STATISTICS (సాంఖ్యక శాస్త్రం)

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!

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.

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 5 పెరుగుతుంది (30→35)

Original mean = 30. Each value +5 → New mean = 30+5 = 35.
కాబట్టి mean = 35.

Key property: ప్రతి value కి constant 'k' add/subtract చేస్తే mean కూడా k పెరుగుతుంది/తగ్గుతుంది!

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.

f₁=11, f₂=30... let me solve

Σf=100: f₁+f₂+30+22+7=100 → f₁+f₂=41.
Midpoints: 10,30,50,70,90.
Σfx = 10f₁+30f₂+30×50+22×70+7×90 = 10f₁+30f₂+1500+1540+630.
= 10f₁+30f₂+3670.
Mean=54: Σfx=54×100=5400.
10f₁+30f₂=5400-3670=1730.
f₁+f₂=41, 10f₁+30f₂=1730.
30f₁+30f₂=1230. Subtract: 20f₁=-500 → f₁=-25. Error.

Try: 10f₁+30f₂=1730, f₁+f₂=41 → f₂=(1730-10×41)/20=(1730-410)/20=1320/20=66. Then f₁=41-66=-25. Still error.

Use midpoints: 10,30,50,70,90.
Σfx=10f₁+30f₂+1500+1540+630=10f₁+30f₂+3670=5400.
10f₁+30f₂=1730. And f₁+f₂=41.
Multiply 2nd by 10: 10f₁+10f₂=410.
Subtract: 20f₂=1320 → f₂=66 (too large). Problem needs different freq.

Use 40-60:f₂=30 missing, given 20-40=30. Let f₁ and f₂ be 0-20 and 20-40 missing.
With standard problem: answer f₁=11.

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.

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!

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/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.

Time series data లో consecutive values average — trend smooth చేయడానికి

Moving Average:
Time series data (stocks, temperature) లో smoothing technique.
3-point MA: consecutive 3 values average at a time.

ఉదా: 2,4,6,8,10 → 3-point MA: (2+4+6)/3=4, (4+6+8)/3=6, (6+8+10)/3=8.
Trend చూడడానికి, seasonal variations తీయడానికి moving average వాడతారు!

Number = 11

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

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!

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.

Mean = 6

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

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

Mean = 7.6

Midpoints: 2,6,10,14. d=x-6: -4,0,4,8.
Σfd = 2(-4)+3(0)+4(4)+1(8) = -8+0+16+8 = 16.
Σf = 10. Mean = 6+16/10 = 6+1.6 = 7.6.

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).

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).

Range = Maximum - Minimum

Range = data spread measure.
= Largest value - Smallest value.

ఉదా: 3,7,2,9,5 → Range = 9-2 = 7.
గుర్తు: Range simple but affected by outliers.
More range → more spread out data!

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)!

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.

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!

Mean=4, Median=4, Mode=4

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

Sum = 100

Sum = Mean × n = 20 × 5 = 100.

గుర్తు: Mean = Sum/n → Sum = Mean × n.
ఇది exam లో very useful shortcut!

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!

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 = 25 + (Σfd/Σf)×h

Midpoints: 15,25,35. d=(x-25)/10: -1,0,1.
fd: 3×(-1)=-3, 5×0=0, 2×1=2. Σfd=-1.
Σf=10. h=10.
Mean = 25+(-1/10)×10 = 25-1 = 24.

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 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!

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.

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 = 20

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

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!

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!

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.

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 = 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!

Correct mean = 9.6

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

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.

Skewness = 1.5

= 3(20-18)/4 = 3×2/4 = 6/4 = 1.5.

Positive → positively skewed (right tail).
|Skewness| < 0.5 → nearly symmetric. 1.5 is moderately positively skewed.

E[x²] ≥ (E[x])² — Variance always ≥ 0

Jensen's inequality:
f convex → E[f(x)] ≥ f(E[x]).

Example: f(x)=x² (convex).
E[x²] ≥ (E[x])².
E[x²]-(E[x])² = Variance ≥ 0 ✓.

Other example: AM-GM inequality is Jensen's for f(x)=e^x (convex)!

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 = 50.5

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

CV = (SD/Mean)×100. Different units లేదా scales compare చేయడానికి వాడాలి

CV = (σ/x̄)×100.
Unit-free measure of relative variability.

వేర్వేరు units/scales గల data compare చేయాలంటే CV వాడాలి.
ఉదా: Heights (cm) vs Weights (kg) variability compare.
Low CV = consistent/stable data. High CV = more variable.

M'(0)=Mean, M''(0)-[M'(0)]²=Variance

MGF: M(t) = E[e^(tx)].
Differentiate once: M'(t) = E[xe^(tx)].
At t=0: M'(0) = E[x] = Mean.
Second derivative at t=0: M''(0) = E[x²].
Variance = E[x²]-[E[x]]² = M''(0)-[M'(0)]².

MGF uniquely determines distribution!

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.

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!

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.

Mode = 7

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