Statistics for Data Science Interviews

Statistics provides the framework for making decisions from data. This section covers the concepts and methods commonly tested in data science interviews.

Descriptive Statistics

Measures of Central Tendency

Measure	Description	Properties
Mean	Sum divided by count	Sensitive to outliers
Median	Middle value when sorted	Robust to outliers
Mode	Most frequent value	Applicable to categorical data

Selection guidance: Use median for skewed distributions (e.g., salary data). Use mean for symmetric distributions.

Measures of Spread

Measure	Description	Properties
Variance	Average squared deviation from mean	Units are squared
Standard deviation	Square root of variance	Same units as data
Interquartile range (IQR)	Distance between 25th and 75th percentile	Robust to outliers

Hypothesis Testing

Framework

1. State hypotheses

   • H0 (null): No effect or no difference
   • H1 (alternative): Effect exists

2. Select significance level (alpha)

   • Standard value: 0.05 (5% false positive rate)

3. Collect data and compute test statistic

4. Calculate p-value

   • Probability of observing data this extreme if H0 is true

5. Decision

   • p < alpha: Reject H0
   • p >= alpha: Fail to reject H0

P-Value Definition

The p-value is the probability of observing data at least as extreme as the observed data, assuming the null hypothesis is true.

P-value is NOT:

• Probability that the null hypothesis is true
• Probability of making an error
• Probability that the effect is real

Example interpretation: p = 0.03 means "If there were truly no effect, data this extreme would occur only 3% of the time."

Error Types

	H0 True (No Effect)	H0 False (Effect Exists)
Reject H0	Type I error (alpha)	Correct decision
Fail to reject H0	Correct decision	Type II error (beta)

Error Type	Description	Controlled By
Type I (false positive)	Detecting effect that does not exist	alpha (significance level)
Type II (false negative)	Missing effect that exists	Power (1 - beta)

Trade-off: Decreasing alpha reduces false positives but increases false negatives. Both cannot be minimized simultaneously.

Common Statistical Tests

Test	Application
One-sample t-test	Compare sample mean to known value
Two-sample t-test	Compare means of two independent groups
Paired t-test	Compare means of paired observations (before/after)
Chi-square test	Test independence of categorical variables
ANOVA	Compare means across 3+ groups

Confidence Intervals

A confidence interval provides a range of plausible values for a parameter.

Formula (95% CI for mean)

sample mean +/- 1.96 x (standard deviation / square root of n)

Interpretation

"95% confident" means: If the experiment were repeated 100 times, approximately 95 of the resulting confidence intervals would contain the true parameter value.

Note: This does NOT mean "95% probability that the true value is in this specific interval." The true value is fixed; it either is or is not in the interval.

Factors Affecting Width

Factor	Effect on Width
Sample size (n)	Larger n = narrower interval
Data variability	More spread = wider interval
Confidence level	Higher confidence = wider interval

Effect Size

Statistical significance does not indicate practical importance. Effect size measures the magnitude of an effect.

Cohen's d

d = (mean1 - mean2) / pooled standard deviation

d Value	Interpretation
0.2	Small effect
0.5	Medium effect
0.8	Large effect

Example: A treatment may produce a statistically significant (p < 0.001) reduction of 0.5 mmHg in blood pressure. The effect is statistically significant but may not be clinically meaningful.

Correlation vs Causation

Concept	Definition
Correlation	Two variables move together
Causation	One variable directly influences another

Reasons correlation does not imply causation:

• Confounding variable affects both
• Reverse causality
• Coincidental correlation

Requirements for establishing causation:

1. Correlation exists (necessary but not sufficient)
2. Temporal order (cause precedes effect)
3. No confounders (or adjusted for confounders)
4. Ideally, randomized experiment

Regression

Linear regression models the relationship between variables:

Y = intercept + slope x X + error term

Component	Interpretation
beta0 (intercept)	Predicted Y when X = 0
beta1 (slope)	Change in Y per unit increase in X
R-squared	Proportion of variance explained (0 to 1)

Assumptions

Assumption	Verification Method
Linear relationship	Scatter plot
Independence	Study design
Homoscedasticity	Residual plot (constant variance)
Normality of residuals	Q-Q plot

When assumptions are violated, standard errors and p-values may be unreliable.

Regularization

Method	Penalty	Use Case
Ridge (L2)	Sum of squared coefficients	All features potentially relevant
Lasso (L1)	Sum of absolute coefficients	Feature selection desired
Elastic Net	Combination of L1 and L2	Balance between ridge and lasso

Multiple Testing Problem

Testing multiple hypotheses inflates the overall false positive rate.

Example: With 20 tests at alpha = 0.05, the probability of at least one false positive is approximately 64%.

Corrections

Method	Approach	Properties
Bonferroni	alpha / number of tests	Conservative (reduces power)
False Discovery Rate (FDR)	Controls expected proportion of false discoveries	Less conservative

Common Interview Topics

Explain p-value to non-technical audience

"The p-value indicates how surprising the data would be if there were no real effect. A small p-value suggests the data would be unlikely to occur by chance alone, so we conclude there is probably a real effect."

Type I vs Type II error distinction

Type I error is a false alarm: concluding an effect exists when it does not. Type II error is a miss: failing to detect an effect that exists.

Application: In medical testing, Type I may lead to unnecessary treatment of healthy individuals. Type II may result in missed diagnoses.

When to use non-parametric tests

Non-parametric tests are appropriate when normality assumptions are violated, particularly with small samples or highly skewed data. The trade-off is reduced statistical power, requiring larger samples to detect the same effect size.

A/B test shows p = 0.06

At alpha = 0.05, the result is not statistically significant, but it is close to the threshold. Relevant considerations:

• Effect size magnitude
• Confidence interval range
• Sample size adequacy (power analysis)
• Option to extend test duration or acknowledge uncertainty

Common Pitfalls

Pitfall	Description
P-hacking	Running multiple analyses until finding significant result
Simpson's Paradox	Aggregate trend reverses when data is segmented
Survivorship Bias	Analyzing only successful cases, missing failures
HARKing	Hypothesizing After Results are Known

Quick Reference

Scenario	Approach
Compare two means	t-test (Mann-Whitney if non-normal)
Compare 3+ means	ANOVA
Relationship between variables	Correlation / regression
Categorical variable independence	Chi-square test
Before/after same subjects	Paired t-test
Feature selection with many variables	Lasso
Multiple hypothesis tests	Bonferroni or FDR correction