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Standard deviation is a statistical measure of how spread out data is. Specifically, we use standard deviation to see how much our data varies. The lowercase Greek letter sigma, , represents the standard deviation.

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The idea of how much our data varies is specified by variance, which is actually how we define standard deviation.

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Variance is defined as the average of squared differences from the mean of the data set and is represented by the lowercase greek letter .

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Consider a set of data that represents the number of siblings that each student in a class has. It could look something like {1, 0, 2, 1, 0, 1, 3, 3, 1, 2, 1, 0, 0}.

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This means that to calculate variance, there are three steps:

- Calculate the mean of the data
- Calculate the sum of squared differences from the mean.
- Find the average of the value found in step 2.

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First we calculate the mean. Add up all of the values and divide by the total number of values to find that the mean is . This means that on average, there is 1 sibling for each student.

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Second, we calculate the sum of squared differences from the mean. This means that we subtract each data point from the mean, square it, and then add all of these values together. To illustrate this process, we will simply move through our original data set left to right and apply that process to each term.

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The total sum of all of our squared differences from the mean is 13.

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Now, we must tackle the final step of averaging this. However, it is not as straightforward as just dividing the sum of squared differences by 13! We must take into consideration whether we are dealing with the data from either:

- A population
- A sample

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A population would be if we collected data from every person that we are interested in. This is like the U.S. Census, which aims to collect data about every single person in the United States.

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Sample data is data that is gained from a smaller group of the population. For example, a sample of people from Texas would not necessarily represent the whole United States if that’s what we’re trying to measure.

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If we have the population data, we could simply divide the variance by the total number of data points. For instance, in the class of 13 students that we’re dealing with, if we are only interested in describing the data from that class, we’d have the entire population. However, if we were trying to generalize the data to all of the classes in that school, this group of 13 students would just be a sample!

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In a sample, we actually divide by 1 less than the number of data points. This is an attempt to “correct” our data since the sample is likely to be a little bit off from the actual population value. For our example, we’d divide the sum of our squared differences by 12 instead of 13.

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This means that if the 13 students represent a population, the variance is and if the students represent a sample, the variance is .

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You may be wondering why on earth we’ve spent all of this time calculating variance for a lesson on standard deviation! This is because just like how standard deviation is represented by sigma, the variance is represented by sigma squared. So, to calculate the standard deviation, we just take the square root of variance!

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For our population data, this is and we have a standard deviation of 1. This means that on average, we could expect many of the values in our data set to be 1 value away from the mean. Since our mean is 1, this means that we expect many of the values to be between 0 and 2, since this is subtracting or adding 1 to the mean. This is exactly what our data looks like with many students having 0, 1, or 2 siblings and only one student having 3 siblings!

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Calculating standard deviation is a nightmarish experience for most students, but thankfully calculators can usually do most of the work. The simpler skill to develop is determining how data is spread out on a bell curve.

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A normal distribution looks like this:

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The mean is the value in the middle of the curve and the data is spread out underneath the curve evenly. How large the standard deviation is will affect how spread out our curve becomes!

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If the standard deviation is large, then the data will become very spread out and our curve will look a bit more flat:

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If the standard deviation is low, then the data is not too spread out and will be concentrated around the center:

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Being able to simply look at a graph of data and determine if it has a large or small standard deviation is integral to understanding the main idea of standard deviation, which is simply: how spread out is our data?

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