STATISTICS (సాంఖ్యక శాస్త్రం)

n-1 (Bessel's correction) makes it unbiased estimator of population variance

s² = Σ(xᵢ-x̄)²/(n-1).
E[s²] = σ² (unbiased).

Why n-1? Because x̄ estimated from data → one degree of freedom lost.
With n: E[Σ(xᵢ-x̄)²/n] = (n-1)σ²/n < σ² (biased!). With n-1: unbiased.Degrees of freedom concept: n observations - 1 constraint (mean) = n-1.

n→∞: Sample median ~ N(m, 1/(4n[f(m)]²))

Sample median asymptotically normal:
m̂ → N(m, σ²_median).
σ²_median = 1/(4n[f(m)]²).
f(m) = pdf at median m.

Compare to mean: σ²_mean = σ²/n.
Efficiency = σ²_mean/σ²_median = 4[f(m)]²σ²/1.
For normal: efficiency of median = 2/π ≈ 0.637 (mean is more efficient!)

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!

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!

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.

Mode = 7

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

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.

Weighted Mean = Σ(wᵢxᵢ)/Σwᵢ

ప్రతి value కి weight ఉన్నప్పుడు weighted mean వాడతాం.
W.M. = (w₁x₁+w₂x₂+...)/(w₁+w₂+...).

ఉదా: Marks: Maths=80 (weight=4), English=70 (weight=2).
W.M. = (80×4+70×2)/(4+2) = (320+140)/6 = 460/6 ≈ 76.7.
CGPA calculate చేసేటపుడు weighted mean వాడతారు!

Median = 6

n=6 (even). Middle terms: 3rd=5, 4th=7.
Median = (5+7)/2 = 6.

MLE of μ = Sample mean x̄

For normal distribution N(μ,σ²):
Likelihood = Π f(xᵢ|μ,σ²).
Log-likelihood = -n/2 log(2πσ²) - Σ(xᵢ-μ)²/(2σ²).
∂L/∂μ = Σ(xᵢ-μ)/σ² = 0 → μ̂ = Σxᵢ/n = x̄.

MLE of mean = sample mean — unbiased estimator!

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.

Distribution peak ని measure చేసే statistic

Kurtosis = distribution యొక్క peak/tail heaviness measure.
Mesokurtic: Normal distribution (kurtosis=3 లేదా excess=0).
Leptokurtic: Thin, tall peak (kurtosis>3). Heavy tails.
Platykurtic: Flat, broad distribution (kurtosis<3). Light tails.గుర్తు: Lepto = thin → tall peak. Platy = flat → broad peak. Meso = middle → normal.

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!

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.

Original sample నుండి with replacement resample చేసి distribution estimate చేయడం

Bootstrap:
1. Original sample నుండి n samples with replacement గా draw చేయాలి.
2. ప్రతి bootstrap sample కి mean calculate చేయాలి.
3. 1000+ bootstrap means distribution → CI for population mean.
4. 2.5th మరియు 97.5th percentiles = 95% CI.

Distribution assumption అక్కరలేదు — non-parametric method!

Mean = 6

Sum = 2+4+6+8+10 = 30.
Count = 5.
Mean = 30/5 = 6.

గుర్తు: Consecutive even numbers mean = middle value = 6 ✓.

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!

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.

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.

Variance = 2

Mean = 3.
Σ(x-x̄)² = 4+1+0+1+4 = 10.
Variance = 10/5 = 2. SD = √2.

Or: Σx²/n-(x̄)² = (1+4+9+16+25)/5-9 = 55/5-9 = 11-9 = 2 ✓

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!

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!

Mean = 5.5

Sum = n(n+1)/2 = 10×11/2 = 55.
Mean = 55/10 = 5.5.

Formula: Mean of 1 to n = (n+1)/2.
n=10: (10+1)/2 = 5.5 ✓

GM = √(4×16) = √64 = 8

GM = √(x₁×x₂) = √(4×16) = √64 = 8.

Geometric mean = nth root of product of n numbers.
Check: 4, 8, 16 → geometric progression with ratio 2 ✓.
GM always ≤ AM! AM=(4+16)/2=10≥8=GM ✓

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.

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!

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!

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.

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!

Linear time (O(n)) worst-case guaranteed selection algorithm

Median of Medians:
kth smallest element O(n) worst-case గా find చేయడానికి.
Steps:
1. n elements ని groups of 5 గా divide చేయాలి.
2. ప్రతి group median find చేయాలి.
3. ఆ medians మళ్ళీ median find చేయాలి (pivot).
4. Quickselect apply చేయాలి.

O(n) worst-case guarantee! Regular quickselect O(n²) worst-case.
BFPRT algorithm అని కూడా అంటారు.

Extreme values ని trim చేసి replace చేసిన తర్వాత calculate చేసే mean

Winsorized mean:
Extreme values (top/bottom %) ని trim చేయడం కాదు — replace చేస్తారు.
ఉదా: 10% Winsorized mean → bottom 10% values = 10th percentile తో replace, top 10% = 90th percentile తో replace.

Trimmed mean: అన్నీ remove చేస్తారు.
Winsorized mean: replace చేస్తారు.
Both are robust to outliers.

Variance = 50

Mean = Σfx/Σf = 800/40 = 20.
Variance = Σfx²/Σf - (mean)² = 18000/40 - 400 = 450-400 = 50.

≈68 students

Normal distribution:
Mean±1SD covers 68.27% data.
40-60 = 50±10 = Mean±1SD.
≈68% × 100 = 68 students.

68-95-99.7 rule:
±1SD → 68%, ±2SD → 95%, ±3SD → 99.7%!

Mode ≈ 22

Empirical formula: Mode = 3 Median - 2 Mean.
= 3×24 - 2×25 = 72-50 = 22.

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!

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!

n large గా వెళ్ళినప్పుడు sample means distribution → Normal Distribution (any population)

CLT: ఏ population distribution అయినా, sample size (n) పెద్దగా ఉంటే sample means distribution approximately normal అవుతుంది.

Conditions: n ≥ 30 (rule of thumb).
Sample mean: μ_x̄ = μ (population mean).
Sample SD: σ_x̄ = σ/√n (standard error).

Statistics లో most important theorem!

Mean=4, Median=4, Mode=4

All values = 4.
Mean = 20/5 = 4.
Median = middle value = 4.
Mode = 4 (5 times).
All three = 4!

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).

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.

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!

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.

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.

HM = 3.2

HM = n/Σ(1/xᵢ) = 2/(1/2+1/8) = 2/(4/8+1/8) = 2/(5/8) = 16/5 = 3.2.

AM=5, GM=4, HM=3.2. AM≥GM≥HM ✓.
HM useful for rates, speeds (average speed problems)!

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!

n/2 frequency వద్ద horizontal line గీసి ogive తో కలిసే point నుండి x-axis కి vertical drop చేయాలి

Steps:
1. Less than Ogive draw చేయాలి (cumulative frequency vs upper class boundary).
2. y-axis పై n/2 mark చేయాలి.
3. Horizontal line గీసి ogive తో intersect చేయాలి.
4. Intersect point నుండి x-axis కి vertical line వేయాలి.
5. x-axis పై వచ్చే value = Median!

Ogive = Cumulative frequency curve.

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!

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).

Mode < Median < MeanPositively skewed (right skewed): Tail on right → Mean pulled right most. Mode < Median < Mean.Negatively skewed: Mean < Median < Mode. Memory: Skew direction = direction of tail = where mean is pulled!

Median = 20

n=6 (even). 3rd=19, 4th=21.
Median = (19+21)/2 = 40/2 = 20.