Measures of dispersion — variance, standard deviation: Worked examples

2.7.3 · D3 · Maths › Statistics & Probability — Intermediate › Measures of dispersion — variance, standard deviation

Hum sirf woh do engines use karte hain jo parent ne di hain. Main unhe simple words mein restate karta hoon taaki yahan koi bhi symbol mystery na rahe.

Scenario matrix

Neeche "case classes" ki poori list hai. Har cell mein kam se kam ek example hai — har [!example] mein tag batata hai ki woh kaunsi cell hit karta hai.

Case class	Kya tricky hai	Covered by
A. Definition, plain data	bas steps follow karo	Ex 1
B. Shortcut on same data	A se exactly match karna chahiye	Ex 2
C. Degenerate — all equal	$σ$ should be $0$	Ex 3
D. Shift invariance ( $x + c$ )	variance nahi hilna chahiye	Ex 4
E. Scaling ( $c x$ , incl. negative $c$ )	variance $\times c^{2}$ , SD $\times	c
F. Sample vs population ( $n$ vs $n - 1$ )	kaunsa divisor?	Ex 6
G. Frequency / grouped data	weights, raw list nahi	Ex 7
H. Word problem, real units	SD ko context mein interpret karo	Ex 8
I. Exam twist — combine sets	summary stats se derive karo	Ex 9
J. Limiting behaviour ( $n \to \infty$ , one outlier grows)	$σ$ kaise react karta hai	Ex 10

Figure — Measures of dispersion — variance, standard deviation

Upar wali figure mental map hai: har example ek dart-scatter picture hai, aur $σ$ measure karta hai ki scatter kitna wide hai.

A & B — definition vs shortcut agree karte hain

Worked example Example 1 — Case A (definition engine)

Data (ek population): $3, 5, 5, 7, 10$ .

Forecast: padhne se pehle guess karo ki $σ$ $1$ , $2.5$ , ya $5$ ke zyada kareeb hai.

Mean. $\overset{x}{ˉ} = \frac{3 + 5 + 5 + 7 + 10}{5} = \frac{30}{5} = 6$ . Yeh step kyun? Spread centre se measure hoti hai, isliye pehle centre chahiye.
Deviations $d_{i} = x_{i} - \overset{x}{ˉ}$ : $- 3, - 1, - 1, 1, 4$ . Kyun? Middle se har point ki raw signed distance.
Square karo: $9, 1, 1, 1, 16$ . Kyun? Squaring signs khatam karta hai (warna woh $0$ pe cancel ho jaate hain) aur door ke points ko zyada punish karta hai.
Sum & average: sum $= 28$ , toh $σ^{2} = 28/5 = 5.6$ . Kyun? Variance average squared distance hai.
Square-root: $σ = 5.6 \approx 2.37$ . Kyun? Hume data ke apne units mein wapas le jaata hai.

Verify: deviations ka sum zero hona chahiye: $- 3 - 1 - 1 + 1 + 4 = 0$ ✓. Answer $2.37$ "closer to $2.5$ " wale forecast se match karta hai.

C — degenerate case (sab values equal)

D & E — shifting aur scaling

Yeh dono exam questions ke workhorse hain: yeh allow karte hain ki jo variance tumne already compute ki ho, use reuse karo.

F — sample vs population divisor

G — frequency / grouped data

Worked example Example 7 — Case G (weighted data)

Marks aur kitne students ko mile:

mark $x_{i}$	$2$	$4$	$6$
frequency $f_{i}$	$1$	$3$	$2$

Forecast: zyaatar students $4$ par hain; guess karo ki $σ$ $2$ se neeche hai ya upar.

Total count $N = 1 + 3 + 2 = 6$ .
Weighted mean: $\sum f_{i} x_{i} = 1 \cdot 2 + 3 \cdot 4 + 2 \cdot 6 = 2 + 12 + 12 = 26$ , toh $\overset{x}{ˉ} = 26/6 \approx 4.33$ . Weight kyun? Ek mark jo $3$ baar appear karta hai, centre ko teen guna zyada pull karta hai.
Squares ka weighted mean: $\sum f_{i} x_{i}^{2} = 1 \cdot 4 + 3 \cdot 16 + 2 \cdot 36 = 4 + 48 + 72 = 124$ ; $N$ se divide karo: $124/6 \approx 20.67$ .
Shortcut: $σ^{2} = 20.67 - (4.33)^{2} = 20.67 - 18.78 \approx 1.89$ .
$σ \approx 1.37$ .

Verify: exact fractions — $\overset{x}{ˉ} = 13/3$ , $\overline{x^{2}} = 62/3$ , toh $σ^{2} = \frac{62}{3} - \frac{169}{9} = \frac{186 - 169}{9} = \frac{17}{9} \approx 1.89$ ✓. $2$ se neeche, jaise forecast tha, kyunki zyaatar students $4$ ke paas cluster karte hain.

H — real-world word problem

Worked example Example 8 — Case H (units interpret karo)

Ek machine bottles bharta hai. Paanch bottles measure ki gayi (ml), is batch ki population maano: $498, 502, 500, 501, 499$ .

Forecast: yeh $500$ ke paas tightly bunched hain; $σ$ ml mein guess karo.

Mean $= \frac{498 + 502 + 500 + 501 + 499}{5} = \frac{2500}{5} = 500$ ml.
Deviations: $- 2, 2, 0, 1, - 1$ ml.
Squares: $4, 4, 0, 1, 1$ ; sum $= 10$ .
$σ^{2} = 10/5 = 2$ ml². "ml²" kyun? Deviations ko square karne se units bhi square ho gayi.
$σ = 2 \approx 1.41$ ml. Square-root kyun? Spread real millilitres mein report karne ke liye, ml² mein nahi.

