Sum of Squares Calculator
Compute sum of squared deviations for data sets quickly. Paste numbers, see mean, deviations, Σx², and SS with steps clearly.
Equation Preview
Helping Notes
- SS (sum of squares): SS = Σ(x − x̄)². Computational: SS = Σx² − (Σx)² / n.
- At least two values are required to compute SS for a dataset.
- You can paste values directly from spreadsheets; separators like comma, space, or newline are accepted.
- Results show n, mean, Σx, Σx², and a step-by-step deviations table.
Results
Sum of Squares (SS)
Basic Stats
Error
Steps (Deviations Table)
What is Sum of Squares Calculator?
A Sum of Squares Calculator finds the total of squared values in a sequence or data set. In arithmetic contexts, it evaluates the series of squared integers; in statistics, it measures total variation around a reference value (often the mean). The idea is simple: square each value and add the results. For a list of numbers x1, …, xn:
When the task is to sum the squares of the first n positive integers, there is a closed‑form formula:
In statistics, the sum of squares around the sample mean \bar{x} captures dispersion:
About the Sum of Squares Calculator
This tool accepts either (1) a single integer n to compute the series \(1^2+2^2+\cdots+n^2\) using the closed‑form expression, or (2) a custom list of values to square and sum. It shows step‑by‑step working, including each square, partial totals, and the final result. For data analysis, it can also compute the sum of squared deviations about the mean and report useful companions (count, mean, and optional variance based on the same sum of squares). All formulas display clearly and resize with your screen for comfortable reading.
How to Use this Sum of Squares Calculator
- Choose mode: series (1 to n) or custom data list.
- Enter n (series mode) or paste numbers separated by commas/spaces (data mode).
- Optionally select “about mean” to compute \(\sum (x_i-\bar{x})^2\).
- Press calculate to see squares, partial sums, and the final total.
- Copy the steps and result for homework, reports, or quality checks.
Examples
Example 1: Series 1² + 2² + … + 10²
Apply the closed‑form formula with n=10.
Example 2: Custom list [3, 5, 7]
Square each value and add.
Example 3: Sum of squares about the mean for [3, 5, 7]
The mean is 5; subtract the mean before squaring.
FAQs
What does “sum of squares” mean in simple terms?
It’s the total you get by squaring each number in a set and adding those squares together.
Is there a quick formula for 1²+2²+…+n²?
Yes, use the closed‑form \(n(n+1)(2n+1)/6\) for any positive integer n.
Can the calculator handle negative numbers?
Yes. Squaring removes the sign, so −4 and +4 both contribute 16 to the total.
How do I compute “about the mean” sums?
Select the option to subtract the mean first; the tool squares each deviation and adds them up.
What’s the difference between SS and variance?
Variance scales sum of squares by a divisor (e.g., n−1 for samples); SS itself is the unscaled total.
Does this work for decimals or fractions?
Absolutely. The numbers are squared as real values, and the total reflects the same precision.
Can I paste a long list of numbers?
Yes. Paste values separated by commas or spaces; the calculator parses them and computes the total efficiently.
Will I see each step?
Yes. You’ll see every square and the running partial sums before the final total.
How big can n be for the series formula?
Very large. The formula is constant‑time; limits depend mainly on device memory and numeric range.
What if I only want the total, not the steps?
You can ignore the step table and read the final result line—it’s displayed prominently.
Is this useful for regression analysis?
Yes. Sums of squares underpin measures like residual sum of squares and total sum of squares in modeling.
Does order of numbers matter?
No. Addition is commutative; the total is the same regardless of input order.