STATISTICS (సాంఖ్యక శాస్త్రం)

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!

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!

Mode = 7

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

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.

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.

≈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%!

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.

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 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!

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!

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.

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!

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!

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.

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.

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.

Median = 50.5

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

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=4, Median=4, Mode=4

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

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.

Sum = 100

Sum = Mean × n = 20 × 5 = 100.

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

Median = arranged data లో మధ్య విలువ

Data ని ascending order లో arrange చేసినప్పుడు మధ్యలో వచ్చే విలువ = Median.

Odd n: Median = ((n+1)/2)th term.
Even n: Median = average of (n/2)th and (n/2+1)th terms.

గుర్తు: Median outliers వల్ల affect కాదు! Income data కోసం median వాడతారు.

Number = 11

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

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!

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!

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 = 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!

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!

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 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.

Estimator infinite/meaningless అవ్వడానికి minimum contamination fraction

Breakdown point:
Data లో extreme values replace చేసినప్పుడు estimator complete breakdown అవ్వడానికి minimum fraction.

Mean breakdown point = 1/n ≈ 0 (one outlier spoils it).
Median breakdown point = 0.5 (50% data corrupt అయినా works).
Trimmed mean: trim% = breakdown point.

Median most robust! This is why we use it for skewed data.

Median = 6

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

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).

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!

Correct mean = 9.6

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

Combined mean = 52.5

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

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 ✓

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!

Removed number = 14

Original sum = 10×5 = 50.
New sum = 9×4 = 36.
Removed = 50-36 = 14.

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!

Mean = 6

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

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

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.

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 ✓

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!

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!)

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!

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 = 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 (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.

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!