Averages: Explained with Examples, A Complete Student Guide

An average is a single number that represents a whole group: add up all the values and divide by how many there are. The average of 70, 80 and 90 is (70 + 80 + 90) ÷ 3 = 80. It tells you the typical value — the number each member would have if the total were shared equally.

Averages run the modern world: cricket batting averages, class test averages, average speed of a journey, average rainfall for a month. Any time someone summarises many numbers with one, they are using an average — so reading them correctly is a life skill, not just an exam topic.

Key Concepts

• Mean (arithmetic average): sum of values ÷ number of values. This chapter's "average" almost always means the mean.
• Sum from average: total = average × count. If the average of 5 numbers is 20, their sum is 100.
• Weighted average: when values count unequally, multiply each by its weight before dividing by the total weight.
• New average after adding or removing a value: recompute the total first, then divide by the new count.
• Outliers: one extreme value can drag the average far from most values.

Formulas

• Average = (x₁ + x₂ + … + xₙ) ÷ n, where n is the count of values.
• Sum = average × n.
• Weighted average = (w₁x₁ + w₂x₂ + … + wₙxₙ) ÷ (w₁ + w₂ + … + wₙ).
• Average of first n natural numbers = (n + 1) ÷ 2.
• Average speed = total distance ÷ total time (a weighted average, not a simple mean of speeds).

Worked Example

Ravi's scores in 5 tests are 78, 85, 92, 88 and 77. Find his average score.

1. Add the scores: 78 + 85 = 163; 163 + 92 = 255; 255 + 88 = 343; 343 + 77 = 420.
2. Count the tests: n = 5.
3. Divide: 420 ÷ 5 = 84.

His average is 84 marks. Check backwards: 84 × 5 = 420, which matches the total. If a sixth test score of 96 is added, the new total is 516 and the new average is 516 ÷ 6 = 86.

Common Misconceptions

• "The average of two averages is their mean." Only if both groups are the same size. Class A (30 students, average 80) and Class B (10 students, average 60) combine to (30×80 + 10×60) ÷ 40 = 3000 ÷ 40 = 75, not 70. Always weight by group size.
• "Average means most people got that score." Not necessarily. In the set {0, 0, 0, 100, 100}, the average is 40 but nobody scored 40. The average is a balancing point, not the most common value.
• "Removing the lowest score always raises the average a lot." It raises it only slightly when there are many values: removing 77 from Ravi's scores gives 343 ÷ 4 = 85.75, up just 1.75 marks.

Real-World Applications

• Teachers and boards report class performance as average marks.
• Sports statistics rank players with batting, bowling and scoring averages.
• Businesses track average daily sales and average customer spending.
• Meteorologists report average temperature and rainfall for planning and records.

An average is total divided by count — and total equals average times count, which unlocks most tricky problems. Weight properly when groups differ in size, and remember that one number can never tell you the full story of the data behind it.