10.1 Averages, Weighted Averages & Age Problems
Key Takeaways
- The mathematical average expresses central tendency (Sum / Count); under the Net Deviation Principle, the algebraic sum of deviations about the true mean is strictly zero (sum(x_i - mean) = 0).
- Weighted average aggregates heterogeneous subgroup means scaled by subgroup frequencies or monetary weights: x_bar_w = (sum w_i * x_i) / (sum w_i).
- When one member joins a group, the newcomer's value equals Old Average + (New Count × Change in Average); when one member leaves, the departing value equals Old Average − (New Count × Change in Average).
- Replacement scenarios preserve total entity count N: New Value - Replaced Value = N * (New Average - Old Average).
- In chronological age problems, the absolute difference in age between two individuals remains perfectly invariant across past, present, and future epochs.
10.1 Averages, Weighted Averages & Age Problems
Quantitative Aptitude in banking examinations tests numerical dexterity, logical deduction, and operational speed. In the SBI Clerk (Junior Associate) preliminary and main examinations, arithmetic problems centered on averages, weighted means, and chronological age relationships appear reliably as individual standalone word problems (1 to 3 questions) and serve as the mathematical foundation for Data Interpretation (DI) caselets and tabular charts. Mastering shortcut formulas and structural balancing techniques eliminates tedious long-division calculations and preserves critical exam time.
1. Mathematical Foundations & The Net Deviation Principle
The arithmetic mean (or average) of a discrete dataset is defined as the total aggregate sum of all observations divided by the total number of observations:
While direct summation works for small numbers, commercial banking datasets often feature large, clustered values (e.g., daily branch transaction counts, customer deposit sizes, or employee productivity logs). In competitive examinations, calculating large sums induces arithmetic errors. The Deviation Method (Assumed Mean Method) bypasses large summations.
The Net Deviation Principle
Let $A$ represent an assumed arbitrary mean chosen close to the center of the observations. The deviation $d_i$ of any observation $x_i$ from $A$ is:
The true arithmetic mean is then obtained by adjusting the assumed mean by the average of the deviations:
[!NOTE] The Fundamental Deviation Theorem: The algebraic sum of deviations of all values taken from their actual arithmetic mean is identically zero: If the sum of deviations about an assumed baseline $A$ is positive, the true average exceeds $A$; if negative, the true average lies below $A$.
Deviation Method Calculation Box
Consider five customer service counters at an SBI administrative hub handling the following daily voucher volumes: 42, 48, 55, 61, and 64.
| Counter | Volume ($x_i$) | Assumed Baseline ($A = 50$) | Deviation ($d_i = x_i - 50$) |
|---|---|---|---|
| Counter 1 | 42 | 50 | $-8$ |
| Counter 2 | 48 | 50 | $-2$ |
| Counter 3 | 55 | 50 | $+5$ |
| Counter 4 | 61 | 50 | $+11$ |
| Counter 5 | 64 | 50 | $+14$ |
| Total / Net | $\sum x_i = 270$ | — | $\sum d_i = (-8 - 2 + 5 + 11 + 14) = +20$ |
By balancing deviations, two-digit addition replaces multi-digit column summation, yielding the answer in under 15 seconds.
2. Weighted Average Framework
When combining distinct groups with different average values and unequal group sizes, a simple average of averages yields incorrect results. The weighted average accounts for the relative weight or frequency of each constituent group.
Mathematical Formulation
If $k$ distinct groups have sizes $w_1, w_2, \dots, w_k$ and respective means $\bar{x}_1, \bar{x}_2, \dots, \bar{x}_k$, the combined weighted average $\bar{x}_w$ is:
Key Ratio Simplification Shortcut
In the weighted average formula, absolute weights ($w_i$) can be replaced by their lowest integer ratio. If an SBI branch holds Rs. 30 crore in retail deposits at 6.5% interest, Rs. 20 crore in senior citizen deposits at 7.25% interest, and Rs. 50 crore in corporate term deposits at 5.75% interest, the weights ratio is:
3. Inclusion, Exclusion, and Replacement Dynamics
Questions involving an individual joining, leaving, or replacing an existing member of a group are standard in the SBI Clerk exam.
1. Inclusion Dynamics (New Member Joins)
When a new person joins a group of $N$ individuals with initial average $\bar{x}{\text{old}}$, altering the group size to $N + 1$ and changing the average to $\bar{x}{\text{new}}$:
Alternative perspective: The incoming entrant must bring their own share equal to the new average, plus or minus the cumulative adjustment contributed to the initial $N$ members: $\text{Value} = \bar{x}{\text{new}} + N \times (\bar{x}{\text{new}} - \bar{x}_{\text{old}})$. Both expressions are mathematically equivalent.
2. Exclusion Dynamics (Existing Member Leaves)
When one individual leaves a group of $N$ members, reducing the count to $N - 1$:
3. Replacement Dynamics (Count Invariant)
When one member leaves and is immediately replaced by a new entrant, the total count $N$ remains strictly invariant:
[!WARNING] The Directional Sign Trap: When an average decreases, $\Delta\bar{x} = (\bar{x}{\text{new}} - \bar{x}{\text{old}})$ is negative. Candidates frequently make sign errors. For instance, if the average weight of 15 clerks decreases by 1.2 kg upon replacement of a 74 kg clerk, the adjustment is $15 \times (-1.2) = -18\text{ kg}$. The new recruit weighs $74 - 18 = 56\text{ kg}$, not $74 + 18 = 92\text{ kg}$.
4. Chronological Age Problems & The Invariant Difference Principle
Age arithmetic problems require translating past, present, and future chronological relationships into linear equations.
The Core Axiom of Age Arithmetic
If Person A is 6 years older than Person B today, Person A was 6 years older 15 years ago and will remain 6 years older 25 years in the future ($A_t - B_t = \text{constant}$). Total elapsed chronological time shifts every individual's age by the exact same scalar: after $t$ years, both individuals have aged by $+t$ years.
Chronological Transition Matrix
| Epoch | Relative Shift | Person A Age | Person B Age | Constant Invariant Difference |
|---|---|---|---|---|
| Past ($k$ years ago) | $-k$ | $A - k$ | $B - k$ | $(A - k) - (B - k) = A - B$ |
| Present | $0$ | $A$ | $B$ | $A - B$ |
| Future ($m$ years hence) | $+m$ | $A + m$ | $B + m$ | $(A + m) - (B + m) = A - B$ |
Ratio Equating Technique
When past and future age ratios are specified, align the unit differences:
- Suppose the ratio of ages of A to B was $a_1 : b_1$ at time $t_1$, and becomes $a_2 : b_2$ at time $t_2$.
- Compute difference in parts: $\Delta_1 = |a_1 - b_1|$ and $\Delta_2 = |a_2 - b_2|$.
- Normalize the ratios by multiplying ratio 1 by $\Delta_2$ and ratio 2 by $\Delta_1$, ensuring that the invariant age difference corresponds to the exact same number of ratio units.
- Equate the change in individual ratio parts directly to the elapsed calendar time $(t_2 - t_1)$.
The average age of 24 customer associates in an SBI administrative branch is 28 years. When the branch manager's age is included, the overall average age increases by 1.5 years. What is the branch manager's age?
The average weight of 15 clerical staff members in a regional clearing office decreases by 1.2 kg when an employee weighing 74 kg is transferred out and replaced by a newly recruited junior associate. What is the weight of the new junior associate?
Four years ago, the ratio of the ages of employee A and employee B was 3 : 4. Eight years hence, the ratio of their ages will become 5 : 6. What is the present age of employee B?