Recall from the previous mountain entry that a symmetrical distribution will have a mean and median that are equal to each other. The most famous, BY FAR, of these "symmetrical" data distributions is the Normal Distribution.

What the hell is that?

The normal distribution is that famous "bell-shaped" curve that we see. In layman's terms, it means that most of the data is "bunched up" toward the middle. As we move either to the left or the right, the datapoints get rarer and rarer and rarer and rarer and rarer and rarer. Think about height. How many people do you know who are between $5$ feet ($\approx 152$ cm) and $6$ feet ($\approx 183$ cm)? A lot, right? How many people over $7$ feet ($\approx 213$ cm) do you know?

The normal distribution arises naturally in many large datasets. Key word there is large. If you were to randomly sample the heights of $10$ individuals and create a distribution chart, it might not look "bell-shaped" at all. It might look like some weird monstrosity of a chart. That's what happens with datasets with very few datapoints. Anything is possible.

But if you were to randomly sample the heights of $1$,$000$ people and plot each datapoint on a distribution chart, you would almost definitely see that bell-shaped curve.

Thus, we say this data is normally distributed. Not all datasets are normally distributed by the way, even when they're large. For example, if you were to randomly sample GRE scores from $500$ people, the results would be roughly normally distributed. But if those $500$ individuals were all world-renowned mathemeticians, the data might consist of nothing but perfect scores, at least in the case of quant.

What Does The Normal Distribution Mean Mathematically?

A lot.

The curve extends infinitely in both directions.
The area under the curve is always equal to $1$.
A normal distribution curve sits on top of a number line.
If you draw a line that intersects both the number line below and the highest point of the chart, you will identify the mean and median values on the number line.

In this case, the mean and median are both equal to $17$.

Equally as important is we can divide the normal distribution into predictable segments. Imagine we have a normally distributed dataset for which the mean is $30$ and the standard deviation is $6$. Look at how this data is plotted below:

Let's start on the number line in the middle. We know the mean is $30$ and the standard deviation is $6$. If we subtract one standard deviation from the mean, we get $24$. If we subtract two standard deviations from the mean, we get $18$. If we one or two standard deviations, we get $36$ and $42$. Okay, so far so good?

Here's the interesting part. The values between $24$ and $30$, for example, comprise $34\%$ of all the values. The values between $30$ and $36$ ALSO comprise $34\%$ of the values. If we continue along these lines, you can see that the values between $18$ and $24$ compries $14\%$ of the values, as do the values between $36$ and $42$. Values less than $18$ only comprise $2\%$ of the values. The same could be said of values greater than $42$.

Pretty sweet, right?!

And these percentages are equal for every normally distributed data set. Obviously in other datasets the mean and standard deviation could be different, but these percentages tied to subtract or adding a certain number of standard deviations holds for all normally distributed datasets.

NOTE: A true normal distribution exists only in theory. That's why you'll often see the words "approximately" in these problems.

The data is approximately normally distributed.
The mean and median are approximately equal.
The proportion of values within one standard deviation of the mean is approximately $68\%$ ($34 + 34$).