What is a normal distribution?

Definition

A normal distribution is a continuous probability distribution that is symmetric about its mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is often called the "bell curve" due to its characteristic shape.

The probability density function (PDF) of a normal distribution with mean $\mu$ and standard deviation $\sigma$ is:

$$ f(x) = \frac{1}{\sigma \sqrt{2pi}} \exp \left( -\frac{(x - \mu)^2}{2sigma^2} \right) $$

where:

• $\mu$ is the mean (center of the distribution)
• $\sigma$ is the standard deviation (spread or width)

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Worked Example

Example:
Suppose the heights of adult men are normally distributed with mean $\mu = 70$ inches and standard deviation $\sigma = 3$ inches. What is the probability density for a man who is 73 inches tall?

Step 1: Plug values into the PDF

$$ f(73) = \frac{1}{3 \sqrt{2pi}} \exp \left( -\frac{(73 - 70)^2}{2 \times 3^2} \right) $$

Step 2: Simplify the exponent

$$ (73 - 70)^2 = 9 \\ 2 \times 3^2 = 18 \\ -\frac{9}{18} = -0.5 $$

Step 3: Substitute and calculate

$$ f(73) = \frac{1}{3 \sqrt{2pi}} \exp(-0.5) $$

Step 4: Approximate values

• $\sqrt{2pi} \approx 2.5066$
• $\exp(-0.5) \approx 0.6065$

So,

$$ f(73) \approx \frac{1}{3 \times 2.5066} \times 0.6065 \approx \frac{1}{7.5198} \times 0.6065 \approx 0.0807 $$

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Takeaways

• The normal distribution is symmetric and bell-shaped, defined by its mean and standard deviation.
• The PDF describes the likelihood of a random variable taking a specific value.
• Many natural phenomena (like heights, test scores) are approximately normally distributed.