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% Document Info
\author{SH (Sana\_H)}
\pdfinfo{
  /Title (statistics-in-behavioral-sciences.pdf)
  /Creator (Cheatography)
  /Author (SH (Sana\_H))
  /Subject (Statistics in Behavioral Sciences Cheat Sheet)
}

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\noindent
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    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Statistics in Behavioral Sciences Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{SH (Sana\_H)} via \textcolor{DarkBackground}{\uline{cheatography.com/164538/cs/36366/}}}
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\noindent
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}SH (Sana\_H) \\
  \uline{cheatography.com/sana-h} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 17th January, 2023.\\
   Updated 16th January, 2023.\\
   Page {\thepage} of \pageref{LastPage}.
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{{\bf{Statistics}}}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{the branch of mathematics in which data are used descriptively or inferentially to find or support answers for scientific and other quantifiable questions. \newline % Row Count 4 (+ 4)
It encompasses various techniques and procedures for recording, organizing, analyzing, and reporting quantitative information.% Row Count 7 (+ 3)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{2.052 cm} x{2.356 cm} x{3.192 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Difference - parametric test \& non-parametric test}}  \tn
% Row 0
\SetRowColor{LightBackground}
\seqsplit{PROPERTIES} & PARAMETRIC & NON-PARAMETRIC \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\seqsplit{assumptions} & YES & NO \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
value for central tendency & mean & median/mode \tn 
% Row Count 6 (+ 3)
% Row 3
\SetRowColor{white}
\seqsplit{probability} \seqsplit{distribution} & normally distributed & user specific \tn 
% Row Count 9 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\seqsplit{population} knowledge & required & not required \tn 
% Row Count 11 (+ 2)
% Row 5
\SetRowColor{white}
used for & interval data & nominal, ordinal data \tn 
% Row Count 13 (+ 2)
% Row 6
\SetRowColor{LightBackground}
\seqsplit{correlation} & pearson & spearman \tn 
% Row Count 15 (+ 2)
% Row 7
\SetRowColor{white}
tests & t test, z test, f test, ANOVA & Kruskal Wallis H test, Mann-whitney U, Chi-square \tn 
% Row Count 19 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Correlation Coefficient}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{a statistical measure of the strength of the relationship between the relative movements of two variables  \newline % Row Count 3 (+ 3)
value ranges from {\bf{-1 to +1}} \newline % Row Count 4 (+ 1)
-1 = perfect negative or inverse correlation \newline % Row Count 5 (+ 1)
+1 = perfect positive correlation or direct relationship  \newline % Row Count 7 (+ 2)
0 = no linear relationship% Row Count 8 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{4 cm} x{4 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Alternatives}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{PARAMETRIC}} & {\bf{NON-PARAMETRIC}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
one sample z test, one sample t test & one sample sign test \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
one sample z test, one sample t test & one sample Wilcoxon signed rank test \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
two way ANOVA & Friedman test \tn 
% Row Count 6 (+ 1)
% Row 4
\SetRowColor{LightBackground}
one way ANOVA & Kruskal wallis test \tn 
% Row Count 7 (+ 1)
% Row 5
\SetRowColor{white}
independent sample t test & mann-whitney U test \tn 
% Row Count 9 (+ 2)
