# Range, variance and standard deviation as measures of dispersion | Khan Academy

The central trend in previous videos and We talked about different ways to determine the average value of information. This video is a bit more detailed than that information to understand the spread by talking we will try. Let's think about this for a moment. Suppose I am given negative numbers 10, 0, 10, 20 and 30. Assume that data set is here. Here is another group: 8, 9, 10, 11 and 12. Let's take the numerical middle of both groups let's calculate. Find the numerical mean. When learning statistics in more detail between the set and the sample you will understand the difference. That this is a plethora of information Suppose. We have to find the numerical middle of the set here. Here is the set size of the spread we have to find. I know this sounds a bit strange.

We will not use all of them in the future. Here to estimate the whole majority from some examples will use. There is no need to think about it yet. But to understand the statistics more deeply we need to clarify this. Let's find the average of the set, that is, the numerical average of it. -10 plus 0 plus 10, plus 20 plus 30, there are 5 here, so we divide by 5. What does this mean? -10 and positive 10 are reduced. 20 + 30 = 50. Dividing this by 5 gives 10. Now let's find the numerical middle of this group. 8 + 9 + 10 + 11 + 12, divide by 5. Let's calculate it. 8 + 12 = 20. 9 + 11 = 20, plus 20 = 40. Here 50 is taken. Plus 10. Here 50 divisions 5 are obtained. Here, too, the same numerical mean is obtained. Here is the plural or sample word the numerical middle of both groups without running you can say it's the same.

He collected them all to find the middle of all these numbers Divide the total by the number of digits, ie 5. Here the answer is 10, when you find the middle of these numbers 10 is taken here as well. But as you can see, these are different numbers. Looking at these numbers, you can say that they are similar to each other. One point when looking at these two groups gets attention. Each of these numbers is about 10. Here, the outermost numbers differ from 10 by 2 units. 12 differs from 10 by 2 units. The numbers here are greater than 10. 20 is the closest to 10 10 units away. Therefore, the prevalence in this group is higher, is that right These numbers are far from numerical, not only these. Now to different methods of measuring the spread take a look. In other words, how much from the center find that we are far away.

This is the simplest way to find out is the intermediate difference. You may not have encountered this method often, but it is spread between the largest and smallest prices is a very simple method of calculation. From the largest number here, ie 30 we have to get the smallest number. 30 – (-10) = 40. So between the largest and smallest numbers the difference is 40. That is, the difference is 40. Here are the largest numbers from 12 the smallest number, that is, if we subtract 8, we get 4. As you can see, do not calculate the spread of the intermediate difference is one of the best methods. Here the numerical average of both groups is 10. But when we look at the gap, we see that it is much larger, that is, there is more prevalence here.

However, the difference does not always give a complete answer. Maybe two different groups the intermediate difference is the same. But the numbers that the distribution is different we can observe. One of the most commonly used methods is a variance. The standard trend in this video we will get acquainted with the concept. It is probably the most widely used method. It has a close relationship with dispersion. The symbol of variance … The dispersion of the majority is considered here. I remembered again here not samples or subgroups, the whole set is intended. The symbol of variance is the sigma square. This is a Greek letter. This is a symbol of variance. Sigma is a standard trend is a symbol.

There is a reason for this. Consider the definition of variance. Taking each of these points, After finding the difference between them and the numerical mean and calculating the squares we find the average of the squares. It looks a bit confusing. But when we calculate, we see that in fact it is not so difficult. The numerical average is 10. Let's take the first point. Let's solve it here. Let's lower the screen a bit. Let's take the first point. Negative 10. Subtract the number from this price, square. Find the difference between the first value and the numerical mean square. Therefore, the answer is yes. Plus the second point, 0 – 10, that is, the numerical mean. We square it. Plus (10 – 10) squares. That's 10 in the middle. Plus (20 – 10) squares, plus (30 – 10) squares. These are each point and numerical mean is the square of the difference. This is the numerical average. We found the difference between each point and the numerical mean, We squared them and found their sum, we must divide by the number of numbers.

That is, these numbers, squares we need to find the average price. When you say it orally, it's confusing sounds. Taken each number, find the difference with its numerical average, then square, we find the average. There are 1, 2, 3, 4, 5 pieces. We have to divide by 5. What will be the answer? -10 – 10 = -20. -20 square = 400. 0 – 10 = -10. -10 squares = 100, plus 100. The square (10 – 10) is equal to 0. Plus (20 – 10) squares, ie 100. Plus (30 – 10) squares, ie 400. We divide this by 5. What is the answer? 400 + 100 = 500, plus 500 = 1000. 1000/5 = 200. In this example, the variance is 200. This is the size of this spread. Let's compare it with another group. Let's make it less common let's compare with the dispersion group. Let's screen a bit let's step aside. There is a little space above. Let's start. Let's calculate the variance of this group. We know the numerical mean of this. Let's calculate the variance: (8 – 10) square + (9 – 10) square, plus (10 – 10) square, plus (11 – 10) squares, plus (12 – 10) square. Remember, the numerical mean is 10. We must first calculate the numerical mean. Here 1, There are 2, 3, 4, 5 difference squares.

Let's calculate: 8 – 10 = -2. -2 squares = 4. 9 – 10 = -1. -1 square = 1. 10 – 10 = 0, 0 square = 0. 11 – 10 = 1. 1 square = 1. 12 – 10 = 2. 2 squares = 4. What does this mean? We have to divide this by 5. This is 10/5. We can simplify this. 10/5 expression Is equal to 2. The variance here … Let's make sure it's true. Here 10/5 is taken. Hence, the variance of the group with the lowest prevalence is smaller. The variance of this group is 2. It is clear. From this we understand that it is less prevalent is a group. To find the variance here we found the difference between these numbers and the numerical average and we squared it.

An arbitrary price here can be obtained. Their distance Assume that. For example, it is -10 meters, 0 meters, 10 meters, 8 meters and so on. When you find the square of them, variance is expressed in square meters. This is a strange group. If we want to express this with a standard tendency, we can fatten the variance. That is, the square of sigma in the abdomen.

A symbol of standard inclination is sigma. Since we found the variance, easily the standard trend price in both groups we can find. The standard tendency of the first group is equal to 200 in fat. What is 200 at the bottom? Shoot 2 at the root 100. This is equal to 2 in 10 fattening. This is the answer of the first group. The standard trend price of the second group is equal to the value of the variance at the bottom, ie 2 at the bottom. That is, the standard trend of the second group is the price of the first group Is equal to 1/10. This is 2 in 10 fat, which is 2 in fat. That is 10 times the standard trend. I hope it became clear. Let's think about it. Its standard tendency is 10 times greater than its standard tendency. Remember how we calculated it.

To find the variance, the numerical mean of each point found the difference, squared the answer, we find the average. When you overwhelm the answer, the standard tendency of the first group 10 times more than the standard tendency of the second group we see that. Let's look at both groups. This is 10 times more than the standard trend. Ok? There are 10 in both groups. 9 differs by 1 unit from 10, 0 differs by 10 units from 10, that is, less than 10 units. 8 differs from 10 by 2 units. These numbers differ by 20 units. That is, on average, it differs 10 times. I think through the standard trend to see how they differ numerically is better. I hope you found this video useful..