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What is Standard Deviation? - Definition, Equation & Sample

What is Standard Deviation? - Definition, Equation & Sample
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  • 0:02 An Experiment in Variation
  • 1:06 What Is Standard Deviation
  • 2:22 Finding the Standard Deviation
  • 4:33 Lesson Summary
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Lesson Transcript
Instructor: Joshua White

Josh has worked as a high school math teacher for seven years and has undergraduate degrees in Applied Mathematics (BS) & Economics/Physics (BA).

This lesson will examine the concepts of standard deviation and variance. It will examine the formula for standard deviation and show how to calculate it for both a population and a sample of a population.

An Experiment in Variation

On your drive to school one day, you decide to start timing how long it takes. You measure the time every day that week and find that the drive took 13, 15, 15, 17 and 15 minutes. Thus, your average time spent driving to school was 15 minutes. The next week you repeat the process and find the times were 10, 12, 22, 19 and 12 minutes. The average time turns out to be 15 minutes - the same as the first week.

However, you probably noticed that during the second week, your likelihood of arriving extremely early or late was much greater, even though it had the same average time. During the first week, the times were all within a small range - two minutes - of the average, whereas the times during the second week were much further from it. This is essentially the concept of standard deviation, which tells us how much the individual values of a data set are spread out from the average, or mean, of that set. The standard deviation is an additional value that can be used to describe a data set.

What Is Standard Deviation and How Do I Find It?

The standard deviation of a data set is a calculated number that tells you how close, or how far, the values of that data set are in relation to the mean. It's important because it can tell us more information about a data set than the mean itself will provide.

To calculate the standard deviation, you must first find the variance of the data set. The variance is defined as the average of the sum of the squared differences for each value in the set. In other words, you find this difference between each point and the mean of the data set and then square that difference. You will do this for every number in the data set and then add all those squared values together.

Finally, you will divide the variance by either the number of items in the data set, which is usually referred to as n, or by one less than the number of items in the set, which is written as n - 1. What you divide by depends on whether you are calculating the variance of the whole population or just a sample. When you are calculating the variance for an entire population, divide by n. For a sample of the entire population, divide by n - 1. Once you have found the variance for the data set, you can then find the standard deviation by taking the square root of the variance.

The formulas for standard deviation are shown below, where n equals the number of items in the data set and Xi represents each value in the data set.

standard deviation formulas

Finding the Standard Deviation - An Example

Let's go back to the driving times you measured and find the standard deviation for each week. Since the five times for the first week are a smaller group of the entire set of times, which is all the times over the two weeks, we will use the sample formula with n = 5 and n - 1 = 4.

Next, you will find the difference between each value and the mean, which is 15. Doing this should give you values of -2, 0, 0, 2 and 0. Next, you will square these values to get 4, 0, 0, 4 and 0, followed by adding them together to get 8. Then, divide 8 by 4 (n - 1) to get the variance, which will be 2. Finally, take the square root of 2, which is approximately 1.414 minutes, to find the standard deviation.

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