Averages Calculation Shortcuts
- Trick / Shortcut
- Averages Calculation Shortcuts
- Topic
- Averages
- How It Works
Several shortcuts can simplify average calculations:
1. New Average After Adding a Value: If you have $n$ numbers with an old average, and a new value is added, the new average can be found with: $\text{New Avg} = \text{Old Avg} + \frac{\text{New Value} - \text{Old Avg}}{n + 1}$
- If the new value is greater than the old average, the average increases.
- If the new value is less than the old average, the average decreases.
2. Weighted Average: For multiple groups with different averages and counts ($n_1$ items with average $A_1$, $n_2$ items with average $A_2$, etc.): $\text{Weighted Avg} = \frac{(n_1 \times A_1) + (n_2 \times A_2) + \dots}{n_1 + n_2 + \dots}$
- Example
Example 1: Adding a new value The average of 5 numbers is 20. A 6th number, 32, is added. Here, Old Avg = 20, New Value = 32, $n=5$.
$\text{New Avg} = 20 + \frac{32 - 20}{5 + 1} = 20 + \frac{12}{6} = 20 + 2 = \textbf{22}$
Verification: Sum of 5 numbers $= 5 \times 20 = 100$. New sum $= 100 + 32 = 132$. New average $= \frac{132}{6} = 22$. $\checkmark$