Statistics in Behavioral Sciences Cheat Sheet

the branch of mathem­atics in which data are used descri­ptively or infere­ntially to find or support answers for scientific and other quanti­fiable questions.
It encomp­asses various techniques and procedures for recording, organi­zing, analyzing, and reporting quanti­tative inform­ation.

a statis­tical measure of the strength of the relati­onship between the relative movements of two variables

value ranges from -1 to +1

-1 = perfect negative or inverse correl­ation
+1 = perfect positive correl­ation or direct relati­onship
0 = no linear relati­onship

to compare means of two related groups
ex. compare weight of 20 mice before and after treatment

two condit­ions:
- pre post treatment
- two diff conditions ex two drugs

ASSUMP­TIONS
- random selection
- normally distri­buted
- no extreme outliers

FORMULA
t= m / s/√n
m= sample mean of differ­ences

df= n-1

aka Studen­t's t-dist­rib­ution = probab­ility distri­bution similar to normal distri­bution but has heavier tails
used to estimate pop parameters for small samples

Tail heaviness is determined by 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

df = n - 1
above 30df, use z-dist­rib­ution

t-score = no of SD from mean in a t-dist­rib­ution
we find:
- upper and lower boundaries
- p value

TO BE USED WHEN:
- small sample
- SD is unknown

ASSUMP­TIONS
- cont or ordinal scale
- random selection
- NPC
- equal SD for indep two-sample t-test

to determine if means of two indepe­ndent popula­tions are equal or different
to find out if there is signif­icant diff bet two pop by comparing sample mean

knowledge of:
SD and sample >30 in each group

eg. compare perfor­mance of 2 students, average salaries, employee perfor­mance, compare IQ, etc

FORMULA:
z= x̄₁ - x̄₂ / √s₁²/n₁ + s₂²/n₂

s= SD

formula:
z= (x̄₁ - x̄₂) - (µ₁ - µ₂) / √σ₁²/n₁ + σ₂²/n₂

(µ₁ - µ₂) = hypoth­esized difference bet pop means

measures relati­onship between two variables

rpbi = correl­ation coeffi­cient

one continuous variable (ratio­/in­terval scale)
one naturally binary variable

FORMULA:
rpb= M1-M0/Sn * √ pq

Sn= SD

to determine if means of two indepe­ndent popula­tions are equal or different
to find out if there is signif­icant diff bet two pop by comparing sample mean

knowledge of:
SD and sample >30 in each group

eg. compare perfor­mance of 2 students, average salaries, employee perfor­mance, compare IQ, etc

FORMULA:
z= x̄

for hypothesis testing
to check whether means of two popula­tions are equal to each other when pop variance is known

we have knowledge of:
- SD/pop­ulation variance and/or sample n=30 or more

if both unknown -> t-test

left-t­ailed
right-­tailed
two-tailed

REJECT NULL HYPOTHESIS IF Z STATISTIC IS STATIS­TICALLY SIGNIF­ICANT WHEN COMPARED WITH CRITICAL VALUE

z-stat­istic/ z-score = no repres­enting result from z-test

z critical value divides graph into acceptance and rejection regions
if z stat falls in rejection region­-> H0 can be rejected

TYPES
One-sample z-test

Two-sample z-test

Analysis of Variance
comparing several sets of scores
to test if means of 3 or more groups are equal
comparison of variance between and within groups
to check if sample groups are affected by same factors and to same degree
compare differ­ences in means and variance of distri­bution

ONE-WAY ANOVA=no of IVs
single IV with different (2) levels­/va­ria­tions have measurable effect on DV
compare means of 2 or more indep groups
aka:
- one-factor ANOVA
- one-way analysis of variance
- between subjects ANOVA

Assump­tions
- indepe­ndent samples
- equal sample sizes in groups­/levels
- normally distri­buted
- equal variance

F test is used to check statis­tical signif­icance
higher F value --> higher likelihood that difference observed is real and not due to chance
used in field studies, experi­ments, quasi-exp
CONDIT­IONS:
- min 6 subjects
- sample no of samples in each group

H0: µ1=µ2=µ3 ... µk i.e. all pop means are equal
Ha: at least one µi is different i.e atleat one of the k pop means is not equal to the others
µi is the pop mean of group

non-pa­ram­etric version of Pearson correl­ation coeffi­cient
named after Charles Spearman
denoted by ρ(rho)

determine the strength and direction of monotonic variables bet two variables measured at ordinal, interval or ratio levels & whether they are correlated or not

monotonic function=one variable never increases or never decreases as its IV changes
- monoto­nically increa­sing= as X increases, Y never decreases
- monoto­nically decrea­sing= as X increases, Y never increases
- not monotonic= as X increases, Y sometimes dec and sometimes inc

for analysis with: ordinal data, continuous data
uses ranks instead of assump­tions of normality
aka Spearman Rank order test

FORMULA:
ρ= 1- 6Σdᵢ²/n(n²-1)

di= difference between two ranks of each observ­ation

-1 to +1
+1 = perfect associ­ation of ranks
0= no associ­ation
-1= perfect negative associ­ation of ranks

closer the value to 0, weaker the associ­ation

Value Ranges
0 to 0.3 = weak monotonic relati­onship
0.4 to 0.6 = moderate strength monotonic relati­onship
0.7 to 1 = strong monotonic relati­onship

