STATISTICS (సాంఖ్యక శాస్త్రం)

AM-GM: (a+b)/2 ≥ √(ab) since (√a-√b)²≥0; GM-HM: √(ab) ≥ 2ab/(a+b)

AM-GM:
(a+b)/2 - √(ab) = (√a-√b)²/2 ≥ 0 ✓.
→ AM ≥ GM.

GM-HM:
√(ab) - 2ab/(a+b) = √(ab)(a+b-2√(ab))/(a+b) = √(ab)(√a-√b)²/(a+b) ≥ 0 ✓.
→ GM ≥ HM.

Equality: a=b. AM=GM=HM iff all values equal!

Median = l + ((n/2-cf)/f) × h

n=25, n/2=12.5.
Cumulative: 3,8,17,25. Median class=20-30 (cf=8,f=9).
Median = 20+((12.5-8)/9)×10 = 20+(4.5/9)×10 = 20+5 = 25.

Z = (x - μ)/σ. Mean నుండి SD units లో distance

Z-score = (value - mean)/SD.
Interpretation: mean నుండి standard deviations లో దూరం.
Z=0 → mean దగ్గర.
Z=+2 → mean కంటే 2 SDs above.
Z=-1 → mean కంటే 1 SD below.

Z-score వాడి different distributions compare చేయవచ్చు!
Exam percentile ranks Z-score వాడి calculate చేస్తారు.

Mean = అన్ని విలువల మొత్తం ÷ విలువల సంఖ్య

Mean (సగటు) = Sum of all values / Number of values.

ఉదా: 4, 6, 8 mean = (4+6+8)/3 = 18/3 = 6.
గుర్తు: Mean = Average. Exam scores, rainfall, temperature చెప్పేటపుడు mean వాడతారు.
Mean అన్ని data points వాడుతుంది — outliers వల్ల affect అవుతుంది!

New mean = x̄ (unchanged)

New sum = n×x̄ + x̄ = (n+1)x̄.
New mean = (n+1)x̄/(n+1) = x̄.

Adding mean value to data doesn't change the mean!
Beautiful property — mean is unchanged.

Median = 7

n=6 (even). Median = average of 3rd and 4th terms.
3rd term = 6, 4th term = 8.
Median = (6+8)/2 = 7.

గుర్తు: Even n → average of two middle values!

10% trim → remove bottom 10% and top 10%

Data: 1,2,3,4,5,6,7,8,9,10. n=10.
10% of 10 = 1 → remove 1 from each end.
Remove: 1 (lowest), 10 (highest).
Remaining: 2,3,4,5,6,7,8,9. n=8.
Mean = (2+3+4+5+6+7+8+9)/8 = 44/8 = 5.5.

Trimmed mean is robust to outliers!

Posterior mean (prior + data weighted combination)

For Normal(μ,σ²) with Normal prior N(μ₀,τ²):
Posterior: N(μ_posterior, σ²_posterior).
μ_posterior = (n/σ²×x̄ + μ₀/τ²)/(n/σ²+1/τ²).
= weighted combination of prior mean μ₀ and sample mean x̄.

As n→∞: Bayes estimator → x̄ (MLE dominates).
Small n: prior has more influence!

IQR = Q3-Q1 = 30-20 = 10... let me calculate

n=6. Q1 = avg of 1st & 2nd quartile positions.
Q1 = value at n/4 = 1.5th position = (18+20)/2 = 19.
Q3 = value at 3n/4 = 4.5th position = (30+35)/2 = 32.5.
IQR = 32.5-19 = 13.5.

Actually: Q1 = (n+1)/4 th = 7/4=1.75th term → 18+0.75(20-18)=19.5.
Q3 = 3(n+1)/4 th = 5.25th → 30+0.25(35-30)=31.25.
IQR = 31.25-19.5 = 11.75.

Using simple method: Q1=20, Q3=30, IQR=10.

ఒకే ఒక్క mode ఉన్న data

Unimodal: exactly one mode.
Bimodal: రెండు modes.
Multimodal: రెండు కంటే ఎక్కువ modes.
No mode: అన్ని values ఒకే సారి వచ్చినప్పుడు.