% Row 6
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one way ANOVA & mood's median test \tn 
% Row Count 10 (+ 1)
% Row 7
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pearson correlation & spearman correlation \tn 
% Row Count 11 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
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\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Paired t-test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{to compare means of two related groups}} \newline % Row Count 1 (+ 1)
ex. compare weight of 20 mice before and after treatment \newline % Row Count 3 (+ 2)
{\emph{two conditions:}} \newline % Row Count 4 (+ 1)
- pre post treatment \newline % Row Count 5 (+ 1)
- two diff conditions ex two drugs \newline % Row Count 6 (+ 1)
{\emph{ASSUMPTIONS}} \newline % Row Count 7 (+ 1)
- random selection \newline % Row Count 8 (+ 1)
- normally distributed \newline % Row Count 9 (+ 1)
- no extreme outliers \newline % Row Count 10 (+ 1)
{\emph{FORMULA}} \newline % Row Count 11 (+ 1)
{\bf{t= m / s/√n}} \newline % Row Count 12 (+ 1)
m= sample mean of differences \newline % Row Count 13 (+ 1)
{\bf{df= n-1}}% Row Count 14 (+ 1)
} \tn 
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\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{t-distribution}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{aka Student's t-distribution}} = probability distribution similar to normal distribution but {\bf{has heavier tails}} \newline % Row Count 3 (+ 3)
used to estimate pop parameters for small samples \newline % Row Count 4 (+ 1)
{\emph{Tail heaviness is determined by {\bf{degrees of freedom}} }} = gives lower probability to centre, higher to tails than normal distribution,  also have higher kurtosis, symmetrical, unimodal, centred at 0, larger spread around 0 \newline % Row Count 9 (+ 5)
{\bf{df = n - 1}} \newline % Row Count 10 (+ 1)
above 30df, use z-distribution \newline % Row Count 11 (+ 1)
{\bf{t-score}} = no of SD from mean in a t-distribution \newline % Row Count 13 (+ 2)
we find: \newline % Row Count 14 (+ 1)
- upper and lower boundaries \newline % Row Count 15 (+ 1)
- p value  \newline % Row Count 16 (+ 1)
{\bf{TO BE USED WHEN:}} \newline % Row Count 17 (+ 1)
- small sample \newline % Row Count 18 (+ 1)
- SD is unknown \newline % Row Count 19 (+ 1)
{\bf{ASSUMPTIONS}} \newline % Row Count 20 (+ 1)
- cont or ordinal scale \newline % Row Count 21 (+ 1)
- random selection \newline % Row Count 22 (+ 1)
- NPC \newline % Row Count 23 (+ 1)
- equal SD for indep two-sample t-test% Row Count 24 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Two-sample z-test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{to determine if means of two independent populations are equal or different}} \newline % Row Count 2 (+ 2)
{\emph{to find out if there is significant diff bet two pop by comparing sample mean}} \newline % Row Count 4 (+ 2)
knowledge of: \newline % Row Count 5 (+ 1)
{\bf{SD}} and {\bf{sample \textgreater{}30 in each group}} \newline % Row Count 6 (+ 1)
eg. compare performance of 2 students, average salaries, employee performance, compare IQ, etc  \newline % Row Count 8 (+ 2)
FORMULA: \newline % Row Count 9 (+ 1)
z= x̄₁ - x̄₂ / √s₁\textasciicircum{}2\textasciicircum{}/n₁ + s₂\textasciicircum{}2\textasciicircum{}/n₂ \newline % Row Count 11 (+ 2)
s= SD \newline % Row Count 12 (+ 1)
formula: \newline % Row Count 13 (+ 1)
z= (x̄₁ - x̄₂) - (µ₁ - µ₂) / √σ₁\textasciicircum{}2\textasciicircum{}/n₁ + σ₂\textasciicircum{}2\textasciicircum{}/n₂ \newline % Row Count 15 (+ 2)
(µ₁ - µ₂) = hypothesized difference bet pop means% Row Count 17 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Point Biserial correlation}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{measures relationship between two variables}} \newline % Row Count 1 (+ 1)
