Statistics

The mean is the total wearing a mask

$\bar{x} = \frac{\sum x_i}{n}$ Every average question is solved the same way: convert to totals, do arithmetic on totals, convert back. Total $= \text{mean} \times n$ — write this on top of your rough sheet and the entire family collapses to one subtraction.

The four recurring moves

1. Missing value: the mean of $n$ numbers is $m$ and $n-1$ are known — the missing one is $mn - (\text{known sum})$.
2. Member joins: including a new member changes the mean from $a$ to $b$ (new count $n+1$). Newcomer $= b(n+1) - an = b + n(b-a)$.
3. Member leaves: excluding a member changes the mean from $a$ to $b$ (new count $n-1$). Leaver $= an - b(n-1) = a + n(a-b)$.
4. Replacement: replacing $x$ by $y$ shifts the mean by $d$: $y = x + nd$ — the replacement carries $n$ times the shift.

Combined mean of two groups

$\bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$ A weighted average: the group means are pulled toward the bigger group. Equal sizes make it the plain average of the two means. Three groups extend the same formula; a missing group mean is one subtraction from the grand total.

Correcting a misread value

"The average of 30 results was 20 but 24 was recorded as 21": the total rises by $24 - 21 = 3$, so the corrected mean is $\bar{x} + \frac{3}{30} = 20.1$. Add the error to the total, re-divide. The error can be negative (a bigger number wrongly recorded as smaller).

Averages of consecutive numbers

For any arithmetic progression, the mean is the middle term (odd count) or halfway between the two middle terms (even count). Five consecutive even numbers averaging 16 are 12, 14, 16, 18, 20 — the largest is $16 + 4$. This trick answers "find the largest/smallest" without algebra.

Weighted mean

When items carry weights (4 subjects with different credit hours): $\bar{x} = \frac{\sum w_i x_i}{\sum w_i}$. Same machinery as the combined mean — weights replace group sizes.

Worked example: the join shortcut

The average age of 24 students is 15 years; including the teacher it becomes 16. Teacher $= 16 + 24(16 - 15) = 40$ years. Check with totals: $25 \times 16 - 24 \times 15 = 400 - 360 = 40$. The shortcut skips both multiplications — use it whenever the options are close together and time is short.

Quick revision

• Total $= \text{mean} \times n$; do all bookkeeping in totals.
• Join: newcomer $= b + n(b-a)$. Leave: leaver $= a + n(a-b)$.
• Replace $x$ by $y$, shift $d$: $y = x + nd$.
• Combined: $\frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1+n_2}$; equal sizes → plain average.
• Misrecorded value: mean shifts by $\frac{\text{error}}{n}$.
• Consecutive numbers: mean = middle term.