Introduction to Normal Distribution

Learn about the bell curve, the most important probability distribution in statistics.

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Definition

The normal distribution (also called the Gaussian distribution or bell curve) is a symmetric, bell-shaped probability distribution that describes how many natural phenomena are spread around an average value.

Key Properties

  1. 1.Symmetric: The distribution is perfectly symmetric around the mean (μ\mu)
  2. 2.Bell-shaped: Most values cluster near the center, with fewer values in the tails
  3. 3.Mean = Median = Mode: All three measures of center are equal
  4. 4.Defined by two parameters:
  • Mean (μ\mu): The center of the distribution
  • Standard deviation (σ\sigma): How spread out the data is

The Standard Normal Distribution

When μ=0\mu = 0 and σ=1\sigma = 1, we have the standard normal distribution. Any value XX can be converted to a z-score:
Z=X−μσZ = \frac{X - \mu}{\sigma}
The z-score tells us how many standard deviations XX is from the mean. The z-scores of a normal distribution follow the standard normal distribution.

Try it now

What shape does the normal distribution have?

Worked Examples

Adult male heights in a country are normally distributed with mean μ=175\mu = 175 cm and standard deviation σ=7\sigma = 7 cm. What percentage of men are between 168 cm and 182 cm tall?
1

Identify the boundaries in terms of standard deviations

168=175−7=μ−1σ168 = 175 - 7 = \mu - 1\sigma 182=175+7=μ+1σ182 = 175 + 7 = \mu + 1\sigma → We are looking at ±1\pm 1 standard deviation from the mean
2

Apply the 68-95-99.7 rule

Within 1 standard deviation of the mean: 68% Within 2 standard deviations: 95% Within 3 standard deviations: 99.7% → 68% of data falls within 1 standard deviation
3

State the answer

Since 168 cm and 182 cm are exactly 1 standard deviation from the mean → Approximately 68% of men are between 168 cm and 182 cm tall

Common Mistakes

Confusing standard deviation with variance

Why it's wrong: Variance (σ2\sigma^2) is the square of standard deviation. Using variance instead of standard deviation in z-score calculations gives wrong results.

Correct: Always use standard deviation (σ\sigma) in the z-score formula: Z=X−μσZ = \frac{X - \mu}{\sigma}

Forgetting that the normal distribution is symmetric

Why it's wrong: Students sometimes calculate probabilities for only one tail and forget to account for symmetry.

Correct: The probability of being more than 2 standard deviations above the mean (2.5%) equals the probability of being more than 2 standard deviations below the mean (2.5%).

Applying the 68-95-99.7 rule to non-normal distributions

Why it's wrong: This rule only applies to normal distributions. Other distributions may have very different percentages.

Correct: First verify that your data follows a normal distribution before applying these rules.

Thinking a negative z-score is bad

Why it's wrong: A negative z-score simply means the value is below the mean, not that it is wrong or invalid.

Correct: Z-scores can be positive (above mean) or negative (below mean). Context determines whether higher or lower is desirable.

Interactive Visual

Normal Distribution

Men's heights: μ = 175 cm, σ = 7 cm

34.1%34.1%13.6%13.6%2.1%2.1%154μ − 3σ161μ − 2σ168μ − σ175μ182μ + σ189μ + 2σ196μ + 3σcm

μ = 175, σ = 7

Shade within

P(168 ≤ X ≤ 182) ≈ 0.6827 = 68.3%z = -1, z = 1

68-95-99.7 rule: about 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ.

Normal Distribution

Test scores: μ = 500, σ = 100

200μ − 3σ300μ − 2σ400μ − σ500μ600μ + σ700μ + 2σ800μ + 3σ

μ = 500, σ = 100

Shade within

P(X ≤ 650) ≈ 0.9332 = 93.3%z = 1.5

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Practice Problems

15 problems
Problem 1 of 15
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What shape does the normal distribution have?

Why It Matters

The normal distribution appears everywhere in the real world:
  • Human characteristics: Heights, weights, blood pressure, IQ scores all follow approximately normal distributions
  • Measurement errors: Random errors in scientific measurements tend to be normally distributed
  • Test scores: Scores on IQ tests and large standardized tests (like the SAT) are scaled so that they are approximately normally distributed
  • Quality control: Manufacturing processes use normal distributions to monitor product quality
  • Central Limit Theorem: Sample means from ANY distribution become approximately normal with large samples
Understanding the normal distribution is essential for making predictions, detecting outliers, and understanding statistical inference.