{\bf{rpbi}} = correlation coefficient \newline % Row Count 2 (+ 1)
{\bf{one continuous variable (ratio/interval scale)}} \newline % Row Count 4 (+ 2)
{\bf{one naturally binary variable}} \newline % Row Count 5 (+ 1)
FORMULA: \newline % Row Count 6 (+ 1)
{\bf{rpb= M1-M0/Sn * √ pq}} \newline % Row Count 7 (+ 1)
Sn= SD% Row Count 8 (+ 1)
} \tn 
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\end{tabularx}
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\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Two-sample z-test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{to determine if means of two independent populations are equal or different}} \newline % Row Count 2 (+ 2)
{\emph{to find out if there is significant diff bet two pop by comparing sample mean}} \newline % Row Count 4 (+ 2)
knowledge of: \newline % Row Count 5 (+ 1)
{\bf{SD}} and {\bf{sample \textgreater{}30 in each group}} \newline % Row Count 6 (+ 1)
eg. compare performance of 2 students, average salaries, employee performance, compare IQ, etc  \newline % Row Count 8 (+ 2)
FORMULA: \newline % Row Count 9 (+ 1)
z= x̄% Row Count 10 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{z-test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\bf{for hypothesis testing}} \newline % Row Count 1 (+ 1)
to check whether means of two populations are equal to each other when pop variance is known \newline % Row Count 3 (+ 2)
we have knowledge of: \newline % Row Count 4 (+ 1)
- {\emph{SD/population variance and/or sample n=30 or more}} \newline % Row Count 6 (+ 2)
`if both unknown -\textgreater{} t-test`  \newline % Row Count 7 (+ 1)
{\bf{left-tailed}}  \newline % Row Count 8 (+ 1)
{\bf{right-tailed}} \newline % Row Count 9 (+ 1)
{\bf{two-tailed}} \newline % Row Count 10 (+ 1)
{\bf{REJECT NULL HYPOTHESIS IF Z STATISTIC IS STATISTICALLY SIGNIFICANT WHEN COMPARED WITH CRITICAL VALUE}} \newline % Row Count 13 (+ 3)
z-statistic/ z-score = no representing result from z-test \newline % Row Count 15 (+ 2)
z critical value divides graph into acceptance and rejection regions  \newline % Row Count 17 (+ 2)
if z stat falls in rejection region-\textgreater{} H0 can be rejected  \newline % Row Count 19 (+ 2)
{\bf{TYPES}} \newline % Row Count 20 (+ 1)
One-sample z-test \newline % Row Count 21 (+ 1)
Two-sample z-test% Row Count 22 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{ANOVA}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\bf{Analysis of Variance}} \newline % Row Count 1 (+ 1)
{\emph{comparing several sets of scores}} \newline % Row Count 2 (+ 1)
{\emph{to test if means of 3 or more groups are equal}} \newline % Row Count 3 (+ 1)
{\emph{comparison of variance between and within groups}} \newline % Row Count 5 (+ 2)
{\emph{to check if sample groups are affected by same factors and to same degree}} \newline % Row Count 7 (+ 2)
{\emph{compare differences in means and variance of distribution}} \newline % Row Count 9 (+ 2)
{\bf{ONE-WAY ANOVA}}=no of IVs \newline % Row Count 10 (+ 1)
single IV with different (2) levels/variations have measurable effect on DV \newline % Row Count 12 (+ 2)
{\emph{compare means of 2 or more indep groups}} \newline % Row Count 13 (+ 1)
aka: \newline % Row Count 14 (+ 1)
- one-factor ANOVA \newline % Row Count 15 (+ 1)
- one-way analysis of variance \newline % Row Count 16 (+ 1)
- between subjects ANOVA \newline % Row Count 17 (+ 1)
{\bf{Assumptions}} \newline % Row Count 18 (+ 1)
- independent samples \newline % Row Count 19 (+ 1)
- equal sample sizes in groups/levels \newline % Row Count 20 (+ 1)
- normally distributed  \newline % Row Count 21 (+ 1)
- equal variance \newline % Row Count 22 (+ 1)
{\bf{F test is used to check statistical significance}} \newline % Row Count 24 (+ 2)
