STATISTICS (సాంఖ్యక శాస్త్రం)

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.

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.

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!

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!)

Combined mean = 52.5

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

Mean = 1

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

Negative numbers కి కూడా same formula!

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.

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!

Variance unchanged

Variance measures spread, not location.
Each value +10 → mean +10, but deviation (x-x̄) unchanged.
(x+10-(x̄+10)) = x-x̄. Same deviations → same variance.

గుర్తు: Constant add/subtract → Variance unchanged.
Constant multiply/divide → Variance multiplied/divided by k²!

Mode ≈ 22

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

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.

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 ✓

Mean=4, Median=4, Mode=4

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

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!

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!)

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!

Median = 6

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

Symmetric unimodal distribution → Mean=Median=Mode

Conditions:
1. Symmetric distribution (not skewed).
2. Unimodal (single peak).

Examples: Normal distribution, Uniform distribution.
Not sufficient: symmetric but bimodal → Mean=Median but two Modes.

In practice: perfect equality only for ideal symmetric distributions!

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.

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 ✓

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!

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!

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!

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.

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.

Mode = 5

3→3 times, 5→4 times, 7→2 times.
5 అత్యధిక సార్లు → Mode = 5.

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!

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 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!

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).

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 = 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!

Sum = 100

Sum = Mean × n = 20 × 5 = 100.

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

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!

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).

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)!

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!

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!

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.

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!

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!

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.

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!

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.

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!

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!

Removed number = 14

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

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!

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.

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!