What is Normal distribution?

A continuous probability distribution with a symmetric bell shape centered on the mean μ, whose shape is determined by the mean μ and the standard deviation σ. It is also called the Gaussian distribution. About 68% lies within the mean ±1σ, about 95% within ±2σ and about 99.7% within ±3σ. A quantity formed by adding many small independent factors approaches a normal distribution (the central limit theorem), so it is widely used as a model of measurement errors and manufacturing variation. Standardizing with z = (x − μ) ÷ σ gives the standard normal distribution with mean 0 and standard deviation 1.

Japanese name: 正規分布

Normal distribution appears in 4 practice questions across 2 topics, including Probability, Statistics and Queueing, Yield and quality control.

Field: Probability, Statistics and Queueing

Shape and properties

The normal distribution is highest at the mean and falls off steeply away from it, in a bell shape. Changing the mean shifts it left or right without changing its shape, and increasing the standard deviation makes it lower and wider. The fractions within the mean ±1σ, ±2σ and ±3σ are about 68.3%, 95.4% and 99.7%, and remembering this "68-95-99.7 rule" allows quick estimates of variation.

Standardization and the standard normal distribution

A normal distribution with any mean and standard deviation becomes the standard normal distribution, with mean 0 and standard deviation 1, under the transformation z = (x − μ) ÷ σ. Because z expresses how many standard deviations a value is from the mean, data with different units or means can be compared. Normal distribution tables (tables of cumulative probability) are made for this standard normal distribution; for example, the two-sided 95% range is ±1.96.

Why it appears so often

A quantity formed by adding many small independent factors approaches a normal distribution regardless of the distributions of the individual factors (the central limit theorem). This is why measurement errors and dimensions built up from small variations in many process steps are close to normal. The fact that the distribution of the sample mean can be approximated by a normal distribution is also the basis of calculations for confidence intervals and hypothesis tests.

Try it: questions about Normal distribution

The dimension of a part follows a normal distribution with a mean of 50.0 mm and a standard deviation of 0.1 mm. Which of the following is closest to the fraction of parts whose dimension lies between 49.7 mm and 50.3 mm?

AAbout 68%
BAbout 95%
CAbout 99.7%

Show answer and explanation

Answer: C. About 99.7%

49.7 to 50.3 mm is the mean ±0.3 mm, that is, ±3 standard deviations (±3σ). For a normal distribution, about 68% lies within the mean ±1σ, about 95% within ±2σ and about 99.7% within ±3σ (the 68-95-99.7 rule). The fraction in this range is therefore about 99.7%.

The delay of a circuit path varies from chip to chip following a normal distribution with a mean of 940 ps and a standard deviation of 30 ps. Taking the fraction of a normal distribution within the mean ±2σ to be about 95.4%, which of the following is closest to the fraction of chips whose delay exceeds the clock period of 1,000 ps?

AAbout 2.3%
BAbout 4.6%
CAbout 95.4%

Show answer and explanation

Answer: A. About 2.3%

Standardizing gives z = (1,000 − 940) ÷ 30 = 2, so the fraction of chips beyond the mean +2σ is needed. The fraction outside ±2σ is 100 − 95.4 = 4.6%, and since the normal distribution is symmetric, the upper side alone is 4.6 ÷ 2 = 2.3%. Treating the delay variation as a probability distribution in this way makes it possible to estimate the fraction of chips that fail timing.

FAQ

How is it related to "3σ" in process control?

On a control chart, if a process is stable and follows a normal distribution, the probability that a point falls outside the mean ±3σ by chance alone is only about 0.27%. The control limits are therefore drawn at ±3σ, and a point outside them is judged likely to indicate an abnormality rather than chance. Six Sigma is likewise based on how many standard deviations of margin are left up to the specification limits.

Can any data be treated as normally distributed?

No. Many kinds of data do not follow a normal distribution: component lifetimes follow lognormal or Weibull distributions, intervals between arrivals follow exponential distributions and counts of rare events follow Poisson distributions. In particular, the probability in the "tails" far from the mean can differ by orders of magnitude depending on the shape of the distribution. Draw a histogram to check the shape before deciding whether to use the normal distribution.

Normal distribution in the explanations

For a normal distribution, about 68% lies within the mean ±1σ, about 95% within ±2σ and about 99.7% within ±3σ (the 68-95-99.7 rule).
The dimension of a part follows a normal distribution with …
The fraction outside ±2σ is 100 − 95.4 = 4.6%, and since the normal distribution is symmetric, the upper side alone is 4.6 ÷ 2 = 2.3%.
The delay of a circuit path varies from chip to chip follow…
Under a normal distribution, a point falls outside by chance only about 0.27% of the time, so a point outside signals an abnormality.
In control charts (Shewhart control charts) used for proces…

Questions that mention Normal distribution (4)

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