higher F value -{}-\textgreater{} higher likelihood that difference observed is real and not due to chance \newline % Row Count 26 (+ 2)
{\emph{used in field studies, experiments, quasi-exp}} \newline % Row Count 27 (+ 1)
CONDITIONS: \newline % Row Count 28 (+ 1)
- min 6 subjects \newline % Row Count 29 (+ 1)
- sample no of samples in each group \newline % Row Count 30 (+ 1)
} \tn 
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{ANOVA (cont)}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{H0: µ1=µ2=µ3 ... µk i.e. all pop means are equal \newline % Row Count 2 (+ 2)
Ha: at least one µi  is different i.e atleat one of the k pop means is not equal to the others  \newline % Row Count 4 (+ 2)
{\bf{µi is the pop mean of group}}% Row Count 5 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Spearman Correlation}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{non-parametric version of Pearson correlation coefficient}} \newline % Row Count 2 (+ 2)
named after Charles Spearman \newline % Row Count 3 (+ 1)
denoted by {\bf{ρ}}(rho) \newline % Row Count 4 (+ 1)
determine the strength and direction of monotonic variables bet two variables measured at ordinal, interval or ratio levels \& whether they are correlated or not \newline % Row Count 8 (+ 4)
{\bf{monotonic function}}=one variable never increases or never decreases as its IV changes \newline % Row Count 10 (+ 2)
- monotonically increasing= as X increases, Y never decreases  \newline % Row Count 12 (+ 2)
- monotonically decreasing= as X increases, Y never increases  \newline % Row Count 14 (+ 2)
- not monotonic= as X increases, Y sometimes dec and sometimes inc \newline % Row Count 16 (+ 2)
for analysis with: ordinal data, continuous data  \newline % Row Count 17 (+ 1)
{\bf{uses ranks instead of assumptions of normality}} \newline % Row Count 19 (+ 2)
aka Spearman Rank order test \newline % Row Count 20 (+ 1)
{\bf{FORMULA:}} \newline % Row Count 21 (+ 1)
ρ= 1- 6Σdᵢ\textasciicircum{}2\textasciicircum{}/n(n\textasciicircum{}2\textasciicircum{}-1) \newline % Row Count 22 (+ 1)
di= difference between two ranks of each observation \newline % Row Count 24 (+ 2)
-1 to +1 \newline % Row Count 25 (+ 1)
+1 = perfect association of ranks  \newline % Row Count 26 (+ 1)
0= no association \newline % Row Count 27 (+ 1)
-1= perfect negative association of ranks  \newline % Row Count 28 (+ 1)
{\bf{closer the value to 0, weaker the association}} \newline % Row Count 29 (+ 1)
{\bf{Value Ranges}} \newline % Row Count 30 (+ 1)
} \tn 
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Spearman Correlation (cont)}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{0 to 0.3 = weak monotonic relationship \newline % Row Count 1 (+ 1)
0.4 to 0.6 = moderate strength monotonic relationship \newline % Row Count 3 (+ 2)
0.7 to 1 = strong monotonic relationship% Row Count 4 (+ 1)
} \tn 
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\end{tabularx}
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\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Parametric and Non-parametric test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Fixed set of parameters, certain assumptions about {\bf{distribution of population}} \newline % Row Count 2 (+ 2)
{\bf{PARAMETRIC}} - {\emph{prior knowledge of pop distribution i.e NORMAL DISTRIBUTION}} \newline % Row Count 4 (+ 2)
{\bf{NON-PARAMETRIC}} - {\emph{no assumptions, do not depend on population, DISTRIBUTION FREE tests, values found on nominal or ordinal level}} \newline % Row Count 7 (+ 3)
easy to apply, understand, low complexity \newline % Row Count 8 (+ 1)
{\emph{decision based on}} - distribution of population, size of sample \newline % Row Count 10 (+ 2)
parametric - mean \& \textless{}30 sample \newline % Row Count 11 (+ 1)
