Equal-share meaning
Add all observations and divide the total equally among them.
\[\bar x=\frac{x_1+x_2+\cdots+x_n}{n}\]
5 SSC CGL · Mathematics
Averages
Handwritten-style notes for SSC CGL Tier I and Tier II.
01MAP IT!
The Big Picture — Graphic Mind Map
An average replaces unequal observations with one equal share while preserving the total.
AVERAGE
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EQUAL SHARE
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EQUAL SHARE
1. Basic Meantotal divided by count
2. Find Missingrecover total or observation
3. Change Groupadd, remove, replace and correct
4. Combineweighted group averages
5. Use Deviationsassumed mean and balancing
6. Applyages, marks, income, runs and prices
\[\bar x=\frac{\sum_{i=1}^{n}x_i}{n}\qquad\sum x_i=n\bar x\]
02FOUNDATION
The Foundation (Basics) — Core Concepts
Start with total and count; every advanced average problem is an adjustment to one of them.
AWhat is an arithmetic mean?
The average need not be observed
The mean of \(2,4,9\) is \(5\), even though \(5\) is not in the list.
\[\frac{2+4+9}{3}=5\]
Units and range
All values need compatible units. For real observations:
\[\min(x_i)\le\bar x\le\max(x_i)\]
The levelling picture
Move surplus from values above the mean to values below it. The total stays unchanged, and every column reaches the same height.
\[\sum_{i=1}^{n}(x_i-\bar x)=0\]
BThe total–count triangle
Find average
\[\text{average}=\frac{\text{total}}{\text{count}}\]
Find total
\[\text{total}=\text{average}\times\text{count}\]
Find count
\[\text{count}=\frac{\text{total}}{\text{average}}\quad(\bar x\ne0)\]
Cover the unknown corner
Total sits above average and count because it is their product. The other two are obtained by division.
CMissing observation and unknown total
One missing value
\[x=n\bar x-\sum\text{known values}\]
Several missing values
Their combined total is
\[\sum\text{missing}=n\bar x-\sum\text{known}\]
Extra information is needed to split that total.
Average of remaining values
\[\bar x_{\rm rem}=\frac{n\bar x-\sum\text{removed}}{n-r}\]
Here \(r\) values are removed.
DWeighted average
Why weights?
When values occur with different frequencies or importance, each value contributes in proportion to its weight.
Master formula
\[\bar x_w=\frac{\sum w_ix_i}{\sum w_i}\]
Weights must represent comparable counts or stated importance.
Frequency mean
If \(x_i\) occurs \(f_i\) times:
\[\bar x=\frac{\sum f_ix_i}{\sum f_i}\]
A heavier group pulls the mean closer
A weighted average lies between the component values and leans toward the value carrying the larger positive weight.
\[x_1\le\bar x_w\le x_2\quad\text{for }x_1\le x_2,\ w_1,w_2>0\]
ECombined average of groups
Two groups
\[\bar x=\frac{n_1a_1+n_2a_2}{n_1+n_2}\]
Many groups
\[\bar x=\frac{\sum n_ia_i}{\sum n_i}\]
Multiply each group average by its group size first.
Equal group sizes
Only when \(n_1=n_2=\cdots\), the combined average equals the simple average of group averages.
\[\text{Do not use }\frac{a_1+a_2}{2}\text{ unless the two groups contain equal numbers of observations.}\]
Combine totals, not naked averages
Each group average hides a total. Reconstruct every group total, add totals and divide by the combined count.
FAddition, removal and replacement
Add one value \(y\)
\[\bar x_{\rm new}=\frac{n\bar x+y}{n+1}\]
Remove one value \(y\)
\[\bar x_{\rm new}=\frac{n\bar x-y}{n-1}\]
Replace \(x\) by \(y\)
\[\bar x_{\rm new}=\bar x+\frac{y-x}{n}\]
The count remains \(n\).
| Event | Change in total | Change in count | Fast relation |
|---|---|---|---|
| Add one item | \(+y\) | \(+1\) | \((n+1)\bar x'=n\bar x+y\) |
| Remove one item | \(-y\) | \(-1\) | \((n-1)\bar x'=n\bar x-y\) |
| Replace one item | \(y-x\) | \(0\) | \(n(\bar x'-\bar x)=y-x\) |
| Add \(m\) items with mean \(a\) | \(+ma\) | \(+m\) | \((n+m)\bar x'=n\bar x+ma\) |
| Remove \(m\) items with mean \(a\) | \(-ma\) | \(-m\) | \((n-m)\bar x'=n\bar x-ma\) |
The average elevator responds to total change
With the count fixed at \(n\), increasing total by \(D\) raises the average by \(D/n\). Replacement questions are fastest through this total-difference view.
\[\Delta\bar x=\frac{\Delta T}{n}\]
GWrong entry and correction
One wrong value
If wrong \(w\) should be correct \(c\):
\[T_{\rm correct}=T_{\rm wrong}-w+c\]
Corrected mean
\[\bar x_c=\bar x_w+\frac{c-w}{n}\]
Several corrections
\[\bar x_c=\bar x_w+\frac{\sum c_i-\sum w_i}{n}\]
The number of observations remains unchanged.
