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In this lesson you will learn how to calculate and interpret the standard deviation of a dataset.


Standard Deviation in R

Now that we're able to compute the standard deviation of a dataset, what can we do with it? 

Now that our units match, our measure of spread is easier to interpret. By finding the number of standard deviations a data point is away from the mean, we can begin to investigate how unusual that datapoint truly is. In fact, you can usually expect around 68% of your data to fall within one standard deviation of the mean, 95% of your data to fall within two standard deviations of the mean, and 99.7% of your data to fall within three standard deviations of the mean. 

<img src="https://content.codecademy.com/courses/statistics/variance/normal_curve.svg" alt="A histogram showing where the standard deviations fall">
If you have a data point that is over three standard deviations away from the mean, that's an incredibly unusual piece of data!


Using Standard Deviation

When beginning to work with a dataset, one of the first pieces of information you might want to investigate is the spread &mdash; is the data close together or far apart? One of the tools in our statistics toolbelt to do this is the descriptive statistic _variance_:

```tex
\sigma^2 = \frac{\sum_{i=1}^{N}{(X_i -\mu)^2}}{N}
```

By finding the variance of a dataset, we can get a numeric representation of the spread of the data. If you want to take a deeper dive into how to calculate variance, check out our <a href = "https://www.codecademy.com/courses/statistics-variance-and-standard-deviation/lessons/variance/exercises/variance" target = "_blank">variance course</a>.

But what does that number really mean? How can we use this number to interpret the spread?

It turns out, using variance isn't necessarily the best statistic to use to describe spread. Luckily, there is another statistic &mdash; standard deviation &mdash; that can be used instead.

In this lesson, we'll be working with two datasets. The first dataset contains the heights (in inches) of a random selection of players from the NBA. The second dataset contains the heights (in inches) of a random selection of users on the dating platform OkCupid.

Variance Recap

In the last exercise you saw that Lebron James was `0.55` standard deviations above the mean of NBA player heights. He's taller than average, but compared to the other NBA players, he's not absurdly tall.

However, compared to the OkCupid dating pool, he is extremely rare! He's almost three full standard deviations above the mean. You'd expect only about `0.15%` of people on OkCupid to be more than 3 standard deviations away from the mean.

This is the power of standard deviation. By taking the square root of the variance, the standard deviation gives you a statistic about spread that can be easily interpreted and compared to the mean.


Side by side histograms of the two datasets and their respective stats lines including standard deviation and mean.

Review

Variance is a tricky statistic to use because its units are different from both the mean and the data itself. For example, the mean of our NBA dataset is `77.98` inches. Because of this, we can say someone who is `80` inches tall is about two inches taller than the average NBA player.

However, because the formula for variance includes _squaring_ the difference between the data and the mean, the variance is measured in _units squared_. This means that the variance for our NBA dataset is `13.32` inches squared.

This result is hard to interpret in context with the mean or the data because their units are different. This is where the statistic _standard deviation_ is useful.

Standard deviation is computed by taking the square root of the variance. `sigma` is the symbol commonly used for standard deviation. Conveniently, `sigma` squared is the symbol commonly used for variance:

```tex
\sigma = \sqrt{\sigma^2} = \sqrt{\frac{\sum_{i=1}^{N}{(X_i -\mu)^2}}{N}}
```

In R, you can take the square root of a number using `^ 0.5` or `sqrt()`, up to you which one you prefer:

```R
num <- 25
num_square_root <- num ^ 0.5
```


Standard Deviation

There is an R function dedicated to finding the standard deviation of a dataset &mdash; we can cut out the step of first finding the variance. The R function `sd()` takes a dataset as a parameter and returns the standard deviation of that dataset:

```R
dataset <- c(4, 8, 15, 16, 23, 42)
standard_deviation <- sd(dataset)
```