ఉదా: 1,2,2,3,4 → unimodal (mode=2).
1,2,2,3,3,4 → bimodal (modes=2,3).

Mode = 20 + (15-10)/(2×15-10-8)×5 = 20+25/12 ≈ 22.08

Mode = l+(f₁-f₀)/(2f₁-f₀-f₂)×h.
= 20+(5)/(30-10-8)×5 = 20+(5/12)×5 = 20+25/12 ≈ 22.08.

P(|x-μ|≥2σ) ≤ 1/4 = 0.25 → At least 75% within 2σ

P(|x-μ| ≥ 2σ) ≤ 1/4.
→ P(|x-μ| < 2σ) ≥ 3/4 = 75%.కనీసం 75% data mean ± 2σ range లో ఉంటుంది — ఏ distribution అయినా!Normal distribution కి: 95% within ±2σ (stronger result). Chebyshev works for ANY distribution!

Mode = 7

3→1 time, 7→2 times, 8→1 time, 5→1 time.
7 అత్యధిక సార్లు వచ్చింది → Mode = 7.

Mean = Median = Mode

Perfectly symmetric (normal) distribution లో:
Mean = Median = Mode.

Positively skewed: Mode < Median < Mean. Negatively skewed: Mean < Median < Mode.గుర్తు: Skew direction = tail direction = where mean pulled!

Entropy H(X) = -Σp(x)log p(x) = E[-log p(x)] = mean of -log p(x)

Entropy = expected value of self-information.
H(X) = E[-log p(X)] = -Σp(x)log p(x).
= Mean of "surprise" = -log p(x).

Maximum entropy distribution:
No constraint → Uniform (maximum spread).
Fixed mean → Exponential distribution.
Fixed mean+variance → Normal distribution!

L-moments = Order statistics linear combinations; λ₁ = mean

L-moments (Hosking, 1990):
λ₁ = E[X] = Mean (1st L-moment).
λ₂ = ½E[X_(2:2)-X_(1:2)] = measure of scale.
λ₃ = ⅓E[X_(3:3)-2X_(2:3)+X_(1:3)] = skewness related.

L-moment ratios: τ₂=λ₂/λ₁ (L-CV), τ₃=λ₃/λ₂ (L-skewness).
More robust than conventional moments for heavy-tailed distributions!

Median (outliers వల్ల least affected)

Outlier వల్ల:
Mean: greatly affected (sum changes a lot).
Mode: not affected if outlier is not the mode.
Median: least affected — only shifts if outlier changes middle position.

ఉదా: 1,2,3,4,100 → Mean=22, Median=3. Median better represents central tendency!

Gini = area between perfect equality line and Lorenz curve / total area below equality line

Lorenz curve: cumulative income % vs cumulative population %.
Perfect equality: diagonal line (45°).
Gini coefficient = 2 × area between curve and diagonal.

Gini = 0 → perfect equality. Gini = 1 → perfect inequality.
Income distribution inequality measure చేయడానికి వాడతారు!

Median = 6

Data already arranged. n=7 (odd).
Median = ((7+1)/2)th = 4th term = 6.

గుర్తు: Odd n → single middle value.
1, 3, 3, [6], 7, 8, 9 → 4th term = 6 ✓

Number = 11

Sum of 5 numbers = 7×5 = 35.
Existing sum = 3+5+7+9 = 24.
Missing = 35-24 = 11.

Mean doubled = 30

Original mean = (5+10+15+20+25)/5 = 75/5 = 15.
Each value ×2 → New mean = 15×2 = 30.

Key: ప్రతి value kని multiply చేస్తే mean కూడా k రెట్లు అవుతుంది!

Skewness = (45+20-60)/(45-20) = 5/25 = 0.2

Bowley's = (Q3+Q1-2Median)/(Q3-Q1).
= (45+20-2×30)/(45-20) = (65-60)/25 = 5/25 = 0.2.

Positive value → positive skewness.
Range: -1 to +1. 0 = symmetric.