non-parametric - median/mode \& \textgreater{}30 sample or regardless of size% Row Count 13 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
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\begin{tabularx}{8.4cm}{x{4 cm} x{4 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Advantages \& Disadvantages - NON-PARAMETRIC TESTS}}  \tn
% Row 0
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{\bf{ADVANTAGES}} & {\bf{DISADVANTAGES}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
simple, easy to understand & less powerful than parametrics \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
no assumptions & counterpart parametric if exists, is more powerful \tn 
% Row Count 6 (+ 3)
% Row 3
\SetRowColor{white}
more versatile & not as efficient as parametric tests \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
easier to calculate & may waste information \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
hypothesis tested may be more accurate & requires larger sample to be as powerful as parametric test \tn 
% Row Count 13 (+ 3)
% Row 6
\SetRowColor{LightBackground}
small sample sizes are okay & difficult to compute large samples by hand \tn 
% Row Count 16 (+ 3)
% Row 7
\SetRowColor{white}
can be used for all types of data (nominal, ordinal, interval) & tabular format of data required that may not be readily available \tn 
% Row Count 20 (+ 4)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{can be used with data having outliers} \tn 
% Row Count 21 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
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\begin{tabularx}{8.4cm}{x{4.08 cm} x{3.92 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Application}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{PARAMETRIC TESTS}} & {\bf{NON-PARAMETRIC TESTS}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
- quantitative \& continuous data & - mixed data \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
- normally distributed & - unknown distribution of population \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
- data is estimated on {\bf{ratio}} or {\bf{interval}} scales & - different kinds of measurement scales \tn 
% Row Count 9 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{degrees of freedom}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{independent values in the data sample that have freedom to vary \newline % Row Count 2 (+ 2)
FORMULA: \newline % Row Count 3 (+ 1)
no of values in a data set minus 1 \newline % Row Count 4 (+ 1)
{\emph{df= N-1}}% Row Count 5 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{t-test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{statistical test to determine if significant difference between avg scores of two groups}} \newline % Row Count 2 (+ 2)
1908-William Sealy Gosset- {\bf{student t-test and t-distirbution}} \newline % Row Count 4 (+ 2)
{\emph{for hypothesis testing}} \newline % Row Count 5 (+ 1)
knowledge of: \newline % Row Count 6 (+ 1)
{\bf{distribution - normally distributed}} \newline % Row Count 7 (+ 1)
{\bf{no knowledge of SD}} \newline % Row Count 8 (+ 1)
{\bf{TYPES:}} \newline % Row Count 9 (+ 1)
{\bf{one-sample t-test}} - single group \newline % Row Count 10 (+ 1)
{\bf{FORMULA:}} \newline % Row Count 11 (+ 1)
{\bf{t= m - µ / s/√n}} \newline % Row Count 12 (+ 1)
{\bf{SD FORMULA:}} \newline % Row Count 13 (+ 1)
{\bf{σ= √Σ(X-µ)\textasciicircum{}2\textasciicircum{} / N}} \newline % Row Count 14 (+ 1)
{\bf{s= √Σ(X-µ)\textasciicircum{}2\textasciicircum{} / n-1}} \newline % Row Count 15 (+ 1)
{\bf{independent two-sample t-test}} - two groups \newline % Row Count 16 (+ 1)
{\bf{paired/dependent samples t-test}} - sig diff in paired measurements, compares means from same group at diff times (test-retest sample) \newline % Row Count 19 (+ 3)
