Sum of Squares Formula: SST, SSR and SSE Explained

The sum of squares is the total of squared distances: add up the squares of some set of numbers. In statistics the phrase almost always means the sum of squared deviations from the mean, written SS or SST, and it is the raw material for variance, standard deviation, analysis of variance and R-squared. This guide gives the formulas, one dataset worked two ways (a regression partition and a one-way ANOVA partition), a table of the naming differences that trip people up, and a reporting example.

What is sum of squares?

Sum of squares is the sum of the squares of a set of values. Two different quantities share the name, and the formulas below separate them:

• Raw (uncorrected) sum of squares: square each value and add them, Σy_i².
• Corrected, or total, sum of squares: square each value’s distance from the mean and add them, Σ(y_i − ȳ)². The NIST/SEMATECH e-Handbook of Statistical Methods calls the numerator of the sample variance the sum of squares of deviations from the mean, also called the corrected sum of squares and abbreviated Total SS or SS(Total).

Dividing the corrected sum of squares by n − 1 gives the sample variance, s² = Σ(y_i − ȳ)² / (n − 1); its square root is the standard deviation.

Sum of squares formula

Total sum of squares (SST)

SST = Σ(y_i − ȳ)², where y_i is the i-th observation, ȳ is the mean of all n observations, and the sum runs over i = 1 to n.

A shortcut form, which NIST calls the raw sum of squares minus the “correction term for the mean”, gives the same number: SST = Σy_i² − (Σy_i)² / n. OpenStax Introductory Statistics 2e writes the same identity as SS_total = Σx² − (Σx)²/n.

The partition: explained plus unexplained

For a regression with an intercept, the total splits into a part the model accounts for and a part it does not:

• SST = SSR + SSE
• SSR (regression, or explained, sum of squares) = Σ(ŷ_i − ȳ)², where ŷ_i is the fitted value.
• SSE (error, or residual, sum of squares) = Σ(y_i − ŷ_i)².
• R² = SSR / SST = 1 − SSE / SST.

One-way ANOVA partition

With k groups, n_i observations in group i and N observations overall, NIST gives SS(Total) = SST + SSE, with the group (treatment) part Σ n_i(ȳ_i − ȳ)² and the error part Σ_iΣ_j(y_ij − ȳ_i)². OpenStax calls these SS_between and SS_within, with MS_between = SS_between / (k − 1) and MS_within = SS_within / (n − k).

Worked example: one dataset, two partitions

Regression partition

Six students record hours studied (x) and exam score (y). This is an illustrative dataset made up for the calculation.

Hours (x)	Score (y)	y − ȳ	(y − ȳ)²	Fitted ŷ	y − ŷ
1	52	−13.5	182.25	51.571	0.429
2	58	−7.5	56.25	57.143	0.857
3	61	−4.5	20.25	62.714	−1.714
4	69	3.5	12.25	68.286	0.714
5	72	6.5	42.25	73.857	−1.857
6	81	15.5	240.25	79.429	1.571

1. Mean: Σy = 393, so ȳ = 393 / 6 = 65.5.
2. SST: 182.25 + 56.25 + 20.25 + 12.25 + 42.25 + 240.25 = 553.5.
3. Check with the shortcut: Σy² = 26,295 and (Σy)²/n = 393²/6 = 25,741.5, so SST = 26,295 − 25,741.5 = 553.5. The raw sum of squares (26,295) and the corrected one (553.5) are very different numbers; the gap is the correction term for the mean.
4. Least-squares line: slope = 97.5 / 17.5 = 5.5714 and intercept = 65.5 − 5.5714 × 3.5 = 46.0, so ŷ = 46.0 + 5.5714x.
5. SSE: the squared residuals are 0.184, 0.735, 2.939, 0.510, 3.449 and 2.469, which sum to 10.286.
6. SSR = SST − SSE = 553.5 − 10.286 = 543.214. Computed directly from Σ(ŷ − ȳ)² it gives the same value.
7. R² = 543.214 / 553.5 = 0.981.

One-way ANOVA partition

Take twelve scores from three groups of four (again illustrative): Group A = 52, 58, 61, 69; Group B = 72, 81, 64, 70; Group C = 55, 60, 66, 63. The group means are 60, 71.75 and 61; the grand mean is 771 / 12 = 64.25.

• SS_between = 4(60 − 64.25)² + 4(71.75 − 64.25)² + 4(61 − 64.25)² = 72.25 + 225 + 42.25 = 339.5.
• SS_within = 150 + 148.75 + 66 = 364.75 (the squared deviations inside groups A, B and C).
• SS_total = 339.5 + 364.75 = 704.25, which matches summing (y − 64.25)² over all twelve scores.
• df_between = k − 1 = 2 and df_within = n − k = 9, so MS_between = 169.75, MS_within = 40.53 and F = 169.75 / 40.53 = 4.19, p = .052.

Interpretation

A sum of squares has no natural scale on its own: it grows with the number of observations and with the square of the measurement unit. It becomes interpretable as a share of the total or after dividing by degrees of freedom.

