# 7.1 - Standard Normal Distribution

A normal distribution is a bell-shaped distribution. Theoretically, a normal distribution is continuous and may be depicted as a density curve, such as the one below. The distribution plot below is a standard normal distribution. A standard normal distribution has a mean of 0 and standard deviation of 1. This is also known as the z distribution. You may see the notation $$N(\mu, \sigma$$) where N signifies that the distribution is normal, $$\mu$$ is the mean of the distribution, and $$\sigma$$ is the standard deviation of the distribution. A z distribution may be described as $$N(0,1)$$.

While we cannot determine the probability for any one given value because the distribution is continuous, we can determine the probability for a given interval of values. The probability for an interval is equal to the area under the density curve. The total area under the curve is 1.00, or 100%. In other words, 100% of observations fall under the curve.

For example, in Lesson 2 we learned about the Empirical Rule which stated that approximately 68% of observations on a normal distribution will fall within one standard deviation of the mean, approximately 95% will fall within two standard deviations of the mean, and approximately 99.7% will fall within three standard deviations of the mean.

## Example: SAT-Math Scores Section

The distribution of SAT-Math scores can be described as $$N(500, 100)$$. Let's apply the Empirical Rule to determine the SAT-Math scores that separate the middle 68% of scores, the middle 95% of scores, and the middle 99.7% of scores.

Middle 68%: $$500\pm1(100)=[400, 600]$$

Middle 95%: $$500\pm2(100)=[300, 700]$$

Middle 99.7%: $$500\pm 3(100)= [200, 800]$$

## z scores Section

In Lesson 2 we wanted to describe one observation in relation to the distribution of all observations. We did this using a z score.

z score

Distance between an individual score and the mean in standard deviation units; also known as a standardized score.

z score
$$z=\dfrac{x - \overline{x}}{s}$$

$$x$$ = original data value
$$\overline{x}$$ = mean of the original distribution
$$s$$ = standard deviation of the original distribution

This equation could also be rewritten in terms of population values: $$z=\dfrac{x-\mu}{\sigma}$$

## Example: IQ Scores Section

IQ scores are normally distributed with a mean of 100 and standard deviation of 15. Compute the z score for an individual with an IQ score of 120.

$$z=\dfrac{x- \mu}{\sigma}$$
Here, $$x=120$$, $$\mu=100$$, and $$\sigma=15$$.
$$z=\dfrac{120-100}{15}=\dfrac{20}{15}=1.333$$