Statistics for Data Analytics

1Probability & key distributions

A distribution describes how likely each value of a variable is. Three show up constantly in analytics.

The Normal distribution follows the 68–95–99.7 rule: about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

from scipy import stats

# In a standard Normal: P(value < 1.5)
print('P(Z < 1.5) =', round(stats.norm.cdf(1.5), 4))

# The 95th percentile (one-sided z critical value)
print('95th percentile z =', round(stats.norm.ppf(0.95), 3))

P(Z < 1.5) = 0.9332
95th percentile z = 1.645

2Sampling & the Central Limit Theorem

You almost never measure a whole population — you study a sample and infer. The risk is sampling error: a different sample would give a slightly different answer.

Standard error: how much a sample mean wobbles

import numpy as np

sample = np.array([1200, 980, 1450, 760, 2100, 1180, 1320, 990])

mean = sample.mean()
std  = sample.std(ddof=1)               # sample standard deviation
se   = std / np.sqrt(len(sample))       # standard error of the mean

print('Mean:', round(mean, 1), ' Standard error:', round(se, 1))

Mean: 1247.5  Standard error: 150.2

3Hypothesis testing: the framework

A hypothesis test answers: “Could this result be just chance?” The logic is always the same.

1. State a null hypothesis H₀ (no effect / no difference) and an alternative H₁.
2. Pick a significance level α (usually 0.05).
3. Compute a test statistic and its p-value.
4. If p < α, reject H₀ — the effect is “statistically significant”.

from scipy import stats

# Did region A and region B have different average order values?
region_a = [1200, 1450, 1320, 1180, 1390, 1275]
region_b = [ 980, 1020,  940, 1100,  995, 1010]

t_stat, p_value = stats.ttest_ind(region_a, region_b)
print('t =', round(t_stat, 2), ' p =', round(p_value, 4))

t = 6.13  p = 0.0001

p = 0.0001 is far below 0.05, so we reject H₀: the two regions really do differ.

Which test do I use?

4Confidence intervals & p-values

A single number hides uncertainty. A confidence interval (CI) reports a range you can trust.

import numpy as np
from scipy import stats

data = np.array([1200, 980, 1450, 760, 2100, 1180, 1320, 990])
mean = data.mean()
se   = stats.sem(data)                       # standard error

ci = stats.t.interval(0.95, len(data)-1, loc=mean, scale=se)
print('Mean:', round(mean, 1))
print('95% CI:', tuple(round(x, 1) for x in ci))

Mean: 1247.5
95% CI: (892.3, 1602.7)

Read it as: “our best estimate is 1247.5, and we are 95% confident the true average lies between 892 and 1603.” A wider CI means more uncertainty (small sample / high variability).

5A/B testing for business decisions

A/B testing is the analyst's superpower: run a controlled experiment to prove a change causes an improvement.

The recipe

1. Change one thing (the variant B) vs the control (A).
2. Randomly assign users to A or B to remove bias.
3. Decide the sample size in advance (a power calculation).
4. Run to the end, then test the difference.

from statsmodels.stats.proportion import proportions_ztest

# A: 120 conversions / 2400 visitors   B: 168 / 2450
conversions = [120, 168]
visitors    = [2400, 2450]

stat, p = proportions_ztest(conversions, visitors)
rate_a = conversions[0] / visitors[0]
rate_b = conversions[1] / visitors[1]
print('A:', round(rate_a*100, 2), '%   B:', round(rate_b*100, 2), '%')
print('p-value:', round(p, 4))

A: 5.0 %   B: 6.86 %
p-value: 0.0071

B's conversion (6.86%) beats A's (5.0%) and p = 0.007 < 0.05 — a statistically significant lift. Ship B.

6Non-parametric tests, errors & ethics

When data is skewed or not normal, the usual tests can mislead. Non-parametric tests make fewer assumptions.

Two ways to be wrong