Measures of dispersion — variance, standard deviation

2.7.3 · Maths › Statistics & Probability — Intermediate

HUM KYA measure kar rahe hain?

Hamare paas pehle se kuch crude spread measures hain: range (max − min) sirf do points use karta hai, aur mean absolute deviation $∣ x - \overset{x}{ˉ} ∣$ use karta hai. Variance dono ko improve karta hai deviations ko square karke.

Variance ko scratch se derive karna

Hum chahte hain "mean se average squared distance" ke liye ek single number.

Step 1. Mean compute karo: $\overset{x}{ˉ} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$ Kyun? Mean balance point hai — woh value jisse hum spread measure karte hain.

Step 2. Har point ka deviation: $d_{i} = x_{i} - \overset{x}{ˉ}$ . Kyun? Yeh kisi value ki centre se raw distance (sign ke saath) hai.

Step 3. Har ek ko square karo: $d_{i}^{2}$ . Kyun? Sign remove karta hai; bade deviations amplify ho jaate hain.

Step 4. Unka average nikalo:

Step 5. Square root lekar units theek karo:

Computational shortcut formula

Har $(x_{i} - \overset{x}{ˉ})$ compute karna tedious hai. Chalo shortcut derive karte hain.

$σ^{2} = \frac{1}{n} \sum (x_{i} - \overset{x}{ˉ})^{2} = \frac{1}{n} \sum (x_{i}^{2} - 2 x_{i} \overset{x}{ˉ} + \overset{x}{ˉ}^{2})$

Sum ko split karo (Kyun? summation addition par distribute ho jaata hai): $= \frac{1}{n} \sum x_{i}^{2} - \frac{2 x ˉ}{n} \sum x_{i} + \frac{1}{n} \sum \overset{x}{ˉ}^{2}$

Ab $\frac{1}{n} \sum x_{i} = \overset{x}{ˉ}$ aur $\sum \overset{x}{ˉ}^{2} = n \overset{x}{ˉ}^{2}$ use karo: $= \frac{1}{n} \sum x_{i}^{2} - 2 \overset{x}{ˉ} \cdot \overset{x}{ˉ} + \overset{x}{ˉ}^{2} = \frac{1}{n} \sum x_{i}^{2} - \overset{x}{ˉ}^{2}$

Figure — Measures of dispersion — variance, standard deviation

Sample variance: $n - 1$ se KYUN divide karte hain?

Worked examples

Properties (inhe achi tarah yaad karo)

Non-negativity: $σ^{2} \geq 0$ ; 0 tab hoga jab sab values identical hoon.
Shift invariance: har value mein constant $c$ add karne se variance nahi badalta (spread move nahi karta). $Var (x + c) = Var (x)$ .
Scaling: har value ko $c$ se multiply karne par variance $c^{2}$ se multiply hota hai aur SD $∣ c ∣$ se. $Var (c x) = c^{2} Var (x)$ .

Common mistakes

Active-recall flashcards

Square kiye bina

(x_{i} - \overset{x}{ˉ})

average KYUN nahi kar sakte?

Kyunki

\sum (x_{i} - \overset{x}{ˉ}) = 0

hamesha hota hai — positives aur negatives cancel ho jaate hain, spread chahe kuch bhi ho, 0 milta hai.

Population variance formula

σ^{2} = \frac{1}{n} \sum (x_{i} - \overset{x}{ˉ})^{2}

Variance ke terms mein standard deviation

σ = σ^{2}

; yeh original units restore karta hai.

Variance ka shortcut formula

σ^{2} = \overline{x^{2}} - (\overset{x}{ˉ})^{2}

= mean of squares minus square of mean.

Sample ke liye

n - 1

se KYUN divide karte hain?

Bessel's correction — sample mean deviations ko bahut chota bana deta hai, isliye

n - 1

downward bias correct karta hai.

Sab data mein constant

c

add karne ka variance par kya effect?

Koi change nahi; variance shift-invariant hai.

Sab data ko

c

se multiply karne ka variance aur SD par kya effect?

Variance ×

c^{2}

, SD ×

∣ c ∣

Variance exactly 0 kab hota hai?

Sirf jab har data value identical ho.

Kya

\overline{x^{2}} \geq (\overset{x}{ˉ})^{2}

hota hai?

Haan, hamesha, kyunki unka difference variance

\geq 0

hai.

Agar data cm mein hai toh variance vs SD ki units?

Variance cm² mein, SD cm mein.

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho tumhari class bullseye par darts phenk rahi hai. Average batata hai ki darts average mein kahan land karte hain. Lekin do classes ka same average landing spot ho sakta hai — ek mein sab darts tightly bunched hain, doosre mein darts poori wall par scattered hain. Standard deviation ek aisa number hai jo kehta hai "darts kitne scattered hain?" Isse pane ke liye: har dart ki middle se doori measure karo, un distances ko square karo (taaki left aur right cancel na ho), unka average nikalo, aur normal distance par wapas aane ke liye square root lo. Chota number = saaf nishan lagate hain; bada number = wild throws.

Connections

Mean, Median, Mode — measures of central tendency — dispersion ko measure karne ke liye ek centre chahiye.
Normal Distribution — SD 68-95-99.7 rule define karta hai.
Coefficient of Variation — $CV = σ / \overset{x}{ˉ}$ , ek unitless relative-spread measure.
Covariance and Correlation — variance do variables tak generalise hota hai.
Least Squares Regression — squared deviations ka sum minimise karna, same idea.