Weighted Average: Formula, Methods and Solved Questions

Weighted Average gives the average value when different observations have different levels of importance or frequencies. It is calculated by multiplying each value by its weight, adding these products, and dividing by the total weight. This page covers the weighted average formula, frequency-based calculations, combined averages, and useful shortcuts.

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What Is Weighted Average?

Weighted Average is an average in which each value is multiplied by its assigned weight before division by the total weight. The weight may represent frequency, quantity, percentage, duration, or importance.

The weighted average formula is Weighted Average = Σ(wx) ÷ Σw, where x represents an observation and w represents its weight. For values 10 and 20 with weights 2 and 3, the weighted average is (10 × 2 + 20 × 3) ÷ (2 + 3) = 80 ÷ 5 = 16. The weighted average always lies between the smallest and largest values when all weights are positive.

Weighted Average Formula & Tricks

Important Formulas

Basic weighted average formula
Weighted Average = Σ(wx) / Σw

Multiply every value x by its corresponding weight w, add the products, and divide by the sum of all weights.

Frequency form
Mean = Σ(fx) / Σf

When f is the frequency of a value x, the arithmetic mean is calculated as a weighted average.

Combined average of groups
Combined Average = (n₁a₁ + n₂a₂ + ... + nₖaₖ) / (n₁ + n₂ + ... + nₖ)

Multiply each group average by its group size, add the products, and divide by the total number of observations.

Weighted average using deviations
Weighted Average = A + Σ[w(x − A)] / Σw

Here A is a convenient assumed value. This reduces calculation when the values are close to A.

Quick Tricks

Use deviations from a convenient value

Choose an assumed value A near the observations. Find each deviation x − A, multiply it by its weight, and add the correction to A.

Example: For values 48 and 52 with weights 3 and 2, take A = 50. The weighted deviation is [3(−2) + 2(2)] ÷ 5 = −2 ÷ 5. Therefore, the weighted average is 50 − 0.4 = 49.6.
Use cross-difference for two values

For two values a and b with weights m and n, the weighted average is (am + bn) ÷ (m + n). If the average is nearer to one value, its weight is larger.

Example: For 20 and 50 with weights 3 and 2, the average is (20 × 3 + 50 × 2) ÷ 5 = 160 ÷ 5 = 32.
Convert percentages to weights

Percentage weights can be used directly if they have the same base. Otherwise, convert them to proportional weights before applying the formula.

Example: A score of 70 has 40% weight and a score of 80 has 60% weight. The weighted average is 70 × 0.4 + 80 × 0.6 = 76.

Weighted Average Concepts

Weighted Average with Frequencies

When a value occurs multiple times, its frequency acts as its weight in the average calculation.

For observations x₁, x₂, ..., xₙ with frequencies f₁, f₂, ..., fₙ, use Mean = Σ(fx) ÷ Σf. The denominator is the total frequency, not the number of distinct values.

Example: If 5 occurs 2 times and 9 occurs 3 times, the mean is (5 × 2 + 9 × 3) ÷ (2 + 3) = 37 ÷ 5 = 7.4.

Weighted Average with Percentage Weights

When components contribute by percentages, multiply each component value by its percentage expressed as a decimal or proportional weight.

The weights must be measured on the same basis and their total is generally 1 or 100%. For weights p₁, p₂, ..., the average is Σ(px) when decimal weights total 1.

Example: If theory marks are 72 with 60% weight and practical marks are 84 with 40% weight, the final score is 72 × 0.60 + 84 × 0.40 = 76.8.

Combined Average of Two Groups

The average of two groups is found by weighting each group average by the number of members in that group.

If group sizes are n₁ and n₂ and their averages are a₁ and a₂, the combined average is (n₁a₁ + n₂a₂) ÷ (n₁ + n₂). A simple average of a₁ and a₂ is valid only when both group sizes are equal.

Example: A group of 20 students has an average of 60 and a group of 30 students has an average of 70. Their combined average is (20 × 60 + 30 × 70) ÷ 50 = 66.

Properties of Weighted Average

With positive weights, a weighted average lies between the minimum and maximum values and gives greater influence to values with greater weights.

If all weights are multiplied by the same non-zero constant, the weighted average remains unchanged because the constant cancels from the numerator and denominator. A value with zero weight does not affect the result.

Example: The weighted average of 10 and 30 with weights 1 and 4 is (10 + 120) ÷ 5 = 26, which is closer to 30 because 30 has the larger weight.

Weighted Average Video Lessons

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3 Lessons
Lesson 1 of 3 Quick Revision

Average of Two Groups

Understand how to calculate and compare the average of two groups using their group sizes and individual averages in quantitative aptitude problems.

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Practice Weighted Average Questions

Practise published questions related to this topic.

1The marked price of a painting was ₹3,800, but the shopkeeper offered a double discount in a successive manner of 22% and 25%. Ultimately for how much did he sell the item?→ 2The difference between cost price and selling price of an article is Rs. 504. Loss percentage is 18 percent. What is the selling price of the article?→ 3Trader marks an item 50% above cost price and offers a 20% discount on the marked price. If the customer pays ₹720, what is the cost price of the item?→ 4A trader sold an article for ₹720 at a 20% loss. What was the cost price?→ 5The LCM of two numbers is 84. The numbers are in the ratio 4 : 3. Find the sum of the numbers.3129→ 6Selling price of an article is Rs. 7540. Profit percentage is 30 percent. What is the cost price of the article?→ 7An article is sold for 30 percent profit. The value of discount given is equal to the (2/5) of the selling price. What is the cost price of the article?→ 8Two successive discounts of 10 percent and 30 percent are given on an article. What is the net discount?→ 9A dealer marks his goods at 37% above the cost price and allows a discount of 16% on the marked price. What is his gain or loss percentage?→ 10The Highest Common Factor of the two numbers is 12. Which of the following can be the Least Common Multiple of the two numbers?→

Weighted Average Quick Quiz

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Quick Revision Notes

Weighted Average Revision Points

Recall these formulas and rules for quick calculations.

  • Use Weighted Average = Σ(wx) ÷ Σw.
  • For frequency data, use Mean = Σ(fx) ÷ Σf.
  • For groups, multiply each average by its group size before adding.
  • Percentage weights must use a common basis; decimal weights usually total 1.
  • With positive weights, the result lies between the smallest and largest values.
  • Multiplying every weight by the same constant does not change the weighted average.
  • A zero-weight observation has no effect on the answer.
  • Use deviations from a convenient assumed value to reduce arithmetic.

Weighted Average FAQs

What is the weighted average formula?

The formula is Weighted Average = Σ(wx) ÷ Σw, where x is each value and w is its corresponding weight.

How is a weighted average different from a simple average?

A simple average gives equal weight to all values. A weighted average gives different values different weights, so values with larger weights influence the result more.

What is the weighted average of 40 and 60 with weights 3 and 2?

Weighted average = (40 × 3 + 60 × 2) ÷ (3 + 2) = 240 ÷ 5 = 48.

How do you calculate a combined average?

Use Combined Average = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂), where n is group size and a is group average.

Can percentages be used directly as weights?

Yes, if all percentages have the same base. For example, 40% and 60% may be used as weights 40 and 60, or as decimal weights 0.4 and 0.6.

What happens if every weight is multiplied by 5?

The weighted average remains unchanged because 5 is a common factor in both Σ(wx) and Σw.

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