scipy.stats — distributions, hypothesis tests

5.4.14 · Coding › Scientific Computing (Python)

YEH module exist kyun karta hai?

KYA tum actually har baar compute karte ho:

"Exactly is value ki kitni probability hai?" → density (PDF).
"Is value tak ki kitni probability hai?" → cumulative (CDF).
"95th percentile par kaun sa value baithta hai?" → quantile (PPF).
"Kya mera observed effect surprising hai?" → p-value.

Char functions, ek idea se derive kiye

from scipy import stats
n = stats.norm(loc=0, scale=1)      # mean 0, sd 1
n.pdf(0)        # 0.3989  height of the bell at center
n.cdf(1.96)     # 0.9750  area to the left of 1.96
n.sf(1.96)      # 0.0250  area to the right (= 1 - cdf)
n.ppf(0.975)    # 1.96    inverse of cdf
n.rvs(size=5)   # 5 random samples

Standardizing: z-score, derive kiya

Hypothesis testing, scratch se

data = [5.1, 4.9, 5.3, 5.0, 4.8]
stats.ttest_1samp(data, popmean=5.0)
# TtestResult(statistic=-0.0, pvalue=1.0)  -> no evidence mean ≠ 5

Worked examples

Worked example (1) "Test scores mein 130 se zyaada kitne fraction hain?" (

$μ = 100, σ = 15$ )

stats.norm(100, 15).sf(130)   # 0.02275

.sf kyun? Hum $P (X > 130)$ chahte hain, upper tail → survival, cdf nahi. Yeh numbers kyun? $130 = μ + 2 σ$ , toh ~2.3% — empirical 95% rule se match karta hai (5%/2 har tail mein, thoda kam kyunki 2σ ≠ 1.96σ).

Worked example (2) Two-sample t-test: kya do groups alag hain?

a = [20, 22, 19, 24, 25]
b = [28, 27, 30, 26, 29]
stats.ttest_ind(a, b)   # statistic≈-5.6, pvalue≈0.0005

Yeh step kyun? $p < 0.05$ → "same mean" ko reject karo: group b genuinely zyaada hai. Bada $∣ t ∣$ within-group spread ke relative ek clear gap se aata hai.

Worked example (3) Chi-square goodness-of-fit: kya ek die fair hai?

obs = [8, 9, 19, 5, 8, 11]      # 60 rolls
exp = [10]*6                    # fair die expects 10 each
stats.chisquare(obs, exp)       # statistic≈12.4, pvalue≈0.029

Chi-square kyun? Statistic $\sum \frac{( O - E ) ^{2}}{E}$ total squared mismatch measure karta hai, expected size se scale karke. $p < 0.05$ yahan → die loaded lagti hai (face 3 zyaada baar aa raha hai).

Worked example (4) Data par ek distribution fit karo

sample = stats.norm(50, 5).rvs(size=1000)
mu_hat, sd_hat = stats.norm.fit(sample)   # ≈ 50, ≈ 5

.fit kyun? Yeh maximum-likelihood parameter estimates return karta hai — samples se truth recover karta hai.

Common mistakes

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho ek bag of marbles jahan kuch sizes common hain aur kuch rare. Ek distribution woh rulebook hai jo batata hai har size kitna common hai. PDF hai "exactly is size ki kitni popularity hai," CDF hai "is size ya usse chhote kitne marbles hain." Ek hypothesis test aisa hai jaise koi dost claim kare ki bag mein "sab medium marbles hain." Tum ek mutthi bharke uthao: agar baar baar giants nikle, tum kaho "yeh sirf luck nahi ho sakta — teri claim galat hai!" p-value hai ki pure luck kitni baar tumhe itni weird mutthi de sakta tha. Tiny p = "luck explain nahi kar sakta" = tumhara dost galat tha.

Flashcards

stats.norm.cdf(x) kya return karta hai?

P (X \leq x)

x

ke left mein PDF ke neeche ka area.

.sf(x) kya compute karta hai aur 1-cdf se zyaada prefer kyun karte hain?

Survival function

P (X > x) = 1 - F (x)

; yeh far right tail mein numerically zyaada accurate hai.

PPF (.ppf(p)) kya hai?

Inverse CDF: woh value

x

jahan

F (x) = p

— yaani

p

-quantile.

P-value precisely define karo.

P (a test statistic at least as extreme as observed ∣ H_{0} true)

Standard error mein

n

se divide kyun karte hain?

Sample mean ka variance

σ^{2} / n

hota hai, toh uska sd

σ / n

hai;

n

values average karne se noise shrink hoti hai.

One-sample test ke liye t-statistic derive karo.

t = (\overset{x}{ˉ} - μ_{0}) / (s / n)

—

μ_{0}

par center karo, estimated standard error se scale karo;

t_{n - 1}

follow karta hai.

z

ki jagah

t

kyun?

Kyunki

σ

s

se estimate kiya jaata hai, jo uncertainty badhata hai;

n - 1

df wala heavier-tailed Student-t iske liye account karta hai.

stats.chisquare kya measure karta hai?

\sum (O - E)^{2} / E

, observed aur expected counts ke beech total scaled squared mismatch.

Common error: kya chhota p

H_{0}

ko false prove karta hai?

Nahi — yeh

P (data ∣ H_{0})

hai,

P (H_{0} ∣ data)

nahi; sirf yeh kehta hai ki

H_{0}

ke under data surprising hai.

stats.expon mein scale ka matlab kya hai?

scale = 1/λ (mean), rate λ nahi.

stats.norm.fit(data) kya return karta hai?

Maximum-likelihood estimates

(\overset{μ}{^}, \overset{σ}{^})

jo data ko best fit karte hain.

T value se two-sided p-value kaise nikalte hain?

2 \cdot

stats.t.sf(abs(t), df).

Connections

Normal Distribution — $N (μ, σ)$ workhorse jo z-score se standardize hota hai.
Central Limit Theorem — kyun sample means ~normal hote hain, t/z tests ko justify karta hai.
p-values and Significance — $α$ aur rejection ki decision logic.
Maximum Likelihood Estimation — .fit under the hood kya karta hai.
numpy.random — .rvs ke comparison mein lower-level sampling.
Chi-square Distribution — squared normals ka sum; goodness-of-fit ka basis.