HAssumed mean and deviation method
Choose a convenient center \(A\)
Write each observation as \(x_i=A+d_i\).
\[\bar x=A+\frac{\sum d_i}{n}\]
Balanced deviations
Positive and negative deviations cancel partially, reducing arithmetic.
\[\sum(x_i-\bar x)=0\]
Step-deviation form
If \(d_i=(x_i-A)/h\):
\[\bar x=A+h\frac{\sum f_id_i}{\sum f_i}\]
Measure only the distance from a friendly center
When values cluster around a convenient number, work with small signed deviations and add their average back to the assumed mean.
IConsecutive and equally spaced observations
Arithmetic progression
For equally spaced terms, the average is the midpoint of the first and last.
\[\bar x=\frac{\text{first}+\text{last}}2\]
Odd number of terms
The middle term equals the average.
\[a-d,\ldots,a,\ldots,a+d\Rightarrow\bar x=a\]
Consecutive integers
\[m,m+1,\ldots,n\Rightarrow\bar x=\frac{m+n}{2}\]
Count is \(n-m+1\).
| Sequence | Count | Average |
|---|---|---|
| First \(n\) natural numbers | \(n\) | \((n+1)/2\) |
| First \(n\) odd numbers | \(n\) | \(n\) |
| First \(n\) even numbers | \(n\) | \(n+1\) |
| Multiples \(d,2d,\ldots,nd\) | \(n\) | \(d(n+1)/2\) |
| Arithmetic progression | \(n\) | \((a_1+a_n)/2\) |
JAge, marks, income, runs, price and group applications
Ages
After \(t\) years, every one of \(n\) people gains \(t\), so
\[\bar a_{\rm new}=\bar a_{\rm old}+t\]
Births, deaths or membership changes alter the count and need total adjustment.
Runs / marks
Required score to reach target mean \(A\) after \(n\) performances:
\[x=nA-(n-1)B\]
Here \(B\) is the previous mean.
Income / price
Use weighted means when quantities purchased, people, months or frequencies differ.
\[\text{average price}=\frac{\text{total cost}}{\text{total quantity}}\]
Average speed warning
Average speed is always total distance divided by total time.
\[v_{\rm avg}=\frac{D_1+D_2}{D_1/v_1+D_2/v_2}\]
For equal distances: \(2v_1v_2/(v_1+v_2)\); for equal times: \((v_1+v_2)/2\).
Average of averages
A simple mean of means is valid only when underlying group sizes are equal. Otherwise use the group sizes as weights.
03SAVE TIME
Short Tricks & Magic Formulas
Think in changes to total; this avoids repeatedly rebuilding long sums.
1Average-change shortcut
Every item rises by \(d\)
\[x_i\to x_i+d\Rightarrow\bar x\to\bar x+d\]
Every item is multiplied by \(k\)
\[x_i\to kx_i\Rightarrow\bar x\to k\bar x\]
One item changes by \(D\)
\[\Delta\bar x=\frac Dn\]
2Replacement in one line
Find incoming value
\[y=x+n(\bar x_{\rm new}-\bar x_{\rm old})\]
Find outgoing value
\[x=y-n(\bar x_{\rm new}-\bar x_{\rm old})\]
3Deviation and alligation view
Two group means
If group means \(a_1,a_2\) combine to \(A\), their sizes satisfy
\[n_1:n_2=(a_2-A):(A-a_1)\]
Target above old mean
To raise an average, the incoming value must exceed the target mean; the exact surplus repairs the old total deficit.
Choose an assumed mean
Pick a central round number, add signed deviations, divide by count and correct the assumed mean.
4Option-first checks
The average speed dashboard
Check count, reconstruct total, record the change in total, then test whether the answer lies between valid extremes.
04DON'T RUSH
The SSC / TCS Traps — Red Flags 🚩
Wrong options usually average averages directly, forget a changed count, or adjust the mean instead of the total.
Trap 1: average of averages
Do not simply average group means unless group sizes are equal. Rebuild totals with \(n_ia_i\).
Trap 2: addition changes the count
When a value enters or leaves, both total and count change. Replacement changes total but keeps count fixed.
Trap 3: average speed is not always mean speed
Use total distance divided by total time. Arithmetic mean works directly only for equal time intervals.
05LOCK IT IN
Memory Hooks & Mnemonics
Attach each average rule to a visual sentence for instant exam recall.
“Average balances the total”
Redistribution changes individual values but preserves their combined sum.
“Total sits on top”
Average equals total divided by count; total equals average times count.
“Heavy groups pull harder”
A combined mean moves closer to the average of the larger group.
“Mean moves by total change over count”
With fixed count, divide the change in total by the number of items.
“Equal spacing finds the middle”
For an arithmetic progression, average is the midpoint of first and last.
“Wrong out, correct in”
Correction changes total by correct minus wrong; the count remains unchanged.
Basic meanTotal ÷ count
CombinedCombine group totals
WeightedValue × frequency
ReplacementCount unchanged
CorrectionCorrect − wrong
ConsecutiveMidpoint of endpoints