Fixed set of parame­ters, certain assump­tions about distri­bution of population

PARAMETRIC - prior knowledge of pop distri­bution i.e NORMAL DISTRI­BUTION

NON-PA­RAM­ETRIC - no assump­tions, do not depend on popula­tion, DISTRI­BUTION FREE tests, values found on nominal or ordinal level
easy to apply, unders­tand, low complexity

decision based on - distri­bution of popula­tion, size of sample
parametric - mean & <30 sample
non-pa­ram­etric - median­/mode & >30 sample or regardless of size

indepe­ndent values in the data sample that have freedom to vary

FORMULA:
no of values in a data set minus 1
df= N-1

statis­tical test to determine if signif­icant difference between avg scores of two groups
1908-W­illiam Sealy Gosset- student t-test and t-dist­irb­ution
for hypothesis testing

knowledge of:
distri­bution - normally distri­buted
no knowledge of SD

TYPES:
one-sample t-test - single group
FORMULA:
t= m - µ / s/√n

SD FORMULA:
σ= √Σ(X-µ)² / N
s= √Σ(X-µ)² / n-1

indepe­ndent two-sample t-test - two groups

paired­/de­pendent samples t-test - sig diff in paired measur­ements, compares means from same group at diff times (test-­retest sample)

H0: no effective difference = measured diff is due to chance
Ha: two-ta­iled/ one-tailed nonequ­ivalent means/­smaller or larger than hypoth­esized mean

PERFORM two-tailed test: to find out difference bet two popula­tions
one-tailed: one pop mean is > or < other

aka unpaired t-test
to compare mean of two indepe­ndent groups
ex. avg weight of males and females

two forms:
- studen­t's t-test: assumes SD is equal
- welch's t-test: less restri­ctive, no assumption of equal SD
both provide more/less similar results

ASSUMP­TIONS:
- normally distri­buted
- SD is same
- indepe­ndent groups
- randomly selected
- indepe­ndent observ­ations
- measured on interval or ratio scale

FORMULA:
t= x̄₁ - x̄₂ / √s₁2/n₁ + s₂2/n₂

df= n1 + n2 - 2

S= √Σ (x1-x̄)² + (x2-x̄)² / n1+n2-2

to check if difference between sample mean & population mean when SD is known

FORMULA:
z=x-µ/SE

SE=σ/√n

z score is compared to a z table (includes % under NPC bet mean and z score), tells us whether the z score is due to chance or not

condit­ions:
knowledge of:
- pop mean
- SD
- simple random sample
- normal distri­bution

two approaches to reject H0:
- p-value approach - p-value is the smallest level of signif­icance at which H0 can be reject­ed...smaller p-value, stronger evidence
-critical value approach - comparing z stat to critical values... indicate boundary regions where stat is highly improbable to lie= critical region­s/r­eje­ction regions
if z stat is in critical region­-> reject H0
based on:
signif­icance level (0.1, 0.05, 0.01), alpha level, Ha

to measure relati­onship between quanti­tative variables and binary variables
given by Pearson - 1909

biserial correl­ation coeff varies bet -1 and 1
0= no associ­ation
ex. IQ scores and pass/fail correl­ation

continuous variable and binary variable (dicho­tomised to create binary variable)

rbis or rb = correl­ation index estimating strength of relati­onship between artifi­cially dichot­omous variable and a true continuous variable

ASSUMP­TIONS:
- data measured on continuous scale
- one variable to be made dichot­omous
- no outliers
- approx normally distri­buted
- equal variances (SD)

FORMULA
rb= M1-M0/SDt * pq/y

M1=mean of grp 1
M2= mean of grp 2
p= ratio of grp 1
q= ratio of grp 2
SDt= total SD
y= ordinate

measures strength and direction of a linear relati­onship between two variables
how two data sets are correlated
gives us info about the slope of the line
r
aka:
- Pearso­n's r
- bivariate correl­ation
- Pearson produc­t-m­oment correl­ation coeffi­cient (PPMCC)

cannot determine dependence of variables & cannot assess nonlinear associ­ations

r value variation:
-0.1 to -.03 / 0.1 to 0.3 = weak correl­ation
-0.3 to -0.5 / 0.3 to 0.5 = averag­e/m­oderate correl­ation
-0.5 to -1.0 / 0.5 to 1.0 = strong correl­ation

FORMULA:
r=n(Σx­y)-­(Σx­)(Σy) / √[nΣx²-(Σx)²] [nΣy²-(Σy)²]

non-pa­ram­etric test to test the signif­icance of difference two indepe­ndently drawn groups OR compare outcomes between two indepe­ndent groups

equi to unpaired t test

CONDIT­IONS:
No NPC assump­tion, small sample size >30 with min 5 in each group, continuous data (able to take any no in range), randomly selected samples,

aka:
Mann-W­hitney Test
Wilcoxon Rank Sum test

H0: the two pop are equal
Ha: the two pop are not equal

denoted by U

FORMULA:
U1=n1n2+ n1(n1+1)/2 - R1

U2=n1n2+ n2(n2+1)/2 - R2

R= sum of ranks of group