Calculate Syy, the sum of squared deviations of y, from y values or from Σy, Σy², and n with step-by-step work.
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Syy Formula
Syy is the sum of squared deviations of the y values from their mean. Two equivalent formulas:
Syy = Σ(yᵢ − ȳ)²
Syy = Σyᵢ² − (Σyᵢ)² / n
- yᵢ — the i-th y value
- ȳ — mean of the y values, Σyᵢ / n
- n — number of y values
- Σyᵢ — sum of the y values
- Σyᵢ² — sum of the squared y values
Both formulas give the same result. The second (computational) form avoids computing the mean first and is what the calculator uses in the Σy / Σy² mode. Syy is always ≥ 0, and Syy = 0 only when every y value is identical. Divide Syy by n − 1 to get the sample variance of y, or by n for the population variance.
Reference values
Syy on its own is just a sum, so its size depends on n and the spread of y. The table below shows how Syy connects to other common statistics.
| Quantity | Formula using Syy |
|---|---|
| Sample variance s² | Syy / (n − 1) |
| Population variance σ² | Syy / n |
| Sample standard deviation s | √(Syy / (n − 1)) |
| Total sum of squares (SST) | SST = Syy |
| Coefficient of determination R² | 1 − SSE / Syy |
| Pearson correlation r | Sxy / √(Sxx · Syy) |
| Regression slope b₁ | Sxy / Sxx (Syy used for R²) |
Quick example: for y = {4, 7, 9, 12, 15}, ȳ = 9.4, and Syy = 30.96 + 5.76 + 0.16 + 6.76 + 31.36 = 75.0.
| Dataset (y values) | n | ȳ | Syy |
|---|---|---|---|
| 5, 5, 5, 5 | 4 | 5 | 0 |
| 2, 4, 6, 8 | 4 | 5 | 20 |
| 10, 20, 30, 40, 50 | 5 | 30 | 1000 |
| 4, 7, 9, 12, 15 | 5 | 9.4 | 75.2 |
Worked example and FAQ
Example using the computational formula. Suppose Σy = 47, Σy² = 515, n = 5.
- (Σy)² = 47² = 2209
- (Σy)² / n = 2209 / 5 = 441.8
- Syy = 515 − 441.8 = 73.2
Is Syy the same as SST? Yes. In simple linear regression, the total sum of squares about ȳ is exactly Syy.
Can Syy be negative? No. It is a sum of squares. If you get a negative value from the computational formula, you have a rounding error in Σy² or Σy.
Difference between Syy and variance? Syy is the numerator. Divide by n − 1 for sample variance, or n for population variance.
When do I use Syy vs Sxx? Sxx uses x values, Syy uses y values. Both appear in the regression slope, correlation, and R² formulas.
