We next examine the idea of measuring the spread of values in a data set, the amount of dispersion or scattering of the data values. We are talking about the characteristic variability of a data set.
The deviation of each individual value, `x`, in a data set `X` from the data set's mean can be described by:
eq 1: `X-barX = x`. These deviation values then represent the variability among a set of raw scores. The larger are the values of the deviation, x, the more the data in `X` varies from the mean and we say the data set, `X` has large or high variability.
Variance is defined as the average of the squared differences from the Mean. So variance is represent as:
eq 2: `sigma^2 = (sum_(i=1)^n(X_i - barX))/n`
We look at ways to express in a single value the amount of variability and we settle on our next vCalc equation for Standard Deviation (SD, or `sigma`). Deriving from our equation 1 for deviation we come to the Standard Deviation (SD) of a population which is expressed as follows:
eq 3:`SD = sigma = sqrt( (sum_(i=1)^N( x_i^2))/N) = sqrt((sum_(i=1)^N(X_i-barX)^2)/N)`
To represent the SD of a sample we use tilde (~) over the `tildesigma`
eq 4: `tildesigma = sqrt((sum_(i=1)^N(X_i-barX)^2)/(N-1)`
See the next page in Elementary Statistics for a discussion of population versus sample statistics.
NOTE: many vCalc equations are embedded throughout vCalc descriptive pages like this page. Even though they may not stand out in the text, if you hover over the name of an equation it will likely be linked to an actual, pop-up executable equation. For example: Arithmetic Mode