Result	What it says	In the examples
SSE / SST near 0	The fitted values sit close to the data; R² near 1	10.286 / 553.5 = 0.019, so R² = 0.981
SSR / SST near 0	The model explains little beyond the mean	Not the case in the regression example
SSbetween / SStotal	Share of variability attributable to group membership (eta-squared)	339.5 / 704.25 = 0.482
SSwithin / dfwithin	Average spread inside groups (MSwithin), the denominator of F	364.75 / 9 = 40.53

The F ratio of mean squares is covered in how to interpret the F-statistic, and the model-fit reading of SSR/SST in how to interpret R-squared. For the difference between eta-squared and partial eta-squared, see eta-squared vs partial eta-squared. Note the ANOVA example: an eta-squared of 0.48 sits beside p = .052, so a large share of variability does not by itself make the F test significant when groups are this small.

Assumptions and common mistakes

• Mixing up raw and corrected sums. Σy² (26,295 above) is not SST (553.5). Spreadsheets and calculators expose both; use the corrected one for variance and partitions.
• Expecting SST = SSR + SSE without an intercept. The partition above is for models that include a mean or intercept term. statsmodels reports both a centered total sum of squares (about the mean) and an uncentered one, and R-squared depends on which is used.
• Reading “SSR” the same in every source. Naming differs, as the table shows. In statsmodels, ssr is the sum of squared residuals, which many regression texts call SSE.
• Comparing sums of squares across datasets of different size. Compare ratios or mean squares instead.
• Ignoring the model’s assumptions. The sums are arithmetic and always computable. The F test built on them assumes independent observations, roughly normal errors and similar variance across groups.

Naming differences between sources

Source	Total	Explained / between groups	Unexplained / within groups
NIST/SEMATECH e-Handbook, one-way ANOVA	SS(Total)	SST (treatments)	SSE
OpenStax Introductory Statistics 2e	SStotal	SSbetween	SSwithin
statsmodels RegressionResults	centered_tss (about the mean) or uncentered_tss	ess, the explained sum of squares	ssr, the sum of squared residuals

Note that NIST’s SST means the treatment sum of squares, while this page and most regression material use SST for the total. Always check the definition before comparing output.

How to report it

Sums of squares are usually shown in an ANOVA table, while the text reports the test statistic, degrees of freedom, exact p value and an effect size. The two examples above could be written up as follows:

• ANOVA: A one-way ANOVA found no significant difference among the three groups, F(2, 9) = 4.19, p = .052, η² = .48.
• Regression: Hours studied predicted exam score, F(1, 4) = 211.25, p < .001, R² = .98.

How to calculate sum of squares in Excel, R or Python

Excel

=SUMSQ(B2:B7)            raw sum of squares: 26295
=DEVSQ(B2:B7)            corrected sum of squares (SST): 553.5
=INDEX(LINEST(B2:B7,A2:A7,TRUE,TRUE),5,1)   SSR: 543.214
=INDEX(LINEST(B2:B7,A2:A7,TRUE,TRUE),5,2)   SSE: 10.286

R

x <- 1:6; y <- c(52,58,61,69,72,81)
sum((y - mean(y))^2)     # SST = 553.5
anova(lm(y ~ x))         # "Sum Sq" column: x = 543.21, Residuals = 10.29

Python

import numpy as np, statsmodels.api as sm
x = np.arange(1, 7); y = np.array([52,58,61,69,72,81])
m = sm.OLS(y, sm.add_constant(x)).fit()
print(m.centered_tss)   # SST 553.5
print(m.ess)            # explained SS 543.21
print(m.ssr)            # residual SS 10.29 (statsmodels names this ssr)

For a worked statsmodels ANOVA table, see ANOVA in Python with statsmodels.

FAQ

What is the formula for the sum of squares?

For deviations from the mean it is SST = Σ(y_i − ȳ)², equivalently Σy_i² − (Σy_i)²/n. The plain sum of squares of the values is Σy_i².

What is the difference between SST, SSR and SSE?

SST is total variability around the mean, SSR the part explained by the regression and SSE the residual part, with SST = SSR + SSE. Source naming varies, so check the definition (see the naming table above).

Can a sum of squares be negative?

No. Every term is a square, so each is zero or positive. A negative value in output points to a different quantity or a calculation error.

How does sum of squares relate to variance?

The sample variance is the corrected sum of squares divided by n − 1.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, section 7.4.3.1, One-way ANOVA: https://www.itl.nist.gov/div898/handbook/prc/section4/prc431.htm (read 2026-10-29).
• OpenStax, Introductory Statistics 2e, section 13.2, The F Distribution and the F Ratio: https://openstax.org/books/introductory-statistics-2e/pages/13-2-the-f-distribution-and-the-f-ratio (read 2026-10-29).
• statsmodels documentation, RegressionResults: https://www.statsmodels.org/stable/generated/statsmodels.regression.linear_model.RegressionResults.html (read 2026-10-29).
• The numerical examples use illustrative data made up for this page; every sum of squares in them can be reproduced from the tables above.