Averages: Concepts, Formulas, and Clear Examples

Average is total divided by count: $\bar{x}=\frac{\sum x_i}{n}$. Convert every average statement to a sum first—most errors are forgetting to update $n$.

Weighted averages appear as marks, mixtures, and combined groups with different sizes. Never average two averages unless group sizes match.

Replacement and missing-value questions are one formula: $\Delta\bar{x}=(b-a)/n$.

Mean and total

Arithmetic mean

$$\bar{x} = \frac{\sum x_i}{n}$$

Recover sum

$$\sum x_i = n\bar{x}$$

Example — Missing observation

Average of 8 numbers is 35 → sum = 280. Known seven sum to 241. Missing = $280-241=39$.

Effect of replacement

Swap one value

$$\Delta \bar{x} = \frac{b - a}{n}$$

Replace $a$ by $b$ in $n$ numbers.

Weighted average

Weights w_i

$$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$$

Merging two groups

Combined mean

$$\bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1+n_2}$$

• Use totals, not $\frac{\bar{x}_1+\bar{x}_2}{2}$ unless $n_1=n_2$.

Run of integers

• Average of 1..n is $(n+1)/2$.
• Adding a number above current mean pulls mean up but by less than that number.

Harmonic mean teaser (equal distance)

Not the arithmetic mean of $u$ and $v$ unless times equal, not distances.

Two speeds same distance

$$S_{\text{avg}} = \frac{2uv}{u+v}$$

Step-by-step method for exam questions

1. Write sum = $n \times$ average.
2. Track $n$ through add/remove operations.
3. Apply weighted formula when frequencies/weights differ.
4. For merged groups, combine sums with sizes.
5. Sanity-check mean lies between min and max.
6. Select option.

Formula cheat sheet

Frequently asked questions

Average of averages?

Wrong unless equal group sizes; use weighted merge.

New member raises average by 2?

New value = old average + $2\times$ new count (after addition).

Can mean be non-integer?

Yes even if data are integers.

Missing value trick?

Missing = total sum − partial sum.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.