STATISTICS (సాంఖ్యక శాస్త్రం)

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 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!

Median = 20

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

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!

Removed number = 14

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

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 ✓

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!

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)!

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!

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.

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)!

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=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.

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.

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 = 50

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

Variance = 50

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

Σxᵢ (sum) is sufficient statistic for normal distribution mean

Sufficient statistic: contains all information about parameter from sample.
For Normal(μ,σ²): T = Σxᵢ is sufficient for μ.
Factorization theorem: f(x|μ) = g(T,μ)×h(x).

Sample mean x̄ = T/n is also sufficient (one-to-one function of T).

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!

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!

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.

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.

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!

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 = 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 = 1

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

Negative numbers కి కూడా same formula!

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!

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!

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!)

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.

రెండు peaks ఉన్న distribution — రెండు modes

Bimodal: exactly two modes / two peaks in frequency curve.
ఉదా: Heights of men and women combined → two peaks (one around 165cm women, one around 175cm men).

Example data: 1,2,2,3,4,5,5,6 → Modes 2 and 5.
Bimodal suggests two distinct groups in data!

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 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!

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.

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).

Combined mean = 52.5

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

Median = 50.5

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

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!

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!

Number = 11

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

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!

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!

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.

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!

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 = 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.

Correct mean = 9.6

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

Mode = 7

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