Data Interpretation

Averages inside a table

An average asks: "if everything were shared equally, how much would each get?" In DI the "things" are cells of a table — production of a factory across days, marks of students, salaries of departments. The rule never changes: $\text{average} = \frac{\text{sum of the cells}}{\text{number of cells}}$ So every average question is really two questions: find the total, count the items, divide.

Row or column average

A row (or column) average sums one row and divides by the count. Production over six days is 480, 520, 460, 540, 500, 500: the sum is 3,000, so the average is $3000 \div 6 = 500$ units. With friendly DI numbers the division is usually exact — if you get a messy decimal, recheck the cells you picked.

Deviations: average without big sums

Instead of adding, pick any round number near the values and add the differences. For 480, 520, 460, 540, 500, 500 around 500: deviations $-20, +20, -40, +40, 0, 0$ add to $0$, so the average is exactly 500. This is fast and self-checking.

Combined (weighted) average

Two groups with different sizes cannot be averaged plainly. Weight each group's average by its size: $\bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2 + \dots}{n_1 + n_2 + \dots}$ Department A: 40 employees at ₹25,000; department B: 30 at ₹32,000. Combined $= \frac{40 \times 25000 + 30 \times 32000}{70} = \frac{1960000}{70} = ₹28{,}000$ — closer to A's salary because A has more people. The plain average $\frac{25000 + 32000}{2} = 28{,}500$ is the planted trap.

Per-period values

"Average sale per day" from a weekly total: divide the total by the number of periods, not by the number of rows of some other table. Total sales ₹84,000 in 6 days means ₹14,000 per day. Read the question: per day, per month, per employee — each divisor is different.

Excluding or adding an item

• Excluding: new average $= \frac{\text{old total} - \text{removed value}}{\text{count} - 1}$. Dropping the lowest day (460) from our week: $\frac{3000 - 460}{5} = 508$.
• Adding: needed value $= \text{new average} \times (n+1) - \text{old total}$. To lift the 6-day average of 500 to 520 with a 7th day: $7 \times 520 - 3000 = 640$ units.

Missing cell from an average

If the average is given, rebuild the total: total $= \text{average} \times n$. Then the missing cell is one subtraction. Six months average 2,500 units, so the half-year total is 15,000; if March is missing, March $= 15000 - (\text{other five})$.

Quick revision

• average = total ÷ count; total = average × count.
• Deviation trick: pick a round base, add the differences, divide by n.
• Weighted average $\frac{\sum n_i \bar{x}_i}{\sum n_i}$ — plain average of the group means is the trap.
• Per day / per month: divide by periods, not rows.
• Exclude: $(T - x)/(n-1)$. Add: new avg × $(n+1) - T$.
• "More than the average" does not include cells equal to the average.