Statistics & Probability for Data Science

1Descriptive vs inferential statistics

Two jobs. Descriptive statistics summarise the data you have. Inferential statistics use a sample to draw conclusions about a larger population you cannot fully measure. Almost all data science is inference: we never have every customer, only some.

Centre and spread, precisely

import numpy as np

sample = np.array([12, 15, 14, 10, 18, 22, 16, 13, 19, 25])

print('Mean    :', sample.mean())
print('Median  :', np.median(sample))
print('Variance:', round(sample.var(ddof=1), 2))    # sample variance
print('Std dev :', round(sample.std(ddof=1), 2))

Mean    : 16.4
Median  : 15.5
Variance: 21.82
Std dev : 4.67

2Probability distributions with SciPy

A distribution describes how likely each outcome is. SciPy's stats module gives every common one a consistent toolkit: pdf/pmf (density), cdf (probability up to a point) and ppf (the inverse — percentiles).

The normal distribution

from scipy import stats

# Daily visitors ~ Normal(mean=500, sd=80)
rv = stats.norm(loc=500, scale=80)

print('P(visitors < 600):', round(rv.cdf(600), 4))
print('95th percentile  :', round(rv.ppf(0.95), 1))
print('P(420 < X < 580) :', round(rv.cdf(580) - rv.cdf(420), 4))

P(visitors < 600): 0.8944
95th percentile  : 631.6
P(420 < X < 580) : 0.6827

That last line is the 68% rule in action: one standard deviation either side of the mean holds ~68.3% of the data.

The binomial distribution (counts of successes)

# Send 10 emails, each opens with probability 0.2
print('P(exactly 3 open):', round(stats.binom.pmf(3, n=10, p=0.2), 4))
print('P(3 or fewer)   :', round(stats.binom.cdf(3, n=10, p=0.2), 4))

P(exactly 3 open): 0.2013
P(3 or fewer)   : 0.8791

3Sampling distributions & the Central Limit Theorem

The single most important idea in inference: even if your data is wildly non-normal, the distribution of the sample mean becomes approximately normal as the sample grows. That is the Central Limit Theorem (CLT), and it is why so much of statistics works.

See it happen

import numpy as np

rng = np.random.default_rng(0)
population = rng.exponential(scale=10, size=100_000)   # very skewed

# Take 1000 samples of size 50 and record each mean
means = [rng.choice(population, 50).mean() for _ in range(1000)]

print('Population mean      :', round(population.mean(), 2))
print('Mean of sample means :', round(np.mean(means), 2))
print('Std of sample means  :', round(np.std(means), 2))
print('Predicted SE (s/√n):', round(population.std() / np.sqrt(50), 2))

Population mean      : 9.98
Mean of sample means : 9.97
Std of sample means  : 1.40
Predicted SE (s/√n): 1.41

4Confidence intervals & estimation

A point estimate (“mean tip = $3.00”) is incomplete without a measure of uncertainty. A confidence interval (CI) gives a plausible range for the true value.

Build a 95% CI for a mean

import numpy as np
from scipy import stats
import seaborn as sns

tips = sns.load_dataset('tips')
x = tips['tip']

mean = x.mean()
sem  = stats.sem(x)                     # standard error of the mean
ci   = stats.t.interval(0.95, len(x) - 1, loc=mean, scale=sem)

print(f'Mean tip : {mean:.2f}')
print(f'95% CI   : ({ci[0]:.2f}, {ci[1]:.2f})')

Mean tip : 3.00
95% CI   : (2.82, 3.17)

Bootstrap: a CI with no formula

# Resample with replacement 10,000 times, recompute the mean each time
rng = np.random.default_rng(1)
boot = [rng.choice(x, len(x), replace=True).mean() for _ in range(10_000)]
lo, hi = np.percentile(boot, [2.5, 97.5])
print(f'Bootstrap 95% CI: ({lo:.2f}, {hi:.2f})')

Bootstrap 95% CI: (2.83, 3.18)

The bootstrap reaches almost the same interval with zero distribution assumptions — just resampling. It is the analyst's Swiss-army knife when formulas get hard.

5Hypothesis testing & p-values

A hypothesis test asks: could this difference be just noise? You assume “no real effect” (the null hypothesis), then compute how surprising your data would be if that were true. That surprise is the p-value.

A two-sample t-test

from scipy import stats

dinner = tips[tips['time'] == 'Dinner']['total_bill']
lunch  = tips[tips['time'] == 'Lunch']['total_bill']

t, p = stats.ttest_ind(dinner, lunch, equal_var=False)
print(f'Dinner mean: {dinner.mean():.2f}')
print(f'Lunch mean : {lunch.mean():.2f}')
print(f't = {t:.3f}, p = {p:.4f}')

Dinner mean: 20.80
Lunch mean : 17.17
t = 2.898, p = 0.0043

p = 0.004 is well below 0.05, so we reject the null: dinner bills are significantly higher than lunch bills.

6A/B testing & a glimpse of Bayes

A/B testing is hypothesis testing applied to product decisions: show variant A to one group, B to another, and test whether the difference in outcomes is real.

Compare two conversion rates

import numpy as np
from statsmodels.stats.proportion import proportions_ztest

# A: 120 / 1000 converted   B: 150 / 1000 converted
conversions = np.array([120, 150])
visitors    = np.array([1000, 1000])

z, p = proportions_ztest(conversions, visitors)
print(f'A rate: {120/1000:.1%}   B rate: {150/1000:.1%}')
print(f'z = {z:.3f}, p = {p:.4f}')

A rate: 12.0%   B rate: 15.0%
z = -1.963, p = 0.0497

p = 0.0497 is just under 0.05 — technically significant, but barely. A careful analyst would want a larger sample before betting the roadmap on a 3-point lift this marginal.

Designing a trustworthy test

• Power & sample size: decide the smallest lift worth detecting and compute the sample size before you start.
• Do not peek: repeatedly checking and stopping when p < 0.05 inflates false positives. Fix the duration up front.
• Randomise properly and check the groups are balanced.
• One metric: testing twenty metrics guarantees a “significant” fluke (multiple-comparisons problem).

The Bayesian alternative

Frequentist tests ask “how surprising is the data under the null?” Bayesian methods ask the question businesses actually want: “given the data, what is the probability B is better than A?” You start with a prior belief and update it with evidence to get a posterior. Both views are valuable; the Bayesian framing is often easier to act on.