Normal Distribution & Statistics: Complete Guide

What Is the Normal Distribution?

The normal distribution, also called the Gaussian distribution or bell curve, is the most important probability distribution in statistics. It describes how data naturally clusters around a central value, with fewer observations appearing as you move further from the center.

Named after mathematician Carl Friedrich Gauss, this distribution appears everywhere in nature and human measurements. Heights, weights, IQ scores, blood pressure readings, measurement errors, and countless other phenomena follow this pattern.

Why Is It So Common?

When many independent random factors influence an outcome, the result tends to be normally distributed. This is why the normal distribution appears so frequently in real-world data—most measurements are affected by numerous small, independent variations.

The mathematical formula for the normal distribution's probability density function is:

f(x) = (1 / (σ√(2π))) × e^{-((x-μ)²/(2σ²))}

Where μ = mean and σ = standard deviation

Don't worry if that formula looks intimidating—you rarely need to calculate it by hand. What matters is understanding what the distribution means and how to use it.

Properties of the Bell Curve

The normal distribution has several distinctive properties that make it recognizable and mathematically useful:

1. Symmetric Shape

The curve is perfectly symmetric around the mean. The left half is a mirror image of the right half. This means 50% of data falls below the mean and 50% above.

2. Mean = Median = Mode

In a perfect normal distribution, all three measures of central tendency are equal. The most common value, the middle value, and the average are all the same.

3. Asymptotic Tails

The curve approaches but never touches the x-axis. Theoretically, values extend infinitely in both directions, though extreme values become increasingly rare.

4. Total Area = 1

The total area under the curve equals 1 (or 100%). This allows us to interpret areas as probabilities—a fundamental concept for statistical analysis.

5. Defined by Two Parameters

Every normal distribution is completely determined by its mean (μ) and standard deviation (σ). Change either value, and you get a different bell curve.

Mean and Standard Deviation Explained

The Mean (μ)

The mean is the center of the distribution—the peak of the bell curve. It determines where the distribution is located on the number line. Shifting the mean moves the entire curve left or right without changing its shape.

A distribution with μ = 100 is centered at 100
A distribution with μ = 500 is centered at 500
The mean represents the expected value or "typical" outcome

The Standard Deviation (σ)

The standard deviation measures the spread or dispersion of the data. A small standard deviation means data points cluster tightly around the mean (narrow, tall bell curve). A large standard deviation means data is spread out (wide, flat bell curve).

Small Standard Deviation

σ = 5 for test scores means most scores are within 5 points of the average. Data is consistent and predictable.

Large Standard Deviation

σ = 20 for test scores means scores vary widely from the average. Data is more spread out and variable.

Calculating Standard Deviation

For a population, standard deviation is calculated as:

Find the mean of all values
Subtract the mean from each value (these are deviations)
Square each deviation
Find the average of the squared deviations (variance)
Take the square root of the variance

The Empirical Rule (68-95-99.7)

The empirical rule, also known as the 68-95-99.7 rule, tells you what percentage of data falls within 1, 2, or 3 standard deviations of the mean in a normal distribution:

68%

Within 1 Standard Deviation

About 68% of data falls within μ ± 1σ

95%

Within 2 Standard Deviations

About 95% of data falls within μ ± 2σ

99.7%

Within 3 Standard Deviations

About 99.7% of data falls within μ ± 3σ

Example: Test Scores

If test scores are normally distributed with μ = 75 and σ = 10:

68% of students score between 65 and 85 (75 ± 10)
95% of students score between 55 and 95 (75 ± 20)
99.7% of students score between 45 and 105 (75 ± 30)
Only 0.3% score below 45 or above 105

Understanding Z-Scores

A z-score (also called a standard score) tells you how many standard deviations a value is from the mean. It standardizes different normal distributions to a common scale, making comparisons possible.

z = (x - μ) / σ

Where x is the raw score, μ is the mean, and σ is the standard deviation

Interpreting Z-Scores

Z-Score	Interpretation	Percentile (approx)
-3.0	3 SDs below mean (very low)	0.13%
-2.0	2 SDs below mean (low)	2.28%
-1.0	1 SD below mean (below average)	15.87%
0	Exactly at the mean	50%
+1.0	1 SD above mean (above average)	84.13%
+2.0	2 SDs above mean (high)	97.72%
+3.0	3 SDs above mean (very high)	99.87%

Example: Comparing Scores from Different Tests

Alice scored 85 on Test A (μ = 75, σ = 5) and Bob scored 92 on Test B (μ = 80, σ = 8). Who performed better relative to their class?

Alice's z-score: (85 - 75) / 5 = 2.0

Bob's z-score: (92 - 80) / 8 = 1.5

Conclusion: Alice performed better relative to her class because her z-score is higher. She was 2 standard deviations above her class mean, while Bob was only 1.5 standard deviations above his.

Calculating Probabilities with normalcdf

The normalcdf function (normal cumulative distribution function) calculates the probability that a randomly selected value falls within a specified range. It finds the area under the normal curve between two bounds.

normalcdf Parameters

normalcdf(lower bound, upper bound, mean, standard deviation)

Returns the probability (as a decimal) that a value falls between the lower and upper bounds.

Common Probability Questions

Finding P(X < value) - "Less than"

What's the probability of scoring less than 80 on a test where μ = 70 and σ = 10?

normalcdf(-∞, 80, 70, 10) = 0.8413

There's an 84.13% chance of scoring below 80.

Finding P(X > value) - "Greater than"

What's the probability of scoring more than 85?

normalcdf(85, ∞, 70, 10) = 0.0668

There's a 6.68% chance of scoring above 85.

Finding P(a < X < b) - "Between"

What's the probability of scoring between 60 and 80?

normalcdf(60, 80, 70, 10) = 0.6827

There's a 68.27% chance of scoring between 60 and 80.

Pro tip: Since no calculator can handle actual infinity, use a very large number like 1E99 (or -1E99) for the bounds when calculating "less than" or "greater than" probabilities.

Real-World Examples

📏 Human Height

Adult male heights in the US are approximately normally distributed with μ = 5'10" (70 inches) and σ = 3 inches.

Application: What percentage of men are over 6'1" (73 inches)?

z = (73 - 70) / 3 = 1.0, so about 15.87% of men are taller than 6'1".

🧠 IQ Scores

IQ scores are designed to be normally distributed with μ = 100 and σ = 15.

Application: What IQ score puts someone in the top 2%?

A z-score of about 2.05 corresponds to the 98th percentile. IQ = 100 + (2.05 × 15) ≈ 131.

🏭 Manufacturing Quality Control

A factory produces bolts with target diameter μ = 10mm and σ = 0.1mm. Bolts outside 9.8-10.2mm are rejected.

Application: What percentage of bolts pass inspection?

normalcdf(9.8, 10.2, 10, 0.1) = 0.9545, so about 95.45% pass.

🎓 SAT Scores

SAT scores are approximately normal with μ = 1050 and σ = 200.

Application: What score is needed for the top 10%?

The 90th percentile corresponds to z ≈ 1.28. Score = 1050 + (1.28 × 200) = 1306.

The Central Limit Theorem

The Central Limit Theorem (CLT) is one of the most important concepts in statistics. It states that when you take sufficiently large random samples from ANY population (regardless of its distribution), the distribution of sample means will be approximately normal.

Central Limit Theorem

For random samples of size n from a population with mean μ and standard deviation σ:

The sampling distribution of the mean approaches normal as n increases
The mean of sample means equals the population mean (μ)
The standard deviation of sample means (standard error) equals σ/√n

Why Is the CLT So Important?

Enables inference: We can make conclusions about population parameters from sample data
Works for any distribution: Even if the population is skewed or non-normal, sample means will be normal
Foundation for confidence intervals: Most statistical tests rely on the CLT
Practical application: Generally works well when n ≥ 30 (rule of thumb)

Example: Dice Rolling

A single die has a uniform distribution (each outcome 1-6 equally likely), not normal at all. But if you roll 50 dice and record the average, then repeat this many times, those averages will form a normal distribution centered around 3.5 (the expected value of a single die).

Summary: Key Takeaways

The normal distribution is a symmetric, bell-shaped curve defined by mean (μ) and standard deviation (σ)
The empirical rule: 68% within 1σ, 95% within 2σ, 99.7% within 3σ
Z-scores standardize values to compare across different distributions
normalcdf calculates probabilities (areas under the curve) between two bounds
The Central Limit Theorem explains why the normal distribution appears so often in statistics
Normal distributions are used in IQ testing, quality control, finance, medical research, and much more