Statistical Analysis & Probability

The Statistical Toolkit

Every data science decision involves uncertainty. Statistics gives you tools to quantify that uncertainty and make defensible decisions.

Descriptive Statistics

Before testing hypotheses, summarize your data. These are the numbers that describe the shape and spread of your dataset.

import numpy as np
import pandas as pd

# Central tendency
mean = df['salary'].mean()       # average (sensitive to outliers)
median = df['salary'].median()   # middle value (robust to outliers)
mode = df['salary'].mode()[0]    # most common value

# Spread
std = df['salary'].std()         # standard deviation
var = df['salary'].var()         # variance (std²)
iqr = df['salary'].quantile(0.75) - df['salary'].quantile(0.25)  # IQR

# Shape
skew = df['salary'].skew()      # >0 = right-skewed, <0 = left-skewed
kurt = df['salary'].kurtosis()   # >0 = heavy tails, <0 = light tails

# Quick summary
print(df['salary'].describe())

Probability Distributions

A distribution describes the possible values a variable can take and how likely each is. Knowing the right distribution lets you model uncertainty correctly.

Normal (Gaussian) Distribution

The bell curve. Defined by mean (μ) and standard deviation (σ). Most natural phenomena follow it (heights, test scores, measurement errors).

from scipy import stats
import numpy as np

# Generate normally distributed data
data = np.random.normal(loc=100, scale=15, size=1000)  # μ=100, σ=15

# 68-95-99.7 rule
within_1_std = np.mean(np.abs(data - 100) < 15)   # ≈ 68%
within_2_std = np.mean(np.abs(data - 100) < 30)   # ≈ 95%
within_3_std = np.mean(np.abs(data - 100) < 45)   # ≈ 99.7%

# Probability of a value > 130
p = 1 - stats.norm.cdf(130, loc=100, scale=15)  # ≈ 0.023 (2.3%)
print(f"P(X > 130) = {p:.4f}")

Other Key Distributions

Hypothesis Testing

Hypothesis testing answers: "Is this result real or just random noise?" You set up two competing hypotheses, collect data, and use a test statistic to decide.

The 5-Step Process

1. State hypotheses — H₀ (null: no effect) vs H_a (alternative: there is an effect)
2. Choose significance level — Usually α = 0.05 (5% chance of false positive)
3. Select the right test — Based on data type and question
4. Calculate p-value — Probability of seeing this result if H₀ is true
5. Decide — If p < α, reject H₀. If p ≥ α, fail to reject H₀

from scipy import stats

# t-test: Is the average salary different from $75,000?
sample_salaries = df['salary'].dropna()
t_stat, p_value = stats.ttest_1samp(sample_salaries, 75000)
print(f"t-statistic: {t_stat:.3f}, p-value: {p_value:.4f}")

if p_value < 0.05:
    print("Reject H0: Average salary IS significantly different from $75K")
else:
    print("Fail to reject H0: No significant difference from $75K")

Choosing the Right Test

Confidence Intervals

A confidence interval gives you a range of plausible values for a parameter, not just a single estimate.

from scipy import stats
import numpy as np

sample = df['conversion_rate'].dropna()
mean = sample.mean()
se = stats.sem(sample)  # standard error of the mean

# 95% confidence interval
ci_95 = stats.t.interval(
    confidence=0.95,
    df=len(sample) - 1,
    loc=mean,
    scale=se
)
print(f"Mean: {mean:.4f}")
print(f"95% CI: ({ci_95[0]:.4f}, {ci_95[1]:.4f})")
# "We're 95% confident the true conversion rate is between X and Y"

Correlation

Correlation measures the strength and direction of a linear relationship between two variables. It ranges from -1 (perfect negative) to +1 (perfect positive).

import pandas as pd

# Pearson correlation (linear relationships, assumes normality)
pearson_corr = df['study_hours'].corr(df['exam_score'])

# Spearman correlation (monotonic relationships, robust to outliers)
spearman_corr = df['study_hours'].corr(df['exam_score'], method='spearman')

# Full correlation matrix
corr_matrix = df.select_dtypes(include='number').corr()

# Statistical significance of correlation
from scipy import stats
r, p_value = stats.pearsonr(df['study_hours'], df['exam_score'])
print(f"r = {r:.3f}, p = {p_value:.4f}")

A/B Testing

A/B testing is hypothesis testing applied to real-world experiments. Show group A the current version, group B the new version, and measure if B is better.

from scipy import stats
import numpy as np

# Example: Testing a new checkout button
# Group A (control): 1000 users, 50 conversions (5.0%)
# Group B (variant): 1000 users, 65 conversions (6.5%)

n_a, conv_a = 1000, 50
n_b, conv_b = 1000, 65

rate_a = conv_a / n_a  # 0.050
rate_b = conv_b / n_b  # 0.065

# Two-proportion z-test
from statsmodels.stats.proportion import proportions_ztest

count = np.array([conv_b, conv_a])
nobs = np.array([n_b, n_a])

z_stat, p_value = proportions_ztest(count, nobs, alternative='larger')
print(f"Z-statistic: {z_stat:.3f}")
print(f"P-value: {p_value:.4f}")

# Effect size (relative lift)
lift = (rate_b - rate_a) / rate_a * 100
print(f"Lift: {lift:.1f}%")

# Sample size calculator (for planning)
from statsmodels.stats.power import NormalIndPower
analysis = NormalIndPower()
sample_size = analysis.solve_power(
    effect_size=0.1,  # expected small effect
    alpha=0.05,
    power=0.8,        # 80% chance of detecting the effect
    ratio=1            # equal group sizes
)
print(f"Required sample per group: {int(np.ceil(sample_size))}")

Interview Questions