8.4: How to Determine if a Result is Significant
- Page ID
- 50674
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Statistically, we use the alpha level to determine if a result is significant.
When you see alpha, think about probability. The alpha level is usually represented with an italicized p; usually, you read p < .05. Sometimes, we use the alpha symbol α, and usually, we write α = .03 to represent the probability associated with a statistical test. The p represents probability. Think of the alpha level as a cut-off probability. Based on the normal distribution, the probability of any outcome purely by chance alone ranges from 1% to 100%.
Based on the normal distribution, recall that results that happen with high frequency and high probability are common, boring, uninteresting, and not significant. Results that happen with low frequency and low probability are not common, interesting, or significant. We want rare results, and by rare, we mean there is a low probability that the result will occur by random chance.
How low is a low probability, or how rare does a result have to be that we are confident that it does not occur by chance?
The alpha level, the level of significance, is a percentage. A percentage of what? A percentage of the normal distribution. Recall that the normal distribution is basically an X-Y graph of the possible values of a given variable on the X-axis, and the Y-axis is the frequency count or the number of times that value occurred. Recall that for a normal distribution, the most frequent occurrence of values occurs around the mean, the highest part of the bell curve. The tails are a) low frequency occurring values and b) far away from the mean. Recall that the distribution is the dispersion of scores, and we use sums of squares, or we square the dispersion of scores, to get the area underneath the curve. The normal curve is shaped such that most of the area underneath the curve is around the mean, hence the wide and tall portion of the curve. That large area underneath the curve means a lot of observations fall in that area because these observations are frequent, close to the mean, and not that interesting. What we want for significance is something greater than the average or that has a small chance of occurring. That area underneath the curve is small, and that area is far away from the mean and short, meaning those observations do not happen that often based on random chance.
When we say alpha, we mean percentage, and the percentage of the area underneath the curve. When you read the ubiquitous, p < .05, what that means is that the alpha level is set at .05, which is a 5% probability that the result occurred by chance. Think “this result will only occur by chance 5% of the time. That’s a pretty low probability.” Think of the alpha level as a cutoff point that we place somewhere along the probability line. In statistical terms, we call that the region of rejection. Rejecting what? Rejecting the null hypothesis that says the results are due to chance and the result is not significant.
BTW, why 5%? Why not 10% or 1%? Who died and set the alpha at 5%? We answer that question by discussing Type I and Type II errors.