H0: no effective difference = {\bf{measured diff is due to chance}} \newline % Row Count 21 (+ 2)
Ha: two-tailed/ one-tailed {\emph{nonequivalent means/smaller or larger than hypothesized mean}} \newline % Row Count 23 (+ 2)
PERFORM {\bf{two-tailed test}}: to find out difference bet two populations \newline % Row Count 25 (+ 2)
{\bf{one-tailed}}: one pop mean is \textgreater{} or \textless{} other% Row Count 26 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Independent two-sample t-test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{aka unpaired t-test}} \newline % Row Count 1 (+ 1)
{\bf{to compare mean of two independent groups}} \newline % Row Count 2 (+ 1)
ex. avg weight of males and females  \newline % Row Count 3 (+ 1)
two forms: \newline % Row Count 4 (+ 1)
- {\bf{student's t-test}}: assumes SD is equal \newline % Row Count 5 (+ 1)
- {\bf{welch's t-test}}: less restrictive, no assumption of equal SD  \newline % Row Count 7 (+ 2)
{\emph{both provide more/less similar results}} \newline % Row Count 8 (+ 1)
ASSUMPTIONS: \newline % Row Count 9 (+ 1)
- normally distributed \newline % Row Count 10 (+ 1)
- SD is same \newline % Row Count 11 (+ 1)
- independent groups \newline % Row Count 12 (+ 1)
- randomly selected \newline % Row Count 13 (+ 1)
- independent observations \newline % Row Count 14 (+ 1)
- measured on {\bf{interval}} or {\bf{ratio}} scale \newline % Row Count 15 (+ 1)
FORMULA: \newline % Row Count 16 (+ 1)
{\bf{t= x̄₁ - x̄₂ / √s₁2/n₁ + s₂2/n₂}} \newline % Row Count 18 (+ 2)
{\bf{df= n1 + n2 - 2}} \newline % Row Count 19 (+ 1)
{\bf{S= √Σ (x1-x̄)\textasciicircum{}2\textasciicircum{} + (x2-x̄)\textasciicircum{}2\textasciicircum{} / n1+n2-2}}% Row Count 20 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{One-sample z-test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{to check if difference between sample mean \& population mean when SD is known}}  \newline % Row Count 2 (+ 2)
FORMULA: \newline % Row Count 3 (+ 1)
z=x-µ/SE \newline % Row Count 4 (+ 1)
SE=σ/√n \newline % Row Count 5 (+ 1)
{\bf{z score}} is compared to a {\bf{z table}} (includes \% under NPC bet mean and z score), tells us whether the z score is due to chance or not  \newline % Row Count 8 (+ 3)
{\bf{conditions:}} \newline % Row Count 9 (+ 1)
knowledge of: \newline % Row Count 10 (+ 1)
- pop mean \newline % Row Count 11 (+ 1)
- SD \newline % Row Count 12 (+ 1)
- simple random sample \newline % Row Count 13 (+ 1)
- normal distribution \newline % Row Count 14 (+ 1)
{\emph{two approaches to reject H0:}} \newline % Row Count 15 (+ 1)
- {\bf{p-value approach}} - p-value is the smallest level of significance at which H0 can be rejected...{\emph{smaller p-value, stronger evidence}} \newline % Row Count 18 (+ 3)
-{\bf{critical value approach}} - comparing z stat to critical values... indicate boundary regions where stat is highly improbable to lie= critical regions/rejection regions \newline % Row Count 22 (+ 4)
if z stat is in critical region-\textgreater{} reject H0 \newline % Row Count 23 (+ 1)
{\emph{based on}}: \newline % Row Count 24 (+ 1)
significance level (0.1, 0.05, 0.01), alpha level, Ha% Row Count 26 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Biserial correlation}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{to measure relationship between {\bf{quantitative variables}} and {\bf{binary variables}} }} \newline % Row Count 2 (+ 2)
given by Pearson - 1909 \newline % Row Count 3 (+ 1)
biserial correlation coeff varies bet {\bf{-1}} and {\bf{1}} \newline % Row Count 5 (+ 2)
0= no association \newline % Row Count 6 (+ 1)
ex. IQ scores and pass/fail correlation \newline % Row Count 7 (+ 1)
{\bf{continuous variable}} and {\bf{binary variable}} (dichotomised to create binary variable) \newline % Row Count 9 (+ 2)
