Sum of Squares Calculator
Calculate sum of squared deviations to measure data variability and dispersion
Calculate Sum of Squares
You need at least 2 data points to calculate sum of squares. Current valid points: 0
Sum of Squares Results
Sum of Squares Formula: SS = Σ(yᵢ - ȳ)²
Interpretation
Enter at least 2 data points to see interpretation of results.
Example: Temperature Measurements
Sample Data
Temperature readings (°C): 20, 22, 18
Sample size (n): 3
Step-by-Step Calculation
1. Mean: ȳ = (20 + 22 + 18) ÷ 3 = 20°C
2. Deviations: (20-20) = 0, (22-20) = 2, (18-20) = -2
3. Squared deviations: 0² = 0, 2² = 4, (-2)² = 4
4. Sum of squares: SS = 0 + 4 + 4 = 8
Result Interpretation
The sum of squares of 8 indicates moderate variability in temperature readings. This value is used to calculate variance (SS/(n-1) = 8/2 = 4) and standard deviation (√4 = 2°C).
Key Concepts
Sum of Squares
Total of all squared deviations from the mean
Mean
Average value of all data points
Variance
Average of squared deviations (SS/(n-1))
Types of Sum of Squares
Total SS (TSS)
Total variation in data from grand mean
Explained SS (ESS)
Variation explained by the model
Residual SS (RSS)
Unexplained variation (error)
Relationship: TSS = ESS + RSS
Quick Tips
Higher SS = greater data variability
SS is always non-negative (≥ 0)
Used to calculate variance and standard deviation
Essential for ANOVA and regression analysis
Understanding Sum of Squares
What is Sum of Squares?
The sum of squares (SS) is a statistical measure that quantifies the total variability or dispersion in a dataset. It represents the sum of all squared deviations from the mean, providing insight into how spread out the data points are.
Why is it Important?
- •Measures data variability and dispersion
- •Foundation for calculating variance and standard deviation
- •Essential for ANOVA and regression analysis
- •Helps identify outliers and data patterns
Formula and Calculation
SS = Σ(yᵢ - ȳ)²
Sum of squared deviations from mean
- SS: Sum of squares
- yᵢ: Individual data points
- ȳ: Sample mean
- Σ: Sum of all values
- (yᵢ - ȳ): Deviation from mean
Key Insight: Squaring deviations ensures all values are positive and gives more weight to larger deviations from the mean.
Applications
- •Analysis of Variance (ANOVA)
- •Regression analysis and model fitting
- •Quality control and process monitoring
- •Outlier detection and data validation
- •Comparing variability between groups
Related Measures
Sample Variance
s² = SS / (n - 1)
Standard Deviation
s = √(SS / (n - 1))
Mean Squared Error
MSE = SS / n (for residuals)