6.2: How Do You Determine Variance?
- Page ID
- 50310
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)To determine variance, think, “How do we determine the spread of scores around the mean?” or “How do we determine how far these scores are away from the mean?” To determine variance, we start with deviation scores. A deviation score is a score’s value minus the mean score. We calculate a deviation score from every value in our variable and determine how far away it is from the mean.
The sum of deviation scores is always zero. Deviation scores are always deviations above and below the mean, so those scores will cancel each other out. Hence, the sum of the deviation scores is always zero. That is not helpful for determining variance. To solve the problem of deviation scores cancelling each other out, we take each deviation score, square it so the score takes on a positive value, add them, and then divide by the number of scores.
Notice that we do not take the absolute value of the deviation score; we square the deviation score. Why? Because, for some reason, the absolute value doesn't work in mathematical equations for statistics. Go figure...
The better reason for this is– that variance consists of the mean sum of squares of the deviation scores because we square them because we want the area underneath the curve, otherwise known as the normal distribution. You are going to hear and read a lot about the sum of squares. The sum of squares refers to the deviation scores.
Recall from geometry that area is calculated by length times width. Or, in the case of a square, length squared. So, we square to calculate the area.
Why do we calculate the area? Because we are interested in the area underneath the curve or the normal distribution. The area is the percentage of scores that are expected to fall under that section of the distribution.
Why is this important? Fast forward to chapter eight on significance. Knowing the area underneath the curve is important because:
- Scores that have a high percentage are likely to occur.
- Scores with a low percentage, perhaps less than 5% of the time, are less likely to occur strictly by chance.
- We are interested in those scores that do not occur that often, such as less than 5% of the time.
As you can surmise, calculating the area underneath the curve is part of the process of understanding which results are significant. We know if a result is significant if it does not occur that often. How often? We say an observation is significant if it only occurs 5% of the time or 5% of all of our observations. That percentage sign is basically a proportion of the area underneath the curve. We calculate area by squaring the deviation scores, which is basically the length of each “square” underneath the curve. Fast forward, if you recognize that 5%, it is because it is ubiquitous, p < .05. More on that later in chapter eight on significance.
Knowing that variance is calculated based on the sum of squares and assuming a normal distribution, we know that the following percentages occur underneath the normal distribution.
The reason we make a big deal about determining if our distribution is normal is that these percentages are only valid if the distribution is normal. These percentages are necessary for establishing what observations are significant or not significant. If the distribution is not normal, then we cannot establish these percentages. Some other model of our variable’s distribution is needed, and that means you need a completely different set of statistical analyses, and this book will not address those analyses. And that is when you need a statistical consultant.