Descriptive Statistics
Sum of Squares Calculator
Corrected sum of squares adds the squared distances from the mean, and the raw sum of squares adds the squared values themselves.
Worksheet
Press Calculate to see the result.
Corrected sum of squares adds the squared distances from the mean, and the raw sum of squares adds the squared values themselves.
What this calculator answers
The values 5, 8, 8, and 11 have a mean of 8. Their gaps from 8 are -3, 0, 0, and 3. Squaring those gaps and adding them gives 18, the corrected sum of squares, which is the headline. Squaring the original values and adding those squares gives 274, the raw sum of squares. Both totals are in squared units. The corrected total is the one that later becomes a variance when it is divided by n or by n minus 1. The raw total still contains the size of the mean, which is why 18 plus n times the squared mean recovers 274.
The headline is the corrected sum of squares, the sum of squared deviations from the mean. The raw sum of squares, the sum of each value squared, is the second figure. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Corrected and raw sums of squares uses this relationship.
corrected = sum of (x - mean)^2. raw = sum of x^2.
- mean. The arithmetic mean the deviations are taken from.
- corrected SS. Sum of squared deviations, the headline.
- raw SS. Sum of the squared values, before the mean is removed.
- n * mean^2. The piece that links raw SS to corrected SS.
Centering removes the level of the batch so only spread remains in the corrected total. The algebra raw = corrected + n * mean^2 is a check: if the two routes disagree, a value was skipped. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The values are 5, 8, 8, 11. The mean is 8. The squared deviations from that mean add to 18.
That corrected sum of squares is the headline. The raw squares add to 274. The link is n times the squared mean: 4 * 8^2 = 256.
Corrected plus that link is 18 + 256 = 274, which matches the raw sum of squares. Neither total has been divided by n or by n minus 1. Those quotients are variances.
| Corrected sum of squares | 18 |
|---|---|
| Raw sum of squares | 274 |
| Mean | 8 |
| Count | 4 |
A second situation
The last 11 becomes 20
The values are 5, 8, 8, 20. The mean is 10.25. The squared deviations from that mean add to 132.75.
That corrected sum of squares is the headline. The raw squares add to 553. The link is n times the squared mean: 4 * 10.25^2 = 420.25.
Corrected plus that link is 132.75 + 420.25 = 553, which matches the raw sum of squares. Neither total has been divided by n or by n minus 1. Those quotients are variances.
| Corrected sum of squares | 132.75 |
|---|---|
| Raw sum of squares | 553 |
| Mean | 10.25 |
| Count | 4 |
Replacing 11 with 20 moves the mean and inflates the gaps. The corrected sum of squares rises from 18 to 132.75, and the raw sum of squares rises from 274 to 553. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
11 was replaced by 20. The headline result went from 18 to 132.75.
The nudged inputs
The values are 5, 8, 8, 20. The mean is 10.25. The squared deviations from that mean add to 132.75.
That corrected sum of squares is the headline. The raw squares add to 553. The link is n times the squared mean: 4 * 10.25^2 = 420.25.
Corrected plus that link is 132.75 + 420.25 = 553, which matches the raw sum of squares. Neither total has been divided by n or by n minus 1. Those quotients are variances.
| Corrected sum of squares | 132.75 |
|---|---|
| Raw sum of squares | 553 |
| Mean | 10.25 |
| Count | 4 |
How to read the result
A corrected sum of squares of 18 is not a variance and not a standard deviation. Divide by 3 to get the sample variance of these four numbers, or by 4 to get the population variance. The raw total 274 still includes the squared level of the mean. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- Every value you intend to include is typed once.
- The center for the corrected total is the arithmetic mean of this same list.
- Squared units are acceptable for the total you will carry forward.
- You want both the centered total and the uncentered total.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Dividing by n minus 1 and still calling the result a sum of squares.
- Reporting the raw sum of squares as if the mean had been removed.
- Adding the absolute deviations instead of the squared deviations.
- Mixing values from two groups in one total when each group needs its own mean.
Where a different calculator is the honest choice
Use the variance calculator to divide this corrected total by n minus 1 and by n. Use the standard deviation calculator when you need the square root. Use the mean calculator when the sum itself, not the squared gaps, is the question. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Variance Calculator, Standard Deviation Calculator, Mean Calculator.
Check yourself
Values are 5, 8, 8, and 11. What are the corrected and raw sums of squares?
The corrected sum of squares is 18 and the raw sum of squares is 274.
Using the result in a sentence
ANOVA pieces, regression totals, and the numerator of a variance all start from a corrected sum of squares. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
Squaring makes a single distant value own the total. The corrected sum of squares also changes if you change which observations share a mean, so do not pool groups before you have decided they belong together. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Sum of Squares Calculator commit to?
This page uses Corrected and raw sums of squares: corrected = sum of (x - mean)^2. raw = sum of x^2.. Centering removes the level of the batch so only spread remains in the corrected total. The algebra raw = corrected + n * mean^2 is a check: if the two routes disagree, a value was skipped. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Numbers, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the corrected sum of squares is 18. A corrected sum of squares of 18 is not a variance and not a standard deviation. Divide by 3 to get the sample variance of these four numbers, or by 4 to get the population variance. The raw total 274 still includes the squared level of the mean. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Dividing by n minus 1 and still calling the result a sum of squares. Squaring makes a single distant value own the total. The corrected sum of squares also changes if you change which observations share a mean, so do not pool groups before you have decided they belong together. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the variance calculator to divide this corrected total by n minus 1 and by n. Use the standard deviation calculator when you need the square root. Use the mean calculator when the sum itself, not the squared gaps, is the question. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The corrected sum of squares is 18 and the raw sum of squares is 274. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.