 There are four measures of variation. For now we are going to concentrate only on the range, the variance and the standard deviation. They measure how is your data spread around your mean or what is the spread of your data around the mean. The common or the simplest measure of variation tells you what is your range by taking the largest value subtracting the smallest value from there. Let's say for example we have all these cycles with starts from 1 up to 18. So it means our largest value here is 18 and our smallest value is 1. To calculate the range of your data we say 18 minus 1 which is equals to 12. That was easy. The other measures of variance and your standard deviation. The variance is your average of square deviation of the values from the mean. We normally do not interpret the variance but we can calculate it and we can calculate for the sample variance and the population variance. The sample variance we always use the simple letters of alphabet. So it's s, s squared, which is the sum of your observation minus the sample mean squared divided by n minus 1. For the population we use the Greek letter sigma. The sigma squared is equals to the sum of your observation minus the population mean squared divided by n. The only difference between the two, the sample variance and the population variance is the n. For the sample we say n minus 1 and for this population we say n. That is the formula to calculate the variance. But you can do this also on your calculator. At the later stage I can show you how to use your calculator. For now let's look at what is the standard deviation. Standard deviation is commonly used as a measure of variation. And we're going to use the standard deviation. You will notice we use it when we do confidence interval sampling distribution hypothesis testing. We will constantly be using the standard deviation. And we are able to also interpret the standard deviation because it shows the variation of your data around the mean. So how far apart are your data from your mean. And the standard deviation is the square root of your variance. So from the variance formula if we put the square root you will see that it becomes the standard deviation. And because we put in the square root and we are able to interpret it is because it goes back to the same units as your original data. And that is how we are able to interpret the standard deviation. So let's look at the formula. For the sample standard deviation s is equals to the square root of your variance. Which is the sum of your observation minus the mean square divided by n minus 1. You can see that the formula underneath the square root is the same as the variance. So the standard deviation is the square root of your sample variance. For the population it will be the square root of your population variance. You can see that it is the same. Now let's look at an example on how to apply this formula. Let's say we have this data set. It starts from 10, 12 and ends at 24. The first step of everything is to count how many there are and also to calculate the mean. So we are going to count how many there are which is the sample size 1, 2, 3, 4, 5, there are 8. n is equals to 8 because our sample size is 8. And we calculate the mean which is the sum of all the observations divided by how many there are. Which is 10 plus 12 plus 14 up until 24. We add them all up and divide by 8 and we get the sample mean as 16. Now we look at the formula for calculating the sample standard deviation. Remember it is the square root of the sum of your X observation minus the mean squared divided by n minus 1. Remember the summation, all the summation it tells us minus the mean squared. The summation just tells us that it is everything in the bracket squared again and again and again adding it together 8 times. So it's like it's 8 minus the mean squared plus X minus X1. Let's say it's X1 because we represent this value as X1 and this value as X2 this value as X3 and so forth until X8. So X1, X2 minus the mean squared plus up until we get to X8 minus the mean squared. And that's how we're going to substitute. So it means for every X value we're going to substitute with all these X values. Let's look at when we have a complete formula, when we have substituted. We calculate in the sample standard deviation which is as X1 is 10, X2 is 12, X3 is 14 and X8 is 24. In the next step, we substitute the value of the mean and we say 10 minus 16 squared plus 12 minus 16 squared plus until we get everything n is 8 minus 1. And once we have solved everything at the top, we say it's 130 divided by 7 and we take the square root of it and we get 4,3095. And this only tells us what is the distance around the mean, how far apart are your data around the mean. Now, remember the standard deviation is the square root of your variance. So everything that is underneath the square root is what we call the variance and calculate the variance. We can also stop there by saying 130 divided by 7 which gives us 18.57 which is our sample variance. I've also on the side shown you how to calculate the standard deviation using a calculator which is simple and easy. To calculate the standard deviation or any of the measures of central location especially the mean or the standard deviation, you can use a formula by putting your calculator to state mode. We need to know the steps. So these steps are only for a scientific calculator. I used a sharp scientific calculator. If you are using a sharp financial calculator, therefore it means instead of putting the M plus, I think you put data. There is different buttons on different calculator. If you are using a cashier, also you need to just check the steps to put your calculator to state mode. All calculators are different but they can all calculate the standard deviation especially if it's a scientific calculator. And thank you.