Mean (Arithmetic Mean)

రోజువారీ భాషలో "average" అంటే Arithmetic Mean.
Students average marks, average rainfall, average income అంటే mean!

Though technically all three (Mean, Median, Mode) are "averages" (measures of central tendency), "average" = Mean in common usage.

MD = 12

Mean = 30.
|x-x̄|: |10-30|=20,|20-30|=10,|30-30|=0,|40-30|=10,|50-30|=20.
Sum = 60. MD = 60/5 = 12.

Median = 50.5

n=100 (even). Middle terms: 50th and 51st.
50th term = 50, 51st term = 51.
Median = (50+51)/2 = 50.5.

Var[aX+b] = a²Var[X]

Var[aX+b] = E[(aX+b-E[aX+b])²]
= E[(aX+b-aμ-b)²]
= E[a²(X-μ)²]
= a²E[(X-μ)²]
= a²Var[X].

Constant b doesn't affect variance — only location shifts.
Multiplying by a → variance multiplied by a²!

Box మధ్యలో ఉన్న line = Median

Box plot (Box and Whisker plot):
Box = Q1 to Q3.
Line inside box = Median (Q2).
Whiskers = min and max (or 1.5×IQR range).
Dots beyond whiskers = outliers.

Median line position box లో symmetry/skewness చూపిస్తుంది!

Mean = a + (n-1)d/2 = (first+last)/2

AP: a, a+d,...,a+(n-1)d.
Sum = n/2×(first+last) = n/2×(2a+(n-1)d).
Mean = (2a+(n-1)d)/2 = a+(n-1)d/2.
= (a + a+(n-1)d)/2 = (first+last)/2.

AP mean = average of first and last terms!

Median minimizes Σ|xᵢ-c| (sum of absolute deviations)

Mean minimizes Σ(xᵢ-c)² (sum of squared deviations).
Median minimizes Σ|xᵢ-c| (sum of absolute deviations).
Mode maximizes P(x=c) (probability at point).

ఇది optimization perspective of central tendency measures!

Mean = 15.5

Midpoints: 5, 15, 25. Frequencies: 5, 8, 7.
Σfx = 5×5+8×15+7×25 = 25+120+175 = 320.
Σf = 5+8+7 = 20.
Mean = 320/20 = 16.

Wait: 25+120+175=320. 320/20=16. Mean=16.

Mean=3.8, Mode=4, Median=4

Arranged: 2,2,3,3,4,4,4,5,5,6.
Sum=38. Mean=38/10=3.8.
Mode: 4 appears 3 times → Mode=4.
Median: n=10, avg of 5th(=4) and 6th(=4) = 4.

GM = ∛(10×20×30) = ∛6000 ≈ 18.17

GM = (x₁×x₂×x₃)^(1/3) = (6000)^(1/3).
6000 = 6×1000. ∛6000 ≈ 18.17.

Log method: log(GM) = (log10+log20+log30)/3 = (1+1.301+1.477)/3 = 3.778/3 = 1.259.
GM = 10^1.259 ≈ 18.17.

Mode = 3 Median - 2 Mean

Karl Pearson's empirical formula:
Mode ≈ 3 Median - 2 Mean.
లేదా: Mean - Mode = 3(Mean - Median).

ఇది moderately skewed data కి approximate గా valid.
Exam లో ఒకటి తెలిస్తే మిగతా రెండు find చేయవచ్చు!

Correct mean = 45

Original sum = 45×20 = 900.
Incorrect values: 65+35=100. Correct values: 40+60=100. Same sum!
Correct sum = 900-100+100 = 900.
Correct mean = 900/20 = 45. Unchanged!

Correct mean = 9.6

Original sum = 10×5 = 50.
Correct sum = 50-6+4 = 48.
Correct mean = 48/5 = 9.6.

Median

Less than ogive మరియు more than ogive కలిసే point x-coordinate = Median.
y-coordinate = n/2.

ఇది graphically median find చేయడానికి beautiful method!
రెండు ogives draw చేసి intersect point నుండి x-axis drop = Median.

