scipy.stats — distributions, hypothesis tests

WHY does this module exist?

WHAT you actually compute every time:

"How likely is exactly this value?" → density (PDF).
"How likely is up to this value?" → cumulative (CDF).
"What value sits at the 95th percentile?" → quantile (PPF).
"Is my observed effect surprising?" → p-value.

The four functions, derived from one idea

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: the z-score, derived

Hypothesis testing, from scratch

Formula One-sample t-test, derived

We test $H_0:\mu=\mu_0$ from a sample of size $n$ with mean $\bar x$ and sample sd $s$ .

The sample mean has standard error $\text{SE} = \dfrac{s}{\sqrt{n}}$ . WHY $\sqrt n$ : variance of a mean of $n$ independent values is $\sigma^2/n$ , so its sd is $\sigma/\sqrt n$ . Averaging shrinks noise.
Standardize: $t = \frac{\bar x - \mu_0}{s/\sqrt n}$ WHY: same z-trick — center on $\mu_0$ , scale by SE. Because $s$ estimates $\sigma$ (extra uncertainty), $t$ follows a Student-t with $n-1$ degrees of freedom, not exactly normal.
p-value (two-sided) $= 2\cdot P(T_{n-1} > |t|)$ = 2 * stats.t.sf(abs(t), df=n-1).

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) "What fraction of test scores beat 130?" (

$\mu=100,\sigma=15$ )

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

Why .sf? We want $P(X>130)$ , the upper tail → survival, not cdf. Why these numbers? $130 = \mu + 2\sigma$ , so ~2.3% — matches the empirical 95% rule (5%/2 in each tail, slightly less since 2σ ≠ 1.96σ).

Worked example (2) Two-sample t-test: do two groups differ?

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

Why this step? $p < 0.05$ → reject "same mean": group b is genuinely higher. The big $|t|$ comes from a clear gap relative to within-group spread.

Worked example (3) Chi-square goodness-of-fit: is a die fair?

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

Why chi-square? The statistic $\sum \frac{(O-E)^2}{E}$ measures total squared mismatch, scaled by expected size. $p<0.05$ here → the die looks loaded (face 3 over-appears).

Worked example (4) Fit a distribution to data

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

Why .fit? It returns maximum-likelihood parameter estimates — recovering the truth from samples.

Common mistakes

Recall Feynman: explain to a 12-year-old

Imagine a bag of marbles where some sizes are common and some rare. A distribution is the rulebook saying how common each size is. The PDF is "how popular is exactly this size," the CDF is "how many marbles are this size or smaller." A hypothesis test is like a friend claiming the bag is "all medium marbles." You grab a handful: if you keep pulling giants, you say "no way that's just luck — your claim is wrong!" The p-value is how often pure luck could have given you a handful this weird. Tiny p = "luck can't explain this" = your friend was wrong.

Flashcards

What does stats.norm.cdf(x) return?

P(X \le x)

, the area under the PDF to the left of

x

What does .sf(x) compute and why prefer it over 1-cdf?

The survival function

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

; it's numerically more accurate in the far right tail.

What is the PPF (.ppf(p))?

The inverse CDF: the value

x

where

F(x)=p

— i.e. the

p

-quantile.

Define a p-value precisely.

P(\text{a test statistic at least as extreme as observed} \mid H_0 \text{ true})

Why divide by

\sqrt n

in the standard error?

Variance of a sample mean is

\sigma^2/n

, so its sd is

\sigma/\sqrt n

; averaging

n

values shrinks noise.

Derive the t-statistic for a one-sample test.

t=(\bar x-\mu_0)/(s/\sqrt n)

— center on

\mu_0

, scale by the estimated standard error; follows

t_{n-1}

Why

t

instead of

z

Because

\sigma

is estimated by

s

, adding uncertainty; the heavier-tailed Student-t with

n-1

df accounts for it.

What does stats.chisquare measure?

\sum (O-E)^2/E

, total scaled squared mismatch between observed and expected counts.

Common error: does small p prove

H_0

false?

No — it's

P(\text{data}\mid H_0)

, not

P(H_0\mid\text{data})

; it only says data is surprising under

H_0

For stats.expon, what does scale mean?

scale = 1/λ (the mean), not the rate λ.

What does stats.norm.fit(data) return?

Maximum-likelihood estimates

(\hat\mu,\hat\sigma)

that best fit the data.

Two-sided p-value from a t value?

2\cdot

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

Connections

Normal Distribution — the $N(\mu,\sigma)$ workhorse standardized by the z-score.
Central Limit Theorem — why sample means are ~normal, justifying t/z tests.
p-values and Significance — the decision logic of $\alpha$ and rejection.
Maximum Likelihood Estimation — what .fit does under the hood.
numpy.random — lower-level sampling vs. .rvs.
Chi-square Distribution — sum of squared normals; basis of goodness-of-fit.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, scipy.stats basically ek statistics ka toolbox hai. Iska pehla kaam hai distributions — jaise stats.norm jo normal (bell curve) ko represent karta hai. Iske andar chaar important functions hote hain: .pdf (kisi exact point pe density kitni hai), .cdf (us point tak ka total area, yaani $P(X \le x)$ ), .sf (upar wala tail, $P(X>x)$ ), aur .ppf (ulta — probability do, value milegi, yaani percentile). Yaad rakho: sab kuch CDF se nikalta hai — PDF uska derivative hai, PPF uska inverse hai.

Doosra bada kaam hai hypothesis testing. Idea simple hai: tum maan lo "kuch khaas nahi ho raha" (yeh hai $H_0$ , null hypothesis). Phir tum ek statistic banate ho jiska distribution $H_0$ ke under pata hota hai. Phir p-value nikalta hai — iska matlab hai "agar $H_0$ sach hota, toh itna ya isse zyada extreme data milne ka chance kitna tha." Agar p chhota hai (jaise $<0.05$ ), toh bolte ho "yeh luck se nahi ho sakta", aur $H_0$ ko reject kar dete ho.

Sabse common galti yeh hai ki students sochte hain p-value ka matlab " $H_0$ sach hone ki probability" hai — yeh galat hai. p-value hai $P(\text{data} \mid H_0)$ , ulta nahi. Aur ek aur trick: t-test mein $z$ ki jagah $t$ isliye use karte hain kyunki hum $\sigma$ ko sample se estimate ( $s$ ) karte hain, jisse thodi extra uncertainty aati hai. Standard error mein $\sqrt n$ se divide karte hain kyunki average lene se noise kam hota hai. Yeh module DS, ML, aur research papers — har jagah kaam aata hai, isliye iski feel banana zaroori hai.

scipy.stats — distributions, hypothesis tests

WHY does this module exist?

The four functions, derived from one idea

Standardizing: the z-score, derived

Hypothesis testing, from scratch

Worked examples

Common mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Scientific Computing (Python)

Connections