x̄, ȳ, and n, plug them into the formula. The calculator does not show covariance directly because it is not a built-in function – you must do that one subtraction and division yourself.

Common Mistakes and How to Avoid Them

• Using Σx·Σy instead of Σxy: Σxy is the sum of products (x�y� + x₂y₂ + …), not the product of the sums. Many users mistakenly multiply Σx by Σy, which gives a completely different number.
• Dividing by n instead of n−1: If your data is a sample (which it usually is in statistics), use n−1. If you have the entire population, use n. The calculator’s Σxy does not tell you which one to use – you must decide based on your data.
• Forgetting to press AC after data entry: If you try to access stats while still in the data editor, you will get an error or wrong values.
• Entering data in the wrong columns: On some models, the first column is X and the second is Y. Swapping them changes the sign of the covariance if the variables are not symmetric, though the magnitude stays the same.
• Rounding intermediate values: If you round x̄ and ȳ to a few decimals before computing, the final covariance can be off, especially with small data sets. Keep full precision until the end.

Interpreting the Result

Covariance can be any real number – positive, negative, or zero.

• Positive covariance: When x increases, y tends to increase. The larger the positive value, the stronger the linear relationship (but the scale depends on the units of x and y).
• Negative covariance: When x increases, y tends to decrease.
• Zero covariance: No linear trend. However, zero covariance does not mean the variables are independent – they could have a nonlinear relationship (e.g., a U-shape).

The magnitude of covariance is not standardized – it depends on the units of x and y. If x is measured in meters and y in seconds, the covariance is in meter-seconds. To get a unitless measure of the strength of association, use the correlation coefficient r, which is covariance divided by the product of the standard deviations. Many Casio calculators give r directly in the summary stats, so you can check that instead.

Limitations and When to Use It

Covariance is useful in finance (to see how two assets move together), in physics (to see if two measurements correlate), and in any field where you have paired numeric data. However, it has clear limitations:

• It only captures linear relationships. A strong curved relationship can have near-zero covariance.
• It is sensitive to outliers – a single extreme pair can dominate the calculation.
• It is not comparable across different units or scales. You cannot say a covariance of 100 is “stronger� than a covariance of 50 unless the units and scales are identical.
• It requires at least two data points for sample covariance (n−1 in the denominator). With n=1, the sample covariance is undefined.

Use covariance when you need the raw, unit-dependent measure of co-variation. For a standardized measure, use the correlation coefficient r. For rank-based association, use Spearman’s rank correlation (not available on most Casio scientific calculators).

Frequently Asked Questions

Does my Casio calculator have a direct covariance button?

No. Casio scientific calculators do not have a dedicated covariance key. They provide the building blocks (Σxy, x̄, ȳ, n) in the statistics summary, and you compute covariance manually with the formula. Some graphing Casio models (like the fx-CG50) have a built-in covariance function in the statistics application, but the standard scientific series does not.

What is the difference between sample and population covariance on a calculator?

The only difference is the denominator. Sample covariance divides by n−1, population covariance divides by n. The calculator gives you Σxy, x̄, and ȳ, which are the same for both. You choose the denominator based on whether your data is a sample (a subset) or the entire population. If you are unsure, most statistics courses use sample covariance.

Can I use the correlation coefficient r instead of covariance?

Yes, and often it is more useful. The correlation coefficient r is a standardized version of covariance – it always falls between −1 and 1, making it easy to compare across different datasets. Casio calculators display r in the same summary stats menu. Use r when you want to describe the strength and direction of a linear relationship without worrying about units.

Why does my covariance result seem too large or too small?

Because covariance is not normalized, its magnitude depends on the units of your variables. If your x-values are in thousands and y-values in millions, the covariance will be enormous. To get a sense of relative strength, divide by the product of the standard deviations to get r. Also check that you used the correct denominator (n−1 vs n) and that you did not accidentally use Σx·Σy instead of Σxy.

What if I have more than two variables? Can I compute a covariance matrix on a Casio?

Standard scientific Casio models only handle two variables at a time. To compute a full covariance matrix, you would need to run the 2-Var stats for each pair of variables separately, or use a graphing calculator or spreadsheet software. For example, with variables A, B, and C, you would compute covariance for (A,B), (A,C), and (B,C) separately.

Does the order of data entry matter for covariance?

No. Covariance is symmetric – cov(x,y) = cov(y,x). Swapping the columns will give the same result. However, the sign of the regression slope (if you later compute a regression) would change if you swap the independent and dependent variables, but covariance itself is unaffected.

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in x | X values | - | 2 | 0
in y | Y values | - | 4 | 0
in n | Sample size | - | 3 | 0
out cov_s | Sample covariance | - | ((x*y - n*((x/n)*(y/n)))/(n-1)) | 2
out cov_p | Population covariance | - | ((x*y - n*((x/n)*(y/n)))/n) | 2