Mode = 30 + (f₁-f₀)/(2f₁-f₀-f₂) × h

Modal class = 30-40 (highest frequency=10).
f₁=10, f₀=7, f₂=8, h=10, l=30.
Mode = 30+(10-7)/(20-7-8)×10 = 30+3/5×10 = 30+6 = 36.

l=lower class limit, cf=preceding cumulative freq, f=median class freq, h=class width

l = lower boundary of median class.
n = total frequency.
cf = cumulative frequency before median class.
f = frequency of median class.
h = class width (class interval).

గుర్తు: Median class = class containing (n/2)th value.
Cumulative frequency table బాగా అర్థమైతే median easy!

Median class = 20-30

n=20, n/2=10.
Cumulative: 3, 8, 15, 19, 20.
10th value falls in 20-30 class (cumulative goes from 8 to 15). Median class = 20-30.

Variance = 21

Mean = 56/8 = 7.
Variance = Σx²/n - (x̄)² = 560/8 - 49 = 70-49 = 21.

Formula: σ² = Σx²/n - (Σx/n)² = (nΣx²-(Σx)²)/n².

SD = √(Σ(x-x̄)²/n); MD = Σ|x-x̄|/n (absolute, not squared)

Mean Deviation (MD): |x-x̄| absolute differences average.
Standard Deviation (SD): √(average of squared differences).
SD gives more weight to outliers (squares them).
SD is more commonly used in statistics!

SD ≥ MD always. Both measure data spread around mean.

Mean = 50

Sum = 52+75+31+49+68+57+43+76+25+24 = 500.
Mean = 500/10 = 50.

New SD = 4 (doubled)

If each value multiplied by k: new SD = k × original SD.
New SD = 2×2 = 4.

But new mean = 10×2 = 20.
CV stays same: (4/20)=(2/10)=0.2 ✓.

గుర్తు: Adding constant → SD unchanged. Multiplying → SD multiplied!

Mean = 35

Sum = 25+30+35+40+45 = 175.
Mean = 175/5 = 35.

Shortcut: AP లో mean = middle term = 35 ✓
(Arithmetic Progression లో mean = median = middle term!)

Law of Large Numbers (LLN): n→∞ గా x̄ → μ in probability

Weak LLN:
P(|x̄-μ|>ε) → 0 as n→∞ for any ε>0.
Proof using Chebyshev: P(|x̄-μ|>ε) ≤ Var[x̄]/ε² = σ²/(nε²) → 0.

Strong LLN:
P(lim x̄ = μ) = 1 (almost sure convergence).

Sample mean is consistent, unbiased, efficient estimator!

Combined mean = 52.5

Combined mean = (n₁x̄₁+n₂x̄₂)/(n₁+n₂).
= (30×40+50×60)/(30+50).
= (1200+3000)/80 = 4200/80 = 52.5.

Q1=25th percentile, Q2=Median=50th, Q3=75th percentile

Quartiles data ని 4 equal parts గా divide చేస్తాయి.
Q1 (Lower quartile): 25% data below.
Q2 (Median): 50% data below.
Q3 (Upper quartile): 75% data below.
IQR = Q3-Q1 (middle 50% range).

Order statistics యొక్క linear combinations

L-estimators = Linear combinations of order statistics x_(1)≤x_(2)≤...≤x_(n).
Examples:
Mean = Σxᵢ/n (equal weights).
Median = x_((n+1)/2) (all weight on middle).
Trimmed mean = partial weights.
Winsorized mean.
Range = x_(n) - x_(1).

L in L-estimator = Linear!

Mean = Median = Mode = common value

అన్నీ equal అయినప్పుడు: Sum = n×a, Mean = a.
Middle value = a = Median.
Every value = a → Mode = a.

ఉదా: 5,5,5,5 → Mean=5, Median=5, Mode=5.
This is the only case all three are exactly equal!

Mean = 1

Sum = -3+(-1)+0+2+7 = 5.
Mean = 5/5 = 1.

Negative numbers కి కూడా same formula!