Mean squared deviation
When we talk about mean squared deviation, we are talking about how much different numbers are from each other. Imagine you have some numbers like 2, 4, 6, and 8. These numbers are different from each other, but how much are they different by?
To figure this out, we need to do a few steps. First, we need to find the average of these numbers. The average means adding up all the numbers and then dividing by how many numbers there are. So in our example, we would add 2 + 4 + 6 + 8, which equals 20. Then we would divide that by 4 (since there are 4 numbers), which equals 5. So the average of our numbers is 5.
Next, we want to figure out how much each number is different from the average. We do this by subtracting the average from each number. For example, 2 - 5 = -3, 4 - 5 = -1, 6 - 5 = 1, and 8 - 5 = 3. These are all the differences between each number and the average.
But we don't just want to add up these differences, because some are negative and some are positive, and we don't want them to cancel out. So instead, we square each difference. Squaring means multiplying a number by itself. So when we square -3, we get 9, because (-3) x (-3) = 9. And when we square 1, we get 1, because 1 x 1 = 1.
Finally, we want to find the average of these squared differences. This is the mean squared deviation. To do this, we add up all the squared differences and then divide by how many there are (in our case, 4). So we would add up 9 + 1 + 1 + 9, which equals 20. Then we would divide by 4, which equals 5. So the mean squared deviation of our numbers (2, 4, 6, and 8) is 5.
So what does this really mean? It means that on average, each number is 5 away from the average of 5. And by squaring the differences and finding the mean, we can compare the spread of numbers in a more meaningful way.
To figure this out, we need to do a few steps. First, we need to find the average of these numbers. The average means adding up all the numbers and then dividing by how many numbers there are. So in our example, we would add 2 + 4 + 6 + 8, which equals 20. Then we would divide that by 4 (since there are 4 numbers), which equals 5. So the average of our numbers is 5.
Next, we want to figure out how much each number is different from the average. We do this by subtracting the average from each number. For example, 2 - 5 = -3, 4 - 5 = -1, 6 - 5 = 1, and 8 - 5 = 3. These are all the differences between each number and the average.
But we don't just want to add up these differences, because some are negative and some are positive, and we don't want them to cancel out. So instead, we square each difference. Squaring means multiplying a number by itself. So when we square -3, we get 9, because (-3) x (-3) = 9. And when we square 1, we get 1, because 1 x 1 = 1.
Finally, we want to find the average of these squared differences. This is the mean squared deviation. To do this, we add up all the squared differences and then divide by how many there are (in our case, 4). So we would add up 9 + 1 + 1 + 9, which equals 20. Then we would divide by 4, which equals 5. So the mean squared deviation of our numbers (2, 4, 6, and 8) is 5.
So what does this really mean? It means that on average, each number is 5 away from the average of 5. And by squaring the differences and finding the mean, we can compare the spread of numbers in a more meaningful way.
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