sum of squares calculator

🔢 Sum of Squares Calculator – Instantly Calculate the Sum of Squares

A Sum of Squares Calculator helps you quickly find the sum of squares of a set of numbers — whether you’re dealing with natural numbers, custom data, or statistical applications. It’s commonly used in mathematics, statistics, data analysis, and physics.

📘 What Is the Sum of Squares?

The sum of squares is the sum of each number squared. For a list of values x1,x2,…,xnx_1, x_2, …, x_nx1​,x2​,…,xn​: Sum of Squares (SS)=x12+x22+x32+⋯+xn2\text{Sum of Squares (SS)} = x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2Sum of Squares (SS)=x12​+x22​+x32​+⋯+xn2​

🔧 Sum of First n Natural Numbers (Formula)

∑i=1ni2=n(n+1)(2n+1)6\sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6}i=1∑n​i2=6n(n+1)(2n+1)​

🔹 Example:
For n=5n = 5n=5: 12+22+32+42+52=551^2 + 2^2 + 3^2 + 4^2 + 5^2 = 5512+22+32+42+52=55

📊 Use Cases

• Math: Calculate squares of a series (e.g., 1² + 2² + … + n²)
• Statistics: Used in variance, standard deviation, and regression
• Physics: Kinetic energy, moment of inertia calculations
• Data Analysis: To find squared deviations from the mean

🧠 Example: Custom Set

For values: [3, 4, 5] SS=32+42+52=9+16+25=50\text{SS} = 3^2 + 4^2 + 5^2 = 9 + 16 + 25 = 50SS=32+42+52=9+16+25=50

❓ FAQs – Sum of Squares Calculator

🔹 What is the sum of squares used for?

It’s used to measure variability (in statistics), energy (in physics), and more.

🔹 Is the sum of squares always positive?

Yes — squaring always results in a non-negative number.

🔹 How is it different from variance?

Variance is the average of squared deviations from the mean. The sum of squares is the total before averaging.

🔹 Can I input a list of numbers?

Yes — most calculators let you enter comma-separated values like:
3, 4, 5, 6, 7

🔹 Does it work for negative numbers?

Absolutely — squaring a negative still gives a positive result.