Average

Average

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Average (Arithmetic Mean) of n observations is their total sum divided by n:

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

The defining property is that sum of deviations from the mean equals zero, i.e., Σ(xᵢ − x̄) = 0, so positive and negative departures balance exactly. For SSC CGL Tier-I, the two highest-yield manipulations are:

• Replacement: if an item with value x is swapped for y among n items, the average shifts by (y − x) / n.
• Alligation: when two groups of averages a₁ and a₂ merge to give overall mean x, their quantity ratio is (a₂ − x) : (x − a₁).

For consecutive integers, average equals (first + last) / 2 — no full sum needed.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core Definition and Properties

The Arithmetic Mean of quantities x₁, x₂, …, xₙ is

$$\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}$$

It is a balancing point: deviations above x̄ exactly cancel deviations below x̄, so Σ(xᵢ − x̄) = 0. This identity drives the deviation method — if you know the mean and one observation is missing, the missing value equals n·x̄ minus the sum of the rest.

Shift Rules (most tested in CGL)

1. Equal increment to all items: add k to every observation → mean rises by k.
2. Single replacement: replace x with y in a set of n → new mean = old mean + (y − x) / n.
3. Person/group migration: a member leaves group A of size n to join group B of size m; A’s mean rises by x / n and B’s mean falls by x / m (where x is the migrant’s value). The combined mean is unchanged.
4. Sequential averages: if group 1 has average a₁ over n₁ items and group 2 has average a₂ over n₂ items, the combined average is (n₁·a₁ + n₂·a₂) / (n₁ + n₂) — not (a₁ + a₂) / 2.

Alligation Method

When two ingredients (or groups) of known mean prices a₁, a₂ combine to yield a mixture of mean price x, the quantity ratio is

$$\frac{Q_1}{Q_2} = \frac{a_2 - x}{x - a_1}$$

Cross-multiply to check: Q₁·a₁ + Q₂·a₂ = (Q₁ + Q₂)·x.

Typical CGL Patterns

• 5 batsmen with given averages; one is replaced → new team average.
• Class average shifts after N new students join with known individual marks.
• Two groups of different sizes merge; find the size ratio from overall average.
• “Average of first 10 multiples of 7” — use (first + last) / 2 = (7 + 70) / 2 = 38.5.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Deeper Mechanisms

Weighted vs simple average. A common trap: “Average of averages” is valid only when group sizes are equal. Otherwise weight by count: weighted mean = Σ(nᵢ·aᵢ) / Σnᵢ. This distinction decides answers in mixture–alligation and partnership–profit-sharing hybrids.

Deviation method with missing observations. Given 9 of 10 values and the overall mean, the 10th value = 10·x̄ − (sum of 9). For m missing values, sum of missing values = m·x̄ − Σ(known values). This is faster than forming algebraic equations.

Sequential entry/exit. If k new members with average y join a group of n members with average x̄, the new average becomes

$$\bar{x}_{\text{new}} = \frac{n\bar{x} + ky}{n + k}$$

Symmetric formula when members leave: subtract k·y and reduce n by k.

Migration with size changes. When sizes are unequal, migration does not preserve group means — only the combined mean is invariant. Verify: A had mean 50 over 10 items (sum = 500); B had mean 30 over 10 items (sum = 300). Combined mean = 40. Move a 60-scoring person A→B: A’s new sum = 440 over 9 → 48.89; B’s new sum = 360 over 11 → 32.73; combined still 800/20 = 40.

Common Mistakes

• Using (a₁ + a₂) / 2 instead of the weighted form.
• Forgetting to update n after removal/entry — the divisor changes.
• Alligation ratio inversion: quantities sit opposite to their means (cheap mean goes with the a₂ − x part).
• Treating average of percentages as a simple mean when the bases differ.

Practice Prompts

1. The average weight of 25 students is 40 kg. A new student weighing 46 kg replaces one weighing 38 kg. Find the new average. (Answer: 40.32 kg)
2. Two alloys of copper costing ₹120/kg and ₹180/kg are mixed in ratio 3 : 2. Find the mean price per kg using alligation. (Answer: ₹144/kg)

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