How to Calculate Standard Deviation

Jeremy Jones
Published at : 04 Jan 2021
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Follow these five steps to calculate standard deviation. Also includes the standard deviation formula.

Here's the video transcript:
"How to Calculate Standard Deviation

How many vegetables do you have in your fridge? Is that a common amount or are you an outlier? We can use standard deviation to know whether someone’s behavior is normal or extraordinary.

Standard deviation, often calculated along with the mean of a data set, tells us how spread out the data is. It is used for data that is normally distributed and can be easily calculated using a graphing calculator or spreadsheet software, but it can also be calculated with a few math operations.

We’re going to use an example involving the number of vegetables five of our friends have in their fridges. They have 2, 3, 4, 7, and 9 vegetables.

To calculate the standard deviation, the first step is to calculate the mean of the data set, denoted by x with a line over it, also called x-bar.
In this case, the mean would be (2 + 3 + 4 + 7 + 9) / 5 = 5. Our average friend has 5 vegetables in their fridge.

The second step is to subtract the mean from each data point to find the differences. It’s helpful to use a table like this. 2 - 5 = -3, 3 - 5 = -2, 4 - 5 = -1, 7 - 5 = 2, and 9 - 5 = 4.

The third step is to square each difference. (This makes all the differences positive so they don’t cancel each other out and it also magnifies larger differences and minimizes smaller differences.)
-32 = 9, -22 = 4, -12 = 1, 22 = 4, and 42 = 16.

The fourth step is to calculate the mean of the squared differences.
(9 + 4 + 1 + 4 + 16) / 5 = 6.8.

The final step is to take the square root. (This counteracts the squaring we did earlier and allows the standard deviation to be expressed in the original units.)
The square root of 6.8 is about 2.6 and that’s the standard deviation.

We're done! The mean number of vegetables is 5 with a standard deviation of 2.6 veggies. Knowing that about ⅔ of the data fall within one standard deviation of the mean (assuming the data is normally distributed), we can say that about ⅔ of our friends have between 2.4 and 7.6 vegetables in their fridges.

To recap, these are the five steps for calculating standard deviation:
1. Calculate the mean.
2. Subtract the mean from each data point.
3. Square each difference.
4. Calculate the mean of the squared differences.
5. Take the square root.

Using symbols, the equation for calculating standard deviation looks like this [see video]...

Lower case sigma stands for standard deviation of a population.
Upper case sigma tells us to calculate the sum for each instance.
X is each data point.
X bar is the mean of the data points.
And n is the number of data points.

Keep in mind that there is a similar formula that divides by n-1. That formula is used when you only have data for a sample of the population.

Hope this was helpful. See you next time!"