Averages

1. Meaning, scope and the total-first approach

The arithmetic mean of n observations is their sum divided by n. If five students score 12, 15, 18, 20 and 25 marks, their total is 90 and their average is 18. In CSAT, the difficulty usually lies not in division but in identifying what is being counted: people, days, matches, kilometres or units purchased. Always attach a unit to the average and identify its corresponding denominator.

The most useful rearrangement is total = number × average. Suppose the average age of eight members is 24 years. Their combined age is 192 years. If seven of them have an average age of 23 years, their combined age is 161 years, so the remaining member is 31 years old. Converting averages into totals makes the hidden quantity directly accessible.

For an arithmetic progression, the mean is the average of the first and last terms. Thus, the average of 11, 14, 17, 20 and 23 is 17. The average of the first n positive integers is (n + 1)/2. These shortcuts require the stated structure: the endpoints alone do not determine the average of an arbitrary list.

• For non-negative weights, a weighted average lies between the smallest and largest observations included.
• Check whether zero-valued observations are included; a zero contributes nothing to the total but still increases the count.

2. Weighted averages and combining groups

If two groups contain n₁ and n₂ observations with averages a₁ and a₂, their combined average is (n₁a₁ + n₂a₂)/(n₁ + n₂). A class of 20 students averaging 60 marks and another of 30 students averaging 70 marks together average 66 marks. Simply averaging 60 and 70 gives 65, which incorrectly assigns equal importance to unequal groups.

A useful reverse method is alligation. If a lower average L and a higher average H combine to produce A, then the numbers in the lower and higher groups are in the ratio (H − A):(A − L). For group averages 40 and 70 producing an overall average of 58, the ratio is 12:18, or 2:3. This follows from balancing the lower group's deficit against the higher group's surplus.

Weights must match the quantity measured. Average marks across all students require student counts; average price per kilogram requires quantities purchased; an overall success percentage requires the numbers of candidates in each group. If two districts report literacy rates, their combined literacy rate must use the eligible population underlying each rate, not an unweighted average of the percentages.

• With positive group sizes and different group averages, the combined average lies strictly between them.
• If group sizes are unavailable, a unique combined average may not be determinable.

3. Addition, removal, replacement and correction

When membership changes, calculate both the new total and the new count. If n observations average a and a new observation x is added, the new average b satisfies x = (n + 1)b − na. For example, ten workers average ₹500 in daily earnings. After an eleventh worker joins, the average becomes ₹520. The new worker earns ₹720 because 11 × 520 − 10 × 500 = 720.

Replacement differs from addition because the number of observations remains unchanged. If one person replaces another in a group of n and the average rises by d, the incoming value exceeds the outgoing value by nd. Thus, replacing one member of an eight-person team raises its average weight by 1.5 kg only if the new member is 12 kg heavier than the departing member.

Correction questions use the same total-based logic. If 36 was mistakenly recorded as 63, the recorded total is 27 too high. For nine observations, the corrected average is therefore 3 lower. In overlapping-group questions, add the subgroup totals carefully: an observation included in both groups has been counted twice and must be adjusted using the overall total.

• For removal: removed value = old total − remaining total.
• For a fixed group, adding c to every observation raises its average by c; multiplying every observation by k multiplies its average by k.

4. Average speed, rates and changing denominators

Average speed is total distance divided by total elapsed time. It is not normally the arithmetic mean of the speeds. If a vehicle covers 60 km at 30 km/h and another 60 km at 60 km/h, it travels 120 km in three hours. Its average speed is 40 km/h, not 45 km/h. The slower leg takes longer and consequently has a greater time weight.

For two equal-distance legs at positive speeds u and v, average speed is 2uv/(u + v), the harmonic mean. For two equal-time legs, average speed is (u + v)/2. For unequal distances or times, return to total distance and total time rather than forcing either shortcut. If the question includes a halt in the journey duration, its time belongs in the denominator even though it adds no distance.

The same denominator discipline applies to prices and productivity. Equal quantities purchased at two prices produce an arithmetic mean price per unit. Equal expenditure at two prices produces a harmonic mean price per unit. Likewise, average output per worker-hour requires total output divided by total worker-hours, not an unweighted average of daily productivity figures.

• Before averaging rates, write what each rate means: distance/time, expenditure/quantity or output/worker-hours.
• Distinguish speed from velocity: velocity depends on displacement, whereas speed depends on distance travelled.

5. Efficient CSAT methods and interpretation

The assumed-mean method reduces arithmetic when observations cluster around a convenient number. For 48, 52, 49, 55 and 46, choose 50 as the reference. Their deviations are −2, 2, −1, 5 and −4, which sum to zero; hence the average is exactly 50. More generally, actual mean = assumed mean + average deviation. The sum of deviations from the actual arithmetic mean is always zero.

Use bounds and answer options before lengthy calculation. A combined average of groups averaging 32 and 44 cannot be 46. If the larger group has average 44, the combined average must be closer to 44 than to 32. Such checks can eliminate options, but closeness alone does not replace exact weighting when multiple plausible options remain.

Finally, distinguish calculation from interpretation. The mean uses every observation but is sensitive to extreme values. A few exceptionally high incomes can raise mean income without representing a typical household. The median is the middle value after ordering, and the mode is the most frequent value. Unless the question specifies otherwise, elementary arithmetic questions asking for an average generally intend the arithmetic mean.

• A reliable sequence is: identify the denominator, reconstruct totals, apply the change, divide, and verify units and bounds.
• In age questions, every continuing member ages by the same elapsed time; new entrants and departures must be handled separately.

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