Sum of Squares Calculator Formula

Understand the math behind the sum of squares calculator. Each variable explained with a worked example.

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Formulas Used

Sum of Squares (SS)

ss = pow(v1 - mn, 2) + pow(v2 - mn, 2) + pow(v3 - mn, 2) + pow(v4 - mn, 2)

Mean

mean_val = mn

Population Variance (SS/n)

pop_var = (pow(v1 - mn, 2) + pow(v2 - mn, 2) + pow(v3 - mn, 2) + pow(v4 - mn, 2)) / 4

Variables

VariableDescriptionDefault
v1Value 13
v2Value 25
v3Value 37
v4Value 49
mnDerived value= (v1 + v2 + v3 + v4) / 4calculated

How It Works

How to Calculate the Sum of Squares

Formula

SS = Sum of (xi - mean)^2

For each value, subtract the mean, square the result, and add them up. The sum of squares quantifies total variability in the data and is the numerator in variance calculations.

  • Population Variance = SS / n
  • Sample Variance = SS / (n - 1)

    • Worked Example

    Find the sum of squares for 3, 5, 7, 9.

    v1 = 3v2 = 5v3 = 7v4 = 9
    1. 01Mean = (3 + 5 + 7 + 9) / 4 = 24 / 4 = 6
    2. 02Deviations: 3-6=-3, 5-6=-1, 7-6=1, 9-6=3
    3. 03Squared deviations: 9, 1, 1, 9
    4. 04SS = 9 + 1 + 1 + 9 = 20

    Frequently Asked Questions

    Why square the deviations instead of just summing them?

    If you sum the raw deviations, positive and negative differences cancel out and the sum is always zero. Squaring ensures all terms are positive and gives extra weight to large deviations.

    What is the relationship between SS, variance, and standard deviation?

    Variance = SS / n (or SS / (n-1) for samples). Standard deviation = sqrt(Variance). The sum of squares is the fundamental building block.

    How is SS used in ANOVA?

    In ANOVA, total SS is partitioned into between-group SS and within-group SS to test whether group means differ significantly.

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