Sum of Products Calculator (2024)

Created by Wei Bin Loo

Reviewed by

Rijk de Wet

Last updated:

Jan 18, 2024

This sum of products calculator will help you calculate the sum of the products of numbers in two series. This sum of products calculator can also help you to understand the relationship between the individual elements in the series and how they contribute to the overall result. It provides a convenient way to analyze the combined effect of multiplying corresponding elements from both series and obtaining their sum.

💡 Looking for other ways of summing series together? Check out our digit sum calculator and sum of series calculator for more.

After reading this article, you will understand what sum of products is, what the sum of products formula is, and how to calculate the sum of products.

What is sum of products?

The sum of products refers to the sum obtained by multiplying corresponding elements from two or more series or datasets and then adding up the results. It allows us to measure the combined effect of the variables or data points, providing insights into their collective impact. Whether you're analyzing sales data, studying the behavior of multiple variables, or trying to find patterns in datasets, the sum of products can be a valuable tool.

How to calculate sum of products?

To understand the sum of products calculation, look at the following formula:

  • Dataset A: [2, 4, 6, 8]
  • Dataset B: [1, 3, 5, 7]

To calculate the sum of products, you need to perform the following three steps:

  1. Determine the series of numbers to perform this calculation

    The first step is to determine the series of numbers you want to calculate the sum of products with. For our example, the datasets involved are:

    • Dataset A: [2, 4, 6, 8]
    • Dataset B: [1, 3, 5, 7]
  2. Multiply the corresponding number

    The next step is to compute the product of the corresponding numbers, such as a1×b1a_1 \times b_1a1​×b1​, a2×b2a_2 \times b_2a2​×b2​, and so on and so forth.

    For our example, they are:

    • a1×b1=2×1=2a_1 \times b_1 = 2 \times 1 = 2a1​×b1​=2×1=2
    • a2×b2=4×3=12a_2 \times b_2 = 4 \times 3 = 12a2​×b2​=4×3=12
    • a3×b3=6×5=30a_3 \times b_3 = 6 \times 5 = 30a3​×b3​=6×5=30
    • a4×b4=8×7=56a_4 \times b_4 = 8 \times 7 = 56a4​×b4​=8×7=56

💡 You can use our multiplication calculator to understand more on this topic.

  1. Add up results of the multiplications

    The last step is to sum everything up using the sum of products formula below. Hence, the results for our example is:

    a1×b1+a2×b2+a3×b3+a4×b4a_1 \times b_1 + a_2 \times b_2 + a_3 \times b_3 + a_4 \times b_4a1​×b1​+a2​×b2​+a3​×b3​+a4​×b4​
    =2×1+4×3+6×5+8×7= 2 \times 1 + 4 \times 3 + 6 \times 5 + 8 \times 7=2×1+4×3+6×5+8×7
    =100= 100=100

Restated in a symbol formula — if we have two series a=[a1,...,an]a = [a_1, ..., a_n]a=[a1​,...,an​] and b=[b1,...,bn]b = [b_1, ..., b_n]b=[b1​,...,bn​], then the sum of products is given by:

sumofproducts=∑i=1n(ai×bi)\text{sum of products} = \sum_{i=1}^{n} (a_i \times b_i)sumofproducts=i=1∑n​(ai​×bi​)


What is the sum of products of two lists if one list has all 0 in it?

The sum of products of these two list will be 0. This is because each number in the list without zeros will be multiplied by zero, leading to a sum of zeros.

How can I calculate the sum of products?

You can calculate the sum of products in 3 steps:

  1. Determine the two series of numbers.
  2. Multiply the corresponding number, for example, a₁ × b₁.
  3. Add up results of the multiplications:

sum products = a₁ × b₁ + a₂ × b₂ + a₃ × b₃ + ... + an × bn

How can I apply the sum of products in real-life scenarios?

The sum of products has various applications. It can be used in statistical analysis to measure covariance, in financial analysis to calculate weighted average cost of capital (WACC), and in regression analysis to determine coefficients and intercepts of a regression model.

Can the sum of products be calculated for more than two series?

Yes, the sum of products can be calculated for any number of series. The formula remains the same, and you would multiply and sum the corresponding elements from each series.

Wei Bin Loo

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