Average: Formula, Shortcuts, Tricks and Questions

Average is the total of all observations divided by their number. This page covers the arithmetic mean, standard average formula, combined average, assumed mean method, replacement-based questions and useful average shortcuts. It also explains how averages change when values are added, removed or replaced.

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What is Average?

Average is the arithmetic mean of a set of values. It is calculated by dividing the sum of all values by the total number of values.

The basic formula is Average = Sum of observations ÷ Number of observations. Therefore, Sum of observations = Average × Number of observations. For 8, 12 and 16, the average is (8 + 12 + 16) ÷ 3 = 36 ÷ 3 = 12.

Average Formula & Tricks

Important Formulas

Basic average formula
Average = Sum of observations / Number of observations

Add all the values and divide the sum by the number of values.

Sum from average
Sum of observations = Average × Number of observations

Use this when the average and the number of observations are known.

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

Here, n represents the number of observations in a group and a represents that group's average.

Average after adding a value
New average = (nA + x) / (n + 1)

If a value x is added to n observations having average A, use the old total nA and divide the new total by n + 1.

Average after replacing a value
New average = A + (New value − Old value) / n

For n observations, replacing one value changes the average by the value difference divided by n.

Quick Tricks

Use an assumed average

Choose a convenient value near all observations. The average equals the assumed value plus the average of the deviations from it.

Example: For 48, 52 and 55, take 50 as the assumed average. Deviations are −2, 2 and 5. Average = 50 + (−2 + 2 + 5) / 3 = 51⅔.
Pair values around a central number

If values occur in pairs with the same total, the average can be found directly from the pair total. For equally spaced values, the average is the middle value.

Example: The average of 18, 22, 26, 30 and 34 is the middle value, 26.
Track average change directly

When one value is replaced, do not recalculate the complete sum. Divide the difference between the new and old values by the total number of observations.

Example: If the average of 10 values is 24 and 31 replaces 21, the new average is 24 + (31 − 21) / 10 = 25.
Convert group averages into totals

For combined-average questions, first find each group's total using total = number × average. Add the totals and divide by the combined number.

Example: The combined average of 20 students with average 60 and 30 students with average 70 is (20 × 60 + 30 × 70) / 50 = 66.

Average Concepts

Arithmetic Mean and Total

The arithmetic mean is found by dividing the total of the observations by their count.

If n observations have average A, their total is nA. This relation can also be used to find a missing value when the required total and the known values are given.

Example: The average of five numbers is 18, so their total is 5 × 18 = 90. If four numbers total 67, the fifth number is 90 − 67 = 23.

Combined Average of Two or More Groups

A combined average is the total of all groups divided by the total number of observations.

The ordinary average of the group averages is correct only when all groups have equal sizes. For unequal groups, multiply each average by its group size before combining.

Example: A group of 12 has average 15 and a group of 8 has average 21. Combined average = (12 × 15 + 8 × 21) / 20 = 348 / 20 = 17.4.

Average After Adding or Removing Values

After adding or removing values, first adjust the total and then divide by the new number of observations.

If n values have average A and x is added, the new average is (nA + x) / (n + 1). If x is removed, the new average is (nA − x) / (n − 1), provided n is greater than 1.

Example: The average of 6 numbers is 14. After adding 20, the new average is (6 × 14 + 20) / 7 = 104 / 7 = 14⅘.

Average After Replacing an Observation

When one observation is replaced, the average changes by the difference between the new and old values divided by the number of observations.

If an old value x is replaced by y among n observations, new average = old average + (y − x) / n. The average rises when y is greater than x and falls when y is smaller than x.

Example: The average of 8 values is 32. If 19 is replaced by 27, the new average is 32 + 8 / 8 = 33.

Average of Consecutive and Equally Spaced Numbers

The average of consecutive or equally spaced numbers is the middle value when the number of terms is odd.

For an arithmetic progression, average = (first term + last term) / 2. This applies because terms equidistant from the ends have equal pair sums.

Example: The average of 11, 15, 19, 23 and 27 is (11 + 27) / 2 = 19.

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Average Formula, Sum and Missing Value

Learn the average formula, calculate the total sum using averages, and find missing values in average-based quantitative aptitude problems.

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Practice Average Questions

Practise published questions related to this topic.

1An article is sold for a profit of 25 percent. What is the ratio of the selling price of the article to the cost price of the article?→ 2A pair of earrings is bought for ₹27,650 and is sold at a profit of 5%. Find the profit (in ₹). (Round off your answer to the nearest rupee)→ 3√20.25+√0.36+√6.25-√2.25 = ?→ 4If an item marked at ₹1,675 is sold for ₹1,273, then what is the discount percentage?→ 5A shopkeeper sells an item for ₹324.3 after giving two successive discounts of 60% and 31% on its marked price. Had he not given any discount, he would have earned a profit of 25%. What is the cost price (in ₹) of the item?→ 6An article is sold for a profit of 50 percent. What is the ratio of the selling price of the article to the cost price of the article?→ 7A luxury watch is marked at ₹75,000. During a festival, an 18% discount is offered followed by another ₹5,000 off. What is the final price of the watch?→ 8Ratio of the value of loss incurred on an article to the selling price of the article is 5 : 7. What is the loss percentage?→ 9Ratio of the cost price of an article to the selling price of the article is 3 : 5. Profit is Rs. 720. What is the cost price of the article?→ 10√1156 + √0.36 + √0.0025 = ?→

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Quick Revision Notes

Average Revision Points

Use these formulas and rules for quick revision of average questions.

  • Average = Sum of observations ÷ Number of observations.
  • Sum = Average × Number of observations.
  • For unequal groups, combined average = Total combined sum ÷ Total number of observations.
  • If a value is added, use New average = (Old total + Added value) ÷ New count.
  • If a value is removed, use New average = (Old total − Removed value) ÷ New count.
  • When one value is replaced, average change = (New value − Old value) ÷ Number of observations.
  • The average of equally spaced terms is (First term + Last term) ÷ 2.
  • The average must lie between the smallest and largest observation.

Average FAQs

What is the average formula?

Average = Sum of observations ÷ Number of observations. For example, the average of 10, 20 and 30 is 60 ÷ 3 = 20.

How do you find the total when the average is known?

Multiply the average by the number of observations. If the average is 24 for 15 values, the total is 24 × 15 = 360.

How is the combined average calculated for two groups?

Combined average = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂). The group sizes must be used as weights when they are unequal.

What happens to the average when a value is replaced?

The average changes by (New value − Old value) ÷ Number of observations. If 40 replaces 25 in 5 observations, the average increases by 15 ÷ 5 = 3.

How do you find the average of consecutive numbers?

For consecutive or equally spaced numbers, use (First number + Last number) ÷ 2. The average of 7, 8, 9, 10 and 11 is (7 + 11) ÷ 2 = 9.

Can the average be greater than the largest observation?

No. The average of a finite set of real numbers cannot be less than the smallest value or greater than the largest value.

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