# Formula for standard deviation and variance relationship

### Absolute Deviation & Variance | How and when to use these measures of spread | Laerd Statistics

The standard deviation of the mean (SD) is the most commonly used measure of The unbiased estimate of population variance calculated from a sample is. Both the variance and the standard deviation meet these three criteria for Here is the equation for calculating the mean, μx, of our data set using the. Expected Return, Variance And Standard Deviation Of A Portfolio . asset class. Expected return is calculated by using the following formula: Portfolio variance looks at the covariance or correlation coefficient for the securities in the portfolio.

So that won't work.

### Standard Deviation and Variance

How about we use absolute values? That looks good and is the Mean Deviationbut what about this case: It also gives a value of 4, Even though the differences are more spread out! So let us try squaring each difference and taking the square root at the end: The Standard Deviation is bigger when the differences are more spread out In fact this method is a similar idea to distance between pointsjust applied in a different way.

And it is easier to use algebra on squares and square roots than absolute values, which makes the standard deviation easy to use in other areas of mathematics.

Covariance and Correlation Covariance and correlation describe how two variables are related. Variables are positively related if they move in the same direction.

## Variance & Standard Deviation

Variables are inversely related if they move in opposite directions. Both covariance and correlation indicate whether variables are positively or inversely related. Correlation also tells you the degree to which the variables tend to move together.

You are probably already familiar with statements about covariance and correlation that appear in the news almost daily.

For example, you might hear that as economic growth increases, stock market returns tend to increase as well.

These variables are said to be positively related because they move in the same direction. You may also hear that as world oil production increases, gasoline prices fall. These variables are said to be negatively, or inversely, related because they move in opposite directions. However, the problem is that because we have both positive and minus signs, when we add up all of these deviations, they cancel each other out, giving us a total deviation of zero.

Since we are only interested in the deviations of the scores and not whether they are above or below the mean score, we can ignore the minus sign and take only the absolute value, giving us the absolute deviation. Adding up all of these absolute deviations and dividing them by the total number of scores then gives us the mean absolute deviation see below.

Therefore, for our students the mean absolute deviation is Join the 10,s of students, academics and professionals who rely on Laerd Statistics.

Unlike the absolute deviation, which uses the absolute value of the deviation in order to "rid itself" of the negative values, the variance achieves positive values by squaring each of the deviations instead. Adding up these squared deviations gives us the sum of squares, which we can then divide by the total number of scores in our group of data in other words, because there are students to find the variance see below.

Therefore, for our students, the variance is As a measure of variability, the variance is useful. If the scores in our group of data are spread out, the variance will be a large number. Conversely, if the scores are spread closely around the mean, the variance will be a smaller number.

However, there are two potential problems with the variance. First, because the deviations of scores from the mean are 'squared', this gives more weight to extreme scores. If our data contains outliers in other words, one or a small number of scores that are particularly far away from the mean and perhaps do not represent well our data as a wholethis can give undo weight to these scores.

Secondly, the variance is not in the same units as the scores in our data set: