Average is one of the most reliable scoring topics in CDS and OTA Maths — almost every paper carries two to three questions on it, and they are usually quick if you know the right approach. In this Cavalier lesson you will learn the core formula, weighted and combined averages, the powerful idea of deviation from average, and the speed-average trap that catches most candidates.
Why average is a must-do CDS topic
The average (or arithmetic mean) tells us the single value that would replace every item in a group while keeping the total unchanged. It is the maths of fair sharing — if all members pooled their amounts and redistributed equally, each share is the average.
In the CDS Elementary Mathematics paper, average questions appear in three flavours: a direct mean calculation, a change-when-one-item-is-added-or-removed problem, and a combined-group or weighted mean. None of them needs heavy algebra, so a trained candidate clears them in well under a minute.
What makes the topic so dependable is that the same handful of ideas keep returning, exam after exam. Once you have internalised the sum-and-count relationship and the speed trap, you are effectively guaranteed a few marks with very little risk of error. This is exactly the kind of high-yield, low-effort topic a smart aspirant locks down early in preparation, freeing time for tougher areas such as geometry and trigonometry. Treat average as a banker question and aim to finish each one within forty-five seconds during the actual paper.
Average always lies between the smallest and the largest value of the data. If your answer is outside that range, you have made an arithmetic slip.
The core formula and what it really means
The definition you must never forget is simply the total divided by how many items contributed to that total.
Average = (Sum of all observations) ÷ (Number of observations)
Equivalently: Sum = Average × Number of items. This rearranged form solves most CDS problems faster than the original.
So if five cadets score 60, 72, 68, 80 and 70 marks, the sum is 350 and the average is 350 ÷ 5 = 70. The second form (Sum = Average × n) is the real workhorse: the moment a question mentions an average, immediately convert it into a total. Totals can be added, subtracted and compared; averages on their own cannot.
Whenever you read “the average of n numbers is A”, instantly write total = A × n in the margin. Most data-handling steps then become simple addition or subtraction.
The deviation (assumed-mean) shortcut
For close-together numbers, adding everything is slow. Instead, pick a convenient assumed mean and average the small deviations from it. This is the same idea NCERT uses for the assumed-mean method in statistics, and it saves precious seconds.
Actual average = Assumed mean + (Average of the deviations)
Take 48, 52, 49, 53, 48. Assume a mean of 50. The deviations are −2, +2, −1, +3, −2, summing to 0. So the average of deviations is 0 ÷ 5 = 0, and the actual average is 50 + 0 = 50. No long addition needed.
Choose the assumed mean as a round number near the middle of the data so the deviations stay tiny and easy to add in your head.
Averages of evenly spaced numbers
Many CDS questions use consecutive integers, consecutive even or odd numbers, or any list in arithmetic progression. For such evenly spaced data there is a one-line rule.
For any set of equally spaced numbers, Average = (First term + Last term) ÷ 2 = the middle term.
Average of first n natural numbers = (n + 1) ÷ 2.
For example, the average of the first 20 natural numbers is (20 + 1) ÷ 2 = 10.5. The average of 11, 13, 15, 17, 19 is simply the middle value 15, or (11 + 19) ÷ 2 = 15. Spotting an evenly spaced set lets you skip the summation entirely.
The reason this works is symmetry. In an evenly spaced list, every value below the centre is balanced by an equal value above it, so all the deviations cancel and the mean settles exactly at the midpoint. This is why the average of an arithmetic progression never depends on the common difference — only on the first and last terms. Examiners love to disguise this as a long list of numbers hoping you will add them one by one; recognise the pattern and answer in a single step.
Weighted average
When groups have different sizes or items carry different importance, a plain average is wrong — you must weight each value by its frequency. This is exactly how a final mark is computed when subjects carry unequal credits.
Weighted average = (w1x1 + w2x2 + …) ÷ (w1 + w2 + …)
Suppose a cadet scores 80 in a 3-credit subject and 50 in a 2-credit subject. The naive average (80 + 50) ÷ 2 = 65 is wrong. The correct weighted mean is (3×80 + 2×50) ÷ (3 + 2) = (240 + 100) ÷ 5 = 68.
Never average two averages directly unless the two groups are the same size. Doing so silently assumes equal weights and gives a wrong figure.
Combined average of two groups
The combined average is just a weighted average where the weights are the group sizes. Add the totals, add the counts, then divide.
Combined average = (n1A1 + n2A2) ÷ (n1 + n2)
where A1, A2 are the group averages and n1, n2 their counts.
If 30 men average 62 kg and 20 men average 67 kg, the combined average is (30×62 + 20×67) ÷ 50 = (1860 + 1340) ÷ 50 = 3200 ÷ 50 = 64 kg. Note the answer leans toward the larger group — a quick sanity check.
This leaning is worth understanding intuitively. The combined average is always pulled towards the group that has more members, because that group contributes more to the total. So the result must lie between 62 and 67, and closer to 62 since the 30-member group outnumbers the 20-member group. If your computed value ever falls outside the two original averages, or sits on the wrong side, you have made an error. The technique extends to three or more groups in exactly the same way: add every group total, add every count, then divide once at the end.
