What is Normal Distribution?
The normal distribution (bell curve) is a symmetric probability distribution where most data cluster around the mean. It's used to model heights, test scores, measurement errors and much more.
A normal distribution is a symmetric, bell-shaped curve defined by its mean (μ) and standard deviation (σ), where about 68% of data fall within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.
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Step-by-step worked examples
On an IQ test with mean 100 and standard deviation 15, find the z-score for a score of 130.
z = (x − μ)/σ = (130 − 100)/15 z = 30/15 = 2.0 → 130 is 2 standard deviations above the mean
On a test with mean 70 and standard deviation 8, find the z-score for a score of 62.
z = (62 − 70)/8 = −8/8 = −1.0 → 62 is 1 standard deviation below the mean
Heights have mean 170 cm and standard deviation 10 cm. What % of people are between 160 and 180 cm?
160 = μ − 1σ (170−10), 180 = μ + 1σ (170+10) The range is μ ± 1σ → about 68% of the data (empirical rule)
Flashcards
Quick quiz
Q1.In a normal distribution with mean 50 and SD 5, what % of data falls between 45 and 55?
Q2.What is the z-score of a value equal to the mean?
Q3.Which best describes the shape of a normal distribution?
Q4.With mean 80 and SD 10, what is the z-score of 100?
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Common mistakes
Assuming all data is normally distributed. — Correct: Many real distributions are skewed — always check before applying normal-curve rules.
Confusing the z-score with the raw value. — Correct: The z-score is standardized (unitless); it tells you position in SDs, not the actual value.
Thinking 95% of data lies within 1 SD. — Correct: 68% lies within 1σ; 95% is within 2σ; 99.7% is within 3σ.
Ignoring that mean = median = mode in a normal distribution. — Correct: In a perfectly normal distribution, the three measures of center coincide at the peak.
FAQ
What is normal distribution?
The normal distribution is a symmetric, bell-shaped probability distribution where data cluster around the mean.
What is the formula for normal distribution (z-score)?
z = (x−μ)/σ — it gives how many standard deviations a value is from the mean.
What are examples of normal distribution?
Heights, IQ scores, exam grades and measurement errors are common approximate examples of normal distribution.
How do you calculate a value's position in a normal distribution?
Use the z-score formula: subtract the mean from the value, divide by the standard deviation, then interpret it with the 68-95-99.7 rule.