# Normal distributions review

Normal distributions come up time and time again in statistics. A normal distribution has some interesting properties: it has a bell shape, the mean and median are equal, and 68% of the data falls within 1 standard deviation.

## What is a normal distribution?

Early statisticians noticed the same shape coming up over and over again in different distributions—so they named it the normal distribution.

Normal distributions have the following features:

- symmetric bell shape
- mean and median are equal; both located at the center of the distribution
- $\approx68\%$ of the data falls within $1$ standard deviation of the mean
- $\approx95\%$ of the data falls within $2$ standard deviations of the mean
- $\approx99.7\%$ of the data falls within $3$ standard deviations of the mean

*Want to learn more about what normal distributions are? Check out this video.*

### Drawing a normal distribution example

The trunk diameter of a certain variety of pine tree is normally distributed with a mean of $\mu=150\,\text{cm}$ and a standard deviation of $\sigma=30\,\text{cm}$.

**Sketch a normal curve that describes this distribution.**

**Solution:**

**Step 1**: Sketch a normal curve.

**Step 2**: The mean of $150\,\text{cm}$ goes in the middle.

**Step 3**: Each standard deviation is a distance of $30\,\text{cm}$.

### Finding percentages example

A certain variety of pine tree has a mean trunk diameter of $\mu=150\,\text{cm}$ and a standard deviation of $\sigma=30\,\text{cm}$.

**Approximately what percent of these trees have a diameter greater than $210\,\text{cm}$?**

**Solution**:

**Step 1:**Sketch a normal distribution with a mean of $\mu=150\,\text{cm}$ and a standard deviation of $\sigma=30\,\text{cm}$.

**Step 2:**The diameter of $210\,\text{cm}$ is two standard deviations above the mean. Shade above that point.

**Step 3:**Add the percentages in the shaded area:

**About $2.5\%$ of these trees have a diameter greater than $210\,\text{cm}.$**

*Want to see another example like this? Check out this video.*

*Want to practice more problems like this? Check out this exercise on the empirical rule.*

### Finding a whole count example

A certain variety of pine tree has a mean trunk diameter of $\mu=150\,\text{cm}$ and a standard deviation of $\sigma=30\,\text{cm}$.

A certain section of a forest has $500$ of these trees.

**Approximately how many of these trees have a diameter smaller than $120\,\text{cm}$?**

**Solution**:

**Step 1:**Sketch a normal distribution with a mean of $\mu=150\,\text{cm}$ and a standard deviation of $\sigma=30\,\text{cm}$.

**Step 2:**The diameter of $120\,\text{cm}$ is one standard deviation below the mean. Shade below that point.

**Step 3:**Add the percentages in the shaded area:

About $16\%$ of these trees have a diameter smaller than $120\,\text{cm}.$

**Step 4: Find how many trees in the forest that percent represents.**

We need to find how many trees $16\%$ of $500$ is.

**About $80$ trees have a diameter smaller than $120\,\text{cm}$.**