The discovery of the normal distribution is credited to Carl Gauss. It is also known as the bell curve or Gaussian distribution.

This is the most commonly used distribution in statistics because it closely approximates the distributions of many different measurements.

Height, birth weight, reading ability, job satisfaction, or SAT scores are just a few examples.

It is continuous.

Symmetric about the mean μ.

It is bell-shaped or mound-shaped.

Mean = median = mode = μ .

The curve approaches the horizontal axis on both sides of the mean without ever touching or crossing it.

Nearly all of the distribution (99.73 percent) lies within three standard deviations of the mean.

It has two inflection points : one at \$x = \mu + \sigma \$ and one at \$x = \mu - \sigma \$

The normal distribution is fully determined by two parameters, namely, mean and variance.

The location of the distribution on the number line depends on the mean of the distribution. (See image1)

The shape of the distribution depends on the standard deviation. A normal distribution with a larger standard deviation is more spread out, while one with a smaller standard deviation is more tightly bunched.(See image2)

The X-axis is the symptote to the curve.

If X∼N(\$\mu_1,\sigma_1^2\$) and Y∼N(\$\mu_2,\sigma_2^2\$) and X and Y are independent, then X+Y∼N(\$\mu_1 + \mu_2,\sigma_1^2 + \sigma_2^2\$)

The standard normal distribution, also called the z-distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.

Approximately 68 percent of the area under the curve lies between μ - σ and μ + σ.

Approximately 95 percent of the area under the curve lies between μ + 2σ and μ + 2σ.

Approximately 99.7 percent of the area under the curve lies between μ + 3σ and μ + 3σ.
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Around 68% of values are within 1 standard deviation from the mean

Around 95% of values are within 2 standard deviations from the mean.

Around 99.7% of values are within 3 standard deviations from the mean.