Variance and Standard Deviation

Similar the Standard Deviation.

Variance:

It measures the spread of observations with respect to the mean. IE Calculates the average of the squared differences from the mean.

How could you calculate it?

• Find the mean
• Calculate the difference between each item and the mean
• Square the differences
• Get the average of the squared differences
• ${\sigma }^2=E[X-{\mu }^2]$

What about for the sample variance:

• When finding the mean, divide by $n-1$ and not $n$, as this will help
• Why?
  • Remember the sample variance is often too small when compared than the population bias

Standard Deviation:

The square root of the variance and provides a measure of the spread of the distribution in the same units of data. Gives a sense of the average distance from the mean.

Calculate it:

$\sigma =\sqrt{\sigma {}^2}$

Covariance Vs Variance:

• Variance: The measure of how much the values in the dataset differ from the mean of the data (Are the arrows spread out or tightly packed around the bullseye?)
  • ${\sigma }^2=\frac{1}{n-1}{\sum }_{i=1}^n(x_i-\bar{x}{)}^2$
• Covariance: The measure of how correlated two variables, X and Y, are. (E.G. Collecting the data to see that taller people are heavier - positive covariance).
  • $\text{Cov}(X,Y)=\frac{1}{n-1}{\sum }_{i=1}^n(X_i-\bar{X})(Y_i-\bar{Y})$