Normal distribution

Normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve where values cluster around the mean, with the frequency of values decreasing as distance from the mean increases, described by only two parameters—mean and standard deviation (Rice J.A. 2007, p.182)^[1]. Human height roughly follows this pattern. Most people cluster near average height; very tall and very short people are rare. The distribution is symmetric—the probability of being 6 inches above average equals the probability of being 6 inches below. This simple, elegant shape appears throughout nature and forms the foundation of statistical analysis.

Abraham de Moivre first described the distribution in 1733. Carl Friedrich Gauss developed it further, leading to the alternative name "Gaussian distribution." The central limit theorem explains why normal distributions appear so frequently: whenever many independent factors combine to produce an outcome, the result tends toward normality regardless of the underlying distributions of individual factors. This mathematical property makes the normal distribution fundamental to statistics.

Properties

The normal distribution has distinctive characteristics:

Symmetry

Bell shape. The distribution is perfectly symmetric around its mean. The left and right tails are mirror images^[2].

Mean equals median equals mode. All three measures of central tendency coincide at the distribution's center.

Parameters

Mean (μ). The center of the distribution—where values cluster.

Standard deviation (σ). The spread of the distribution—how far values typically fall from the mean^[3].

Complete specification. These two parameters fully describe any normal distribution.

Empirical rule

68-95-99.7. For any normal distribution:

• 68% of values fall within 1 standard deviation of the mean
• 95% fall within 2 standard deviations
• 99.7% fall within 3 standard deviations

This rule enables quick probability assessments without calculations^[4].

Standard normal distribution

A special case simplifies calculations:

Mean of zero. The standard normal has μ = 0.

Standard deviation of one. The standard normal has σ = 1.

Z-scores. Any normal distribution can be converted to standard normal by subtracting the mean and dividing by the standard deviation: z = (x - μ) / σ.

Tables and software. Standard normal probabilities are tabulated and programmed into software, enabling analysis of any normal distribution through standardization.

Importance

The normal distribution is foundational:

Central limit theorem

Averaging effect. When independent random variables are summed, their distribution approaches normal regardless of individual distributions. This explains why normal distributions appear so frequently^[5].

Sample means. The distribution of sample means is approximately normal for large samples, enabling statistical inference.

Statistical methods

t-tests. Comparing means assumes normal distributions or relies on central limit theorem for large samples.

Regression. Ordinary least squares regression assumes normally distributed errors.

Confidence intervals. Interval estimates use normal distribution properties^[6].

Quality control

Process capability. Six Sigma and process capability analysis assume normal distribution of process output.

Control limits. Control charts typically set limits at 3 standard deviations, based on normal distribution properties.

Applications

Normal distributions appear in:

Natural phenomena. Heights, weights, blood pressure, and IQ scores approximately follow normal distributions.

Measurement error. Random errors in measurement tend to be normally distributed.

Financial returns. While imperfect, normal distributions provide workable approximations for many financial calculations^[7].

Quality metrics. Manufacturing processes often produce normally distributed output.

Limitations

The normal distribution doesn't fit everything:

Bounded variables. Proportions (0-100%) and counts (non-negative integers) cannot be normal.

Skewed data. Income distributions, response times, and many business metrics are right-skewed, not symmetric.

Heavy tails. Some phenomena produce extreme values more frequently than normal distributions predict—stock market crashes, for example^[8].

Multimodality. Distributions with multiple peaks aren't normal.

Normal distribution — recommended articles

Statistics — Probability — Statistical process control — Six Sigma

References

• Rice J.A. (2007), Mathematical Statistics and Data Analysis, 3rd Edition, Duxbury Press.
• Devore J.L. (2016), Probability and Statistics for Engineering and the Sciences, 9th Edition, Cengage Learning.
• NIST (2023), Normal Distribution, Engineering Statistics Handbook.
• Montgomery D.C. (2020), Introduction to Statistical Quality Control, 8th Edition, Wiley.

Footnotes

1. ↑ Rice J.A. (2007), Mathematical Statistics, p.182
2. ↑ Devore J.L. (2016), Probability and Statistics, pp.145-162
3. ↑ NIST (2023), Normal Distribution
4. ↑ Montgomery D.C. (2020), Statistical Quality Control, pp.78-92
5. ↑ Rice J.A. (2007), Mathematical Statistics, pp.212-228
6. ↑ Devore J.L. (2016), Probability and Statistics, pp.245-262
7. ↑ NIST (2023), Applications
8. ↑ Montgomery D.C. (2020), Statistical Quality Control, pp.156-172

Author: Sławomir Wawak