    # What are the percentages of normal distribution within 1/2 and 3 standard deviation? Meet Mason. He's an average American 40-year-old: 5 foot 10 inches tall and earning \$47,000 per year before tax.

How often would you expect to meet someone who earns 10x as much as Mason?

And now, how often would you expect to meet someone who is 10x as tall as Mason?

Your answers to the two questions above are different, because the distribution of data is different. In some cases, 10x above average is common. While in others, it's not common at all.

## So what are normal distributions?

Today, we're interested in normal distributions. They are represented by a bell curve: they have a peak in the middle that tapers towards each edge. A lot of things follow this distribution, like your height, weight, and IQ.

This distribution is exciting because it's symmetric – which makes it easy to work with. You can reduce lots of complicated mathematics down to a few rules of thumb, because you don't need to worry about weird edge cases.

For example, the peak always divides the distribution in half. There's equal mass before and after the peak.

Another important property is that we don't need a lot of information to describe a normal distribution.

Indeed, we only need two things:

1. The mean. Most people just call this "the average." It's what you get if you add up the value of all your observations, then divide that number by the number of observations. For example, the average of these three numbers: `1, 2, 3 = (1 + 2 + 3) / 3 = 2`
2. And the standard deviation. This tells you how rare an observation would be. Most observations fall within one standard deviation of the mean. Fewer observations are two standard deviations from the mean. And even fewer are three standard deviations away (or further).

Together, the mean and the standard deviation make up everything you need to know about a distribution.

### The 68-95-99 rule

The 68-95-99 rule is based on the mean and standard deviation. It says:

68% of the population is within 1 standard deviation of the mean.

95% of the population is within 2 standard deviation of the mean.

99.7% of the population is within 3 standard deviation of the mean.

## How to calculate normal distributions

To continue our example, the average American male height is 5 feet 10 inches, with a standard deviation of 4 inches. This means:

Now for the fun part: Let's apply what we've just learned.

What's the chance of seeing someone with a height between between 5 feet 10 inches and 6 feet 2 inches? (That is, between 70 and 74 inches.)

It's 34%! We leverage both the properties: the distribution is symmetric, which means chances for (66-70) inches and (70-74) inches are both 68/2 = 34%.

Let's try a tougher one. What's the chance of seeing someone with a height between 62 and 66 inches?

It's (95-68)/2 = 13.5%. Both outer edges have the same %.

And now your final (and hardest test): What's the chance of seeing someone with a height greater than 82 inches?

Here, we use also the final property: everything must sum to 100%. So the outer edges (that is, heights below 58 and heights above 82) together make (100% - 99.7%) = 0.3%.

Remember, you can apply this on any normal distribution. Try doing the same for female heights: the mean is 65 inches, and standard deviation is 3.5 inches.

So, the chance of seeing someone with a height between 65 and 68.5 inches would be: ___.

...

...

34%! It's exactly the same as our first example. It's +1 standard deviation.

## Conclusion

Knowing this rule makes it very easy to calibrate your senses. Since all we need to describe any normal distribution is the mean and standard deviation, this rule holds for every normal distribution in the world!

The challenging part, indeed, is figuring out whether the distribution is normal or not.

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### What is the standard deviation of the mean in normal distribution?

Different categories of the rule are: In a normal distribution, 68% of the data values will rest among 1 standard deviation (within 1 sigma) of the mean. In a normal, bell-shaped, distribution 95% of the data will fall into 2 standard deviations (within 2 sigma) of the mean.

### What is the percentage between 3 standard deviations below the mean?

For a standard normal distribution, how do you find the percentage of data that are between 3 standard deviations below the mean and 1 standard deviation above the mean? Statistics Statistical Distributions The Standard Normal Distribution It's about 84%. You could use a TI 80-series calculator under the "DISTR" menu to get the answer.

### What are the characteristics of a normal distribution?

Normal distributions have key characteristics that are easy to spot in graphs: 1 The mean, median and mode are exactly the same. 2 The distribution is symmetric about the mean—half the values fall below the mean and half above the mean. 3 The distribution can be described by two values: the mean and the standard deviation. More ...

### What is the empirical rule for normal distribution?

The empirical rule, or the 68-95-99.7 rule, tells you where most of your values lie in a normal distribution: Around 68% of values are within 1 standard deviation from the mean. Around 95% of values are within 2 standard deviations from the mean.

### What percent of values fall within 1/2 and 3 standard deviations from the mean?

In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively.

### What percent is within 3 standard deviation?

That is, 68 percent of data is within one standard deviation of the mean; 95 percent of data is within two standard deviation of the mean and 99.7 percent of data is within three standard deviation of the mean.

### What percentage of the data is between 2 and 3 standard deviations?

The empirical rule, or the 68-95-99.7 rule, tells you where your values lie: Around 68% of scores are within 1 standard deviation of the mean, Around 95% of scores are within 2 standard deviations of the mean, Around 99.7% of scores are within 3 standard deviations of the mean.

### What percentage of the 50 regions have stores within 1 2 or 3 standard deviations of the mean?

What percentage of the 50 regions have stores within +/- 1, +/- 2 or +/- 3 standard deviations of the mean? +/-1 = 90% +/-2 = 94% +/-3 = 98% c. 