Adding, removing and replacing items
A favourite CDS pattern: an average changes when one person joins, leaves, or is replaced. The fastest method is to track the change in the total, not the individual values.
Change in total = Change in average × (final number of items).
When one item replaces another, New value = Old value + (change in average × number of items).
Example: the average age of 10 players is 22 years. A new player joins and the average rises to 22.5 years. The total went up by 0.5 × 11 = 5.5 years above what 11 players at the old average would give — more usefully, the new total is 22.5 × 11 = 247.5, while the old total was 220, so the new player is 27.5 years old.
For replacement problems, the change in the group total equals (new member − old member). Set this equal to change-in-average × group-size and solve in one line.
Average speed — the famous trap
Average speed is total distance divided by total time, not the average of the two speeds. This single distinction decides at least one CDS question almost every year.
Average speed = Total distance ÷ Total time.
For equal distances at speeds a and b, average speed = 2ab ÷ (a + b) — the harmonic mean.
A car covers the first half of a trip at 40 km/h and the second half at 60 km/h. The wrong answer 50 km/h ignores that more time is spent at the slower speed. The correct value is 2×40×60 ÷ (40 + 60) = 4800 ÷ 100 = 48 km/h.
The plain mean (a + b) ÷ 2 only works when equal time is spent at each speed. For equal distances always use the harmonic mean 2ab ÷ (a + b).
Worked example
The average weight of 24 students in a section is 36 kg. When the class teacher’s weight is included, the average rises to 37 kg. Find the weight of the teacher.
So the teacher weighs 61 kg. Quick check using deviations: the teacher must cover the old average (37) plus lift 24 students by 1 kg each, i.e. 37 + 24 = 61 kg. Both methods agree.
Handy results worth memorising
A few standard outcomes appear repeatedly in CDS and OTA papers. Keep them ready.
- Average of first n natural numbers = (n + 1) ÷ 2.
- Average of first n even numbers = (n + 1).
- Average of first n odd numbers = n.
- Average of squares of first n naturals = (n + 1)(2n + 1) ÷ 6.
- If every number is increased by k, the average also increases by k; the same holds for multiplication and division.
Adding a constant shifts the average by that constant; multiplying every value by a factor multiplies the average by the same factor. This lets you simplify ugly data before averaging.
Previous-year style practice
Q. The average of 11 numbers is 50. The average of the first six numbers is 49 and the average of the last six numbers is 52. What is the sixth number?
Answer: Total of all 11 = 50 × 11 = 550. Sum of first six = 49 × 6 = 294. Sum of last six = 52 × 6 = 312. First six plus last six = 294 + 312 = 606, but this counts the sixth number twice (it lies in both groups). So sixth number = 606 − 550 = 56.
This overlap trick — where the middle term is counted in both halves — is a classic CDS device. Whenever two overlapping groups together exceed the full count, the surplus is the shared term.
To generalise the idea: if a problem splits a set into two parts that share one or more common members, the sum of the two part-totals will exceed the grand total by exactly the value of the shared members. Subtracting the grand total from the combined part-totals isolates that shared portion immediately. Train yourself to spot the phrase “first six” and “last six” from a group of eleven — the moment six plus six exceeds eleven, you know an overlap of one item is in play, and the question is essentially solved.
Quick revision
- Average = Sum ÷ Count; rearrange to Sum = Average × n for almost every problem.
- Use the assumed-mean deviation trick for clustered numbers.
- Evenly spaced data: average = (first + last) ÷ 2 = middle term.
- Weighted and combined averages weight each value by its group size; never average two averages of unequal groups.
- For add/remove/replace, work with the change in the total.
- Average speed = total distance ÷ total time; equal distances give harmonic mean 2ab ÷ (a + b).
Practise five mixed average questions daily from your CDS PYQ book for two weeks and this topic becomes a guaranteed source of quick marks in the exam hall.
Frequently asked questions
What is the basic formula for average in CDS Maths?
Average equals the sum of all observations divided by the number of observations. The most useful rearrangement is Sum = Average times the number of items, which speeds up nearly every problem.
Why is average speed not the simple mean of two speeds?
Because average speed is total distance divided by total time. When equal distances are covered at different speeds, more time is spent at the slower speed, so you must use the harmonic mean 2ab/(a+b) instead of (a+b)/2.
When should I use a weighted average instead of a plain average?
Use a weighted average whenever the groups or values have different sizes, frequencies or importance. Averaging two averages directly is only valid when both groups contain exactly the same number of items.
How do I quickly find the average of consecutive numbers?
For any evenly spaced set, the average is (first term + last term) divided by 2, which also equals the middle term. The average of the first n natural numbers is simply (n + 1)/2.
How many average questions appear in the CDS exam?
The CDS Elementary Mathematics paper typically carries two to three average-based questions, covering direct means, combined or weighted averages, and add-remove-replace situations. They are quick scoring opportunities if the formulas are revised.
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