Question 1

What is a normal distribution?

Accepted Answer

A normal distribution is a symmetric, bell-shaped probability distribution defined by its mean (μ) and standard deviation (σ). The mean determines the centre, and σ determines the spread. It occurs naturally in many phenomena: heights, test scores, measurement errors, and many natural processes. It is called "normal" because it was once considered the standard pattern of natural variation.

Question 2

What is the 68-95-99.7 empirical rule?

Accepted Answer

For any normal distribution: 68.27% of values fall within 1σ of the mean (μ±σ); 95.45% fall within 2σ (μ±2σ); 99.73% fall within 3σ (μ±3σ). These are the bounds most used in quality control (Six Sigma), hypothesis testing, and confidence intervals. Values beyond 3σ are extremely rare — only 0.27% of the data.

Question 3

How do I calculate the probability between two values?

Accepted Answer

P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ). Convert both bounds to z-scores, then subtract the CDF values. Example: heights ~ N(170, 10). P(160 < X < 180) = Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827 ≈ 68.3%.

Question 4

What is the difference between normal and standard normal distribution?

Accepted Answer

The standard normal distribution is a special case where μ = 0 and σ = 1. Any normal distribution can be converted to standard normal by computing Z = (X − μ)/σ. Z-tables (also called standard normal tables) give probabilities for the standard normal, which is why the z-score conversion is always the first step.

Question 5

When does normal distribution not apply?

Accepted Answer

Normal distribution does not apply when: data is strongly skewed (income, house prices — use log-normal); data has natural bounds (proportions 0–1 — use binomial/beta); events are counts (Poisson distribution); sample size is very small without prior knowledge of normality. Always check normality with a histogram or Q-Q plot before assuming normal distribution.

Question 6

What is the Central Limit Theorem?

Accepted Answer

The Central Limit Theorem (CLT) states that the distribution of sample means approaches a normal distribution as sample size increases — regardless of the original population distribution. This is why normal distribution is so powerful: it justifies using normal-based tests (t-tests, z-tests) even on non-normal populations when n ≥ 30. The sample mean distribution has μ_x̄ = μ and σ_x̄ = σ/√n.

Range	% of data	Common use
μ ± 1σ	68.27%	Normal variation range
μ ± 2σ	95.45%	Confidence intervals (95%)
μ ± 3σ	99.73%	Six Sigma quality control
μ ± 1.96σ	95.00%	Exact 95% CI (z = 1.96)
μ ± 2.576σ	99.00%	Exact 99% CI (z = 2.576)

Situation	Use Normal?	Better Alternative
n ≥ 30, population σ unknown	Use t-distribution	T-distribution — accounts for uncertainty in estimating σ
Symmetric, continuous, unimodal data	✓ Yes	Normal distribution is appropriate
Right-skewed data (income, prices, wait times)	No	Log-normal distribution
Counts of events per interval	No	Poisson distribution
Proportions between 0 and 1	No	Beta or binomial distribution
Time until an event (survival data)	No	Exponential or Weibull distribution

Normal Distribution Calculator

Formula

The 68-95-99.7 Empirical Rule

Real-World Example: Heights

Normal Distribution vs Alternatives — Which Model Fits?

Common Mistakes with Normal Distribution

Case Study: Delivery Time Modelling at an E-Commerce Company

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