Common Formulas
- Page ID
- 22217
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The following formulas are in the order in which you learn about them in this textook. Use the Table of Contents to look for a specific equation.
Descriptive Statistics
Mean
\[ \displaystyle \bar{X} = \dfrac{\sum X}{N} \]
Standard Deviation
\[s=\sqrt{\dfrac{\sum(X-\overline {X})^{2}}{N-1}} \]
Which is also: \(s=\sqrt{\dfrac{\sum(X-\overline {X})^{2}}{N-1}}=\sqrt{\dfrac{S S}{d f}} \)
Some instructors prefer this formula because it is easier to calculate (but more difficult to see what's happening):
\[\sqrt{ \dfrac{\left(\sum(X^2) - \dfrac{(\sum{X})^2}{N}\right)}{(N-1)}}\]
z-score
To find the z-score when you have a raw score:
\[z=\frac{X-\bar{X}}{s}\]
To find a raw score when you have a z-score:
\[x=z s+\overline{X} \]
t-tests
One-Sample t-test
These are the same formulas, but formatted slightly differently.
\[t = \cfrac{(\bar{X}-\mu)}{\left(\cfrac{s} {\sqrt{n}}\right)} \]
Confidence Interval
\[\text {Margin of Error }=t \times \left(\dfrac{s}{\sqrt{N}}\right) \nonumber \]
\[\text { Confidence Interval }=\overline{X} \pm (t \times \left(\dfrac{s}{\sqrt{N}}\right)) \]
Independent Sample t-test
Unequal N
You can always use this formula:
\[t=\dfrac{(\bar{X}_{1}-\bar{X}_{2})}{\sqrt{\left[\dfrac{\left(n_{1}-1\right) \times s_{1}^{2} + \left(n_{2}-1\right) \times s_{2}^{2}}{n_{1}+n_{2}-2}\right] \times \left(\dfrac{1}{n_{1}} + \dfrac{1}{n_{2}}\right)}} \]
Equal N
You should only use this formula when your two independent groups are the same size (N), meaning the same number of people in each group.
\[\dfrac{(\bar{X_1} - \bar{X_2})}{\sqrt{\left(\frac{s_1^2}{N_1}\right)+\left(\frac{s_2^2}{N_2}\right)}}\]
Dependent Sample t-test
Conceptual Formula (symbols)
\[ t = \cfrac{\overline{X}_{D}}{\left(\cfrac{s_{D}}{\sqrt{N}} \right)} \]
Full Formula
\[ t = \cfrac{ \left(\cfrac{\Sigma {D}}{N}\right)} { {\sqrt{\left(\cfrac{\sum\left((X_{D}-\overline{X}_{D})^{2}\right)}{(N-1)}\right)} } /\sqrt{N} } \]
ANOVA
Sums of Squares for Between Groups Designs
Between Groups
\[S S_{B}=\sum_{EachGroup} \left[ \left(\overline{X}_{group}-\overline{X}_{T}\right)^{2} \times (n_{group}) \right] \]
Within Groups
\[S S_{W}=\sum_{EachGroup} \left[ \sum \left(\left(X-\overline{X}_{group}\right)^{2}\right) \right] \]
Total
\[S S_{T}=\sum \left[ \left(X - \overline{X}_{T}\right)^{2} \right] \]
Tukey's HSD for Pairwise Comparison
\[ HSD = q \times \sqrt{\dfrac{MSw}{n_{group}}} \]
Sums of Squares for Repeated Measures Designs
Between Groups
Same as above.
Participants
\[SS_{Ps} = \left[\sum{\left(\dfrac{(\sum{X_{Ps}})^2}{k}\right)}\right] -\dfrac{\left((\sum{X})^2\right)}{N} \]
Within Groups (Error)
\[SS_{WG} = SS_{T} - SS_{BG} - S_{P} \nonumber \]
Total
Same as above.
Pearson's r (Correlation)
The following formulas are the same. Use the first one when you already have the standard deviation calculated.
These are paired data, so N is the number of pairs.
SD Already Calculated:
\[ r= \cfrac{ \left( \cfrac{\sum ((x_{Each} - \bar{X_x})\times(y_{Each} - \bar{X_y}) ) }{(N-1)}\right) } {(s_x \times s_y)} \]
SD Not Calculated:
\[ r = \cfrac{ \left( \cfrac{\sum ((x - \bar{X_x})\times(y - \bar{X_y}) ) }{(N-1)}\right) } {\left( \sqrt{\dfrac{\sum\left((x-\overline {X_x})^{2}\right)}{N-1}} \right) \times \left( \sqrt{\dfrac{\sum\left((y-\overline {X_y})^{2}\right)}{N-1}} \right)} \]
Regression Line Equation
\[\widehat{\mathrm{Y}}=\mathrm{a}+(\mathrm{b}\times{X}) \]
a (intercept):
\[\mathrm{a}=\overline{X_y}- (\mathrm{b} \times \overline{X_x}) \]
b (slope):
\[ \dfrac{\sum(Diff_{x} \times Diff_{y})}{\sum({Diff_{X}}^2)} \]
In which "Diff" means the differences between each score and that variable's mean.
Pearson's \(\chi^2\) (Chi-Square)
\[\chi^{2}=\sum_{Each}\left(\dfrac{\left(E-O\right)^{2}}{E} \right)\]
Expected Frequencies
Goodness of Fit:
\[\dfrac{N}{k}\]
Test of Independence:
\[E_{EachCell}=\dfrac{RT \times CT}{N} \]
In which RT = Row Total and CT = Column Total