2.1: Statistics Involves Comparison and Context
- Page ID
- 48878
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Statistics help us to understand variation by using comparison. Variation helps us to set up a comparison – “This is different than that.” Comparison is basically this question: “This is different compared to what?” We use variation all the time – “This person is taller than this person”; “This person is more depressed than that person”; “This person has severe ADHD compared to that person.” To show that comparison, we need a context. The context, in this case, is comparing something to something.
For example, suppose we want to study eating disorders among different ethnic groups. We might find that Asian American women struggle more with eating disorders compared to Caucasians. So, the variable is eating disorders; the level is more struggle or less struggle with an eating disorder. We find that Asian American women struggle more, but more compared to what? That context is the racial group. Asian American women struggle more compared to Caucasians. What is the basis for that context? Alternatively, what is the conceptual basis for this comparison? Suppose we say the difference we are noticing between Asian Americans and Caucasians is due to cultural values. In Asian American culture, there is more cultural emphasis on thinness and more family criticism of weight gain compared to Caucasian culture.
Why am I making a big deal about this? Because to understand statistics, you must understand the numbers by understanding the context in which those numbers are generated. This context consists of the comparison, which is comparing what to what, and the conceptual basis for that comparison, which is why we are comparing this to that. Students think that the number exists by itself. That is not true. You cannot interpret a number and its accompanying statistical values without understanding the context of the number. The context always involves a comparison and the conceptualization basis for why these numbers vary.
A slight fast forward – the context is necessary to show statistical significance. This finding is significant but compared to what and under what context. Consider these statements -
“This is significantly different from that because the numbers show that this issue is more severe than usual.”
"This person’s depression is more severe than what we typically encounter in this counseling center. May need to hospitalize this person."
Without the numbers, we must rely on verbal descriptions of what entities are different from each other.
We will discuss statistical significance in greater detail. But for now, when you think about statistics, always think in terms of this statement: “What is being compared to what?”
Knowing that statistics involve comparison is important because all statistics involve ratios. Go back to your math classes and recall that a ratio involves a numerator over a denominator. That ratio involves a comparison – a numerator compared to the denominator.
In statistics, these ratios have many forms. The most common form is the following: Actual effects vs. random effects. Actual effect is what happened, or your result, your outcome, versus a random effect, which is a random chance.
Another example consists of changes in scores versus error shifts. An actual change in score can be a depression score before treatment starts compared to a depression score at the end of treatment. This change is compared to error shifts, such as random fluctuations in depression scores that occur over time.
Another example is this group versus that group. A treatment group’s mean score on a depression scale is compared to a control group’s mean score on a depression scale.
In all of these examples, we are comparing variations that are due to what actually happened to variations that are due to random chance. In most cases, we are hoping to find that there are true variations rather than chance variations. We want to see that a decrease in depression from before treatment to after treatment is due to depression actually improving. We want to see that the patients in the treatment group are doing better than the control group because the treatment actually worked, and random chance has nothing to do with the variations in the treatment effects. We are hoping to find true variations rather than chance variations.