Real World Applications

Quality Control in Manufacturing

Factories use normal distributions to monitor product quality and set acceptable tolerance ranges.

Example:

A bolt factory produces bolts with mean diameter μ=10.00\mu = 10.00 mm and σ=0.02\sigma = 0.02 mm. Bolts outside ±3σ\pm 3\sigma (9.94 mm to 10.06 mm) are rejected as defective.

1Try It Yourself

A factory produces bottles that should contain 500 mL of liquid. The filling process is normally distributed with μ=500\mu = 500 mL and σ=5\sigma = 5 mL.

What range contains 95% of all bottles?

Step 1: Choose the right answer

Use the 68-95-99.7 rule with ±2σ\pm 2\sigma:

Medical Testing and Diagnosis

Doctors use normal distributions to interpret medical test results and determine what is considered healthy or abnormal.

Example:

Blood pressure readings are approximately normally distributed. A reading more than 2 standard deviations above the population mean may indicate hypertension.

2Try It Yourself

Cholesterol levels in healthy adults are normally distributed with μ=200\mu = 200 mg/dL and σ=35\sigma = 35 mg/dL. A level above μ+1.5σ\mu + 1.5\sigma is considered elevated.

At what cholesterol level does a result become elevated?

Step 1: Write the mathematical expression

Calculate μ+1.5σ\mu + 1.5\sigma:

Key Takeaways

  • 1The normal distribution is a symmetric, bell-shaped curve defined by mean (μ\mu) and standard deviation (σ\sigma)
  • 2The 68-95-99.7 rule: 68% of data falls within ±1σ\pm 1\sigma, 95% within ±2σ\pm 2\sigma, 99.7% within ±3σ\pm 3\sigma
  • 3Z-score formula: Z=X−μσZ = \frac{X - \mu}{\sigma} tells how many standard deviations a value is from the mean
  • 4The standard normal distribution has μ=0\mu = 0 and σ=1\sigma = 1
  • 5Normal distributions appear in heights, test scores, measurement errors, and many natural phenomena

Frequently Asked Questions

The shape of the normal distribution graph resembles a bell: wide in the middle where most values occur, and tapering off symmetrically at both ends where extreme values are rare.
The shape of the normal distribution graph resembles a bell: wide in the middle where most values occur, and tapering off symmetrically at both ends where extreme values are rare.
Yes! A negative z-score means the value is below the mean. For example, z = -1.5 means the value is 1.5 standard deviations below average. This is not bad in itself: it depends on the context.
Any normal distribution can have any mean and standard deviation. The standard normal distribution specifically has mean = 0 and standard deviation = 1. We convert to standard normal using z-scores.
No. Income distributions are typically skewed (not symmetric). Counts and proportions follow different distributions. Always check if normality is a reasonable assumption for your data.

Glossary

Normal distribution
A symmetric, bell-shaped probability distribution defined by its mean and standard deviation
Mean (μ)
The center of the normal distribution; the average value
Standard deviation (σ)
A measure of spread; indicates how far values typically deviate from the mean
Z-score
The number of standard deviations a value is from the mean: Z=X−μσZ = \frac{X - \mu}{\sigma}
Standard normal distribution
A normal distribution with mean 0 and standard deviation 1
68-95-99.7 rule
In a normal distribution, approximately 68% of data is within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ
Bell curve
Another name for the normal distribution, due to its bell-like shape
Percentile
The percentage of values in a distribution that fall below a given value

Formula Card

Z-score

Z=X−μσZ = \frac{X - \mu}{\sigma}

Convert a raw value to standard units

Raw value from z-score

X=μ+Z⋅σX = \mu + Z \cdot \sigma

Convert a z-score back to the original units

68-95-99.7 Rule

μ±1σ\mu \pm 1\sigma: 68%, μ±2σ\mu \pm 2\sigma: 95%, μ±3σ\mu \pm 3\sigma: 99.7%

Percentages within standard deviations

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