Statistics · medium

Explain the Central Limit Theorem and why it matters.

Asked in Data Scientist interviews, in the Statistics round.

Short answer

Sample means approach normality as n grows, enabling inference regardless of population shape.

How to answer it

The average of many independent draws is approximately normally distributed, whatever the shape of the thing being drawn, once the sample is large enough. Its mean is the population mean and its standard deviation is sigma over root n.

Why it matters is the answer they want: it is the reason you can put a confidence interval or a p-value on a mean without knowing the population's distribution. Revenue per user is wildly skewed, but the average revenue per user across 10,000 users is close to normal, so a t-test on that average is valid.

import numpy as np
rng = np.random.default_rng(0)
pop = rng.exponential(scale=50, size=1_000_000)       # heavily skewed
means = [rng.choice(pop, 200).mean() for _ in range(5_000)]
# means is bell-shaped, centred near 50, sd near 50/sqrt(200)

Two caveats show depth. "Large enough" depends on skew: 30 is the textbook number, but a metric with a long tail, like revenue with a few whales, may need thousands before the average behaves. And the CLT is about the mean; it says nothing about the median or the 95th percentile, which need bootstrap or rank methods.

• What they are checking: that you connect the theorem to why the standard A/B test is valid on skewed metrics.
• Common mistake: stating that "the data becomes normal". The data stays skewed; it is the sample mean that becomes normal.

Related questions

• When would you use a t-test vs a chi-square test?
• What is Simpson's paradox?
• Explain p-value in plain language.
• How do you determine sample size for an A/B test?
• What is the difference between Type I and Type II error?
• What is p-hacking and how do you avoid it?

Practice this for real

• Check your resume against a Data Scientist job description — free, no account needed.
• Every data interview question, by role and topic
• Scored AI mock interviews — two free, no card