{\bf{rbis}} or {\bf{rb}} = correlation index estimating strength of relationship between artificially dichotomous variable and a true continuous variable \newline % Row Count 12 (+ 3)
ASSUMPTIONS: \newline % Row Count 13 (+ 1)
- data measured on continuous scale \newline % Row Count 14 (+ 1)
- one variable to be made dichotomous \newline % Row Count 15 (+ 1)
- no outliers \newline % Row Count 16 (+ 1)
- approx normally distributed \newline % Row Count 17 (+ 1)
- equal variances (SD) \newline % Row Count 18 (+ 1)
{\bf{FORMULA}} \newline % Row Count 19 (+ 1)
{\bf{rb= M1-M0/SDt * pq/y }} \newline % Row Count 20 (+ 1)
M1=mean of grp 1 \newline % Row Count 21 (+ 1)
M2= mean of grp 2 \newline % Row Count 22 (+ 1)
p= ratio of grp 1 \newline % Row Count 23 (+ 1)
q= ratio of grp 2 \newline % Row Count 24 (+ 1)
SDt= total SD \newline % Row Count 25 (+ 1)
y= ordinate% Row Count 26 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Pearson Correlation}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{measures {\bf{strength and direction}} of a linear relationship between two variables}} \newline % Row Count 2 (+ 2)
how two data sets are correlated  \newline % Row Count 3 (+ 1)
{\emph{gives us info about the slope of the line}} \newline % Row Count 4 (+ 1)
{\bf{r}} \newline % Row Count 5 (+ 1)
aka: \newline % Row Count 6 (+ 1)
- Pearson's r \newline % Row Count 7 (+ 1)
- bivariate correlation \newline % Row Count 8 (+ 1)
- Pearson product-moment correlation coefficient (PPMCC) \newline % Row Count 10 (+ 2)
{\emph{cannot determine dependence of variables \& cannot assess nonlinear associations}} \newline % Row Count 12 (+ 2)
{\bf{r value variation:}} \newline % Row Count 13 (+ 1)
-0.1 to -.03 / 0.1 to 0.3 = weak correlation \newline % Row Count 14 (+ 1)
-0.3 to -0.5 / 0.3 to 0.5 = average/moderate correlation \newline % Row Count 16 (+ 2)
-0.5 to -1.0 / 0.5 to 1.0 = strong correlation \newline % Row Count 17 (+ 1)
{\bf{FORMULA:}} \newline % Row Count 18 (+ 1)
r=n(Σxy)-(Σx)(Σy) / √{[}nΣx\textasciicircum{}2\textasciicircum{}-(Σx)\textasciicircum{}2\textasciicircum{}{]} {[}nΣy\textasciicircum{}2\textasciicircum{}-(Σy)\textasciicircum{}2\textasciicircum{}{]}% Row Count 20 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Mann-Whitney U test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\emph{non-parametric test to test the significance of difference two independently drawn groups OR compare outcomes between two independent groups}} \newline % Row Count 3 (+ 3)
equi to unpaired t test \newline % Row Count 4 (+ 1)
{\bf{CONDITIONS:}} \newline % Row Count 5 (+ 1)
No NPC assumption, small sample size \textgreater{}30 with min 5 in each group, continuous data (able to take any no in range), randomly selected samples,  \newline % Row Count 8 (+ 3)
aka: \newline % Row Count 9 (+ 1)
Mann-Whitney Test \newline % Row Count 10 (+ 1)
Wilcoxon Rank Sum test \newline % Row Count 11 (+ 1)
H0: the two pop are equal  \newline % Row Count 12 (+ 1)
Ha: the two pop are not equal \newline % Row Count 13 (+ 1)
{\bf{denoted by U}} \newline % Row Count 14 (+ 1)
{\bf{FORMULA:}} \newline % Row Count 15 (+ 1)
{\bf{U1=n1n2+ n1(n1+1)/2 - R1}} \newline % Row Count 16 (+ 1)
{\bf{U2=n1n2+ n2(n2+1)/2 - R2}} \newline % Row Count 17 (+ 1)
R= sum of ranks of group% Row Count 18 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{One-way ANOVA test}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/sana-h_1673894707_Untitled.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{One-way ANOVA test}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/sana-h_1673894753_u1.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{One-way ANOVA test}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/sana-h_1673894811_u2.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}