Verify: deviations sum $- 2 + 2 + 0 + 1 - 1 = 0$ ✓. Interpretation: typical bottle $500$ ml target se lagbhag $1.41$ ml off hai — machine precise hai. Square-root liye bina " $2$ " report karna classic forgot-to-root mistake hogi, aur woh ml² mein hogi, jo ek bottle ke liye bekar unit hai.

I — exam twist: do datasets ko summaries se combine karo

Kabhi kabhi tumhe raw numbers nahi diye jaate, sirf summary sums. Shortcut engine yahan shine karta hai.

Worked example Example 9 — Case I (

$\sum x$ aur $\sum x^{2}$ use karke merge karo) Group P: $n_{P} = 3$ values with $\sum x = 12$ , $\sum x^{2} = 56$ . Group Q: $n_{Q} = 2$ values with $\sum x = 18$ , $\sum x^{2} = 164$ . Combined population ki variance nikalo.

Forecast: low-mean group ko high-mean group se combine karna aksar spread badhata hai. Dhyan rakhna.

Combined count $N = 3 + 2 = 5$ .
Combined mean: $\overset{x}{ˉ} = \frac{12 + 18}{5} = \frac{30}{5} = 6$ . $\sum x$ kyun add karein? Total sum over total count overall balance point hai.
Squares ka combined mean: $\overline{x^{2}} = \frac{56 + 164}{5} = \frac{220}{5} = 44$ . $\sum x^{2}$ kyun add karein? $\sum x^{2}$ disjoint groups mein additive hai, bilkul $\sum x$ ki tarah.
Shortcut: $σ^{2} = 44 - 6^{2} = 44 - 36 = 8$ .
$σ = 8 \approx 2.83$ .

Verify: consistent example — Group P ho sakta hai $2, 4, 6$ ( $\sum = 12$ , $\sum x^{2} = 4 + 16 + 36 = 56$ ✓) aur Group Q $8, 10$ ( $\sum = 18$ , $\sum x^{2} = 64 + 100 = 164$ ✓). Merged list $2, 4, 6, 8, 10$ : mean $6$ , squared devs $16, 4, 0, 4, 16$ sum $40$ , $σ^{2} = 40/5 = 8$ ✓.

J — limiting behaviour

Worked example Example 10 — Case J (ek value bhaag jaati hai)

$0, 0, 0, 0$ (population) se shuru karo. Ek entry $x$ tak badhti hai: data ban jaata hai $x, 0, 0, 0$ . $x \to \infty$ hone par $σ^{2}$ kaise behave karta hai?

Forecast: ek wild dart spread ko unlimited blow up kar dena chahiye. Kya woh $x$ ki tarah badhta hai ya $x^{2}$ ki tarah?

Mean: $\overset{x}{ˉ} = x /4$ .
Squares ka mean: $\overline{x^{2}} = \frac{x ^{2} + 0 + 0 + 0}{4} = \frac{x ^{2}}{4}$ .
Shortcut: $σ^{2} = \frac{x ^{2}}{4} - (\frac{x}{4})^{2} = \frac{x ^{2}}{4} - \frac{x ^{2}}{16} = \frac{3 x ^{2}}{16}$ .
Limit: $x \to \infty$ hone par, $σ^{2} = \frac{3}{16} x^{2} \to \infty$ , $x^{2}$ ki tarah badhta hai (SD $x$ ki tarah badhti hai). $x^{2}$ kyun, $x$ nahi? Deviations ko square karne ka matlab hai ki ek door ka point apni distance squared contribute karta hai — outliers dominate karte hain. Yahi woh reason hai jis par parent note ne warn kiya tha ki squaring "penalises big deviations more."

Verify: $x = 4$ plug karo: data $4, 0, 0, 0$ , mean $1$ , squared devs $9, 1, 1, 1$ sum $12$ , $σ^{2} = 12/4 = 3$ ; formula $\frac{3 \cdot 16}{16} = 3$ ✓.

Recall Self-test: case batao, phir solve karo

Har mini-prompt ke liye, pehle batao kaunsa matrix cell (A–J) hai. Data $5, 5, 5$ population variance ::: Case C — degenerate, $σ^{2} = 0$ . Tumne already $σ^{2} = 9$ find ki; ab har value $3$ se multiply hoti hai. Naya variance? ::: Case E — $\times c^{2} = 9 \times 9 = 81$ . Same data, lekin ab har value mein $7$ add karo. Naya variance? ::: Case D — unchanged, still $9$ . $n = 3$ sample, sum of squared deviations $= 18$ . Sample variance? ::: Case F — $18/ (3 - 1) = 9$ . Merge: $N = 4$ , $\sum x = 8$ , $\sum x^{2} = 24$ . Variance? ::: Case I — $24/4 - (8/4)^{2} = 6 - 4 = 2$ .

Connections

Measures of Dispersion — Variance & Standard Deviation — woh parent note jis par yeh examples based hain.
Mean, Median, Mode — measures of central tendency — har example centre locate karne se shuru hota hai.
Coefficient of Variation — kisi bhi upar wale $σ$ ko uske $\overset{x}{ˉ}$ se divide karo unitless spread ke liye.
Normal Distribution — yahan ke SD values 68-95-99.7 rule ko feed karte hain.
Covariance and Correlation — Example 9 ka "additive $\sum x^{2}$ " trick do variables tak generalise hota hai.
Least Squares Regression — same square-then-minimise machinery.