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Glacius
ProbabilityConcept reference

Normal distributions

A normal model has mean mu and positive standard deviation sigma, often written N(mu, sigma squared).

2 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. Center, spread, and notation
  3. Read the parameter convention
  4. Specify a normal model
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01Center, spread, and notation#

A normal distribution is a symmetric, bell-shaped model for a numerical quantity. Its mean locates the center and its standard deviation sets the horizontal spread. Specifying these two parameters determines the model, but choosing it for real data still requires justification.

This normal curve is centered at 44. One standard deviation is a horizontal distance of 22.

Normal density with mean 4 and standard deviation 2. The center is at 4; the marked horizontal span from 4 to 6 is one standard deviation. The curve follows exp(-z squared / 2) at x=mean+SD times z.Normal density with mean 4 and standard deviation 2. The center is at 4; the marked horizontal span from 4 to 6 is one standard deviation. The curve follows exp(-z squared / 2) at x=mean+SD times z.
Figure 1Normal density with mean 4 and standard deviation 2. The center is at 4; the marked horizontal span from 4 to 6 is one standard deviation. The curve follows exp(-z squared / 2) at x=mean+SD times z.
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We write N(μ,σ2)N(\mu,\sigma^2) here: the first parameter is the mean, and the second is the variance. The SD is the positive square root of variance.

N(4,4)μ=4,σ=2\begin{gathered}N(4,4)\\\mu=4,\quad\sigma=2\end{gathered}

Under the convention N(μ,σ2)N(\mu,\sigma^2), N(4,4)N(4,4) means mean 4 and variance 4. The standard deviation is 4=2\sqrt4=2. One standard deviation above the mean is 6 and one below is 2. The second argument is not a second center or the curve’s height.

Check your reasoning

N(−1,4)N(-1,4) uses (mean, variance). Give (mean, SD).

  1. A(−1,4)(-1,4)
  2. B(−1,2)(-1,2)
  3. C(1,2)(1,2)
Show answer and explanation
(−1,2)(-1,2)

Mean −1-1; SD 22.

02Read the parameter convention#

The convention belongs to the notation or tool. If a library says its arguments are (mean, SD), its second number is already the SD. Read that declaration before taking a square root.

Check your reasoning

N(2,3)N(2,3) uses (mean, SD). Reported SD 99. Give (mean, SD).

  1. A(2,3)(2,3)
  2. B(2,9)(2,9)
  3. C(5,3)(5,3)
Show answer and explanation
(2,3)(2,3)

Mean 22; SD 33.

03Specify a normal model#

To encode a normal model with mean 1212 and SD 33 in our variance convention, keep the center and square the scale.

σ2=32=9N(12,9)\begin{gathered}\sigma^2=3^2=9\\N(12,9)\end{gathered}
Check your reasoning

Reading model. Normal mean 1010, SD 44. Choose N(μ,variance)N(\mu,\mathrm{variance}).

  1. AN(10,4)N(10,4)
  2. BN(16,10)N(16,10)
  3. CN(10,16)N(10,16)
Show answer and explanation
N(10,16)N(10,16)

Variance is 42=164^2=16.

Normal models appear in measurement-error and sampling calculations. A normal model has support on the whole real line, so it may be a poor literal model for a strictly positive quantity near zero. Read both the modeling assumptions and the parameter convention.

Key takeaway

Read the declared convention. The mean gives the center; SD gives scale, and variance is SD squared.

  • Identify a normal model's location and scale parameters.

Further questions

How can a threshold be measured in standard deviations?
Subtract the mean from the threshold and divide by the positive standard deviation. The resulting sign and size describe its displacement in SD units.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Normal distributions.” Math behind ML. /learn/p-normal