Arithmetic Mean
The arithmetic mean is the total of all values divided by how many there are. For CAT the golden re-statement is total = average × count, because almost every question gives you two of these three and asks for the third. Two shortcuts save real time. First, for any evenly spaced set (an arithmetic progression, consecutive integers, consecutive even numbers) the average equals the middle term, or (first + last)/2 — no summing needed. Second, the deviation method: pick a convenient assumed mean, add up only the small deviations from it, divide by the count, and add back. To average 197, 203, 199, 201, assume 200 and the deviations −3, +3, −1, +1 cancel to 0, so the mean is exactly 200. A key property: the sum of deviations of all values from their own mean is always zero — useful as a sanity check.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.
Formula Reference Sheet
Core average relations
| Arithmetic mean | Average = (Sum of observations) / (Number of observations) |
|---|---|
| Sum from average | Sum = Average × Count |
| Average of first n natural numbers | (n + 1) / 2 |
| Average of an arithmetic progression | (First term + Last term) / 2 |
| Deviation (shift) method | Average = Assumed mean + (Sum of deviations) / Count |
Weighted, speed & replacement tools
| Weighted average | (w₁a₁ + w₂a₂ + …) / (w₁ + w₂ + …) |
|---|---|
| Alligation (ratio of weights) | w₁ : w₂ = (A₂ − Avg) : (Avg − A₁) |
| Average speed (whole journey) | Total distance / Total time |
| Equal-distance two speeds | 2xy / (x + y) (harmonic mean) |
| Change in average on replacement | New value = Old value ± (Change in average × Count) |