Statistics for Psychology Chapters 1-6 Midterm

Statistics Definition

The study of variation — how to measure it, how to model it.

Data Generating Process (DGP)

The hidden, real-world mechanism that produces the data we observe, We never see the true DGP; we use data samples to infer how it works.

Population

The entire group you want to draw conclusions about

Sample

The specific subset of the population from which you collect data.

tally ()

Used to count frequencies and proportions of categorical data.

head ()

Shows the first 6 rows of a data frame to check its structure.

str()

Displays the internal structure of a data frame: data types, number of rows/columns

Tidy Data Structure

A strict data layout where each row = single case/observation, each column = a single variable, and each cell = a single value

Quantitative Variables

Numerical values where mathematical operations make sense

Qualitative (Categorical) Variables

Labels or categories

Quantitative variable R Type

int, num

Categorical data R Type

Factor, chr

as.factpr()

R function used to explicitly convert a quantitative/character variable into a categorical factor

Distribution

How often the different values of a variable occur in a dataset

gf_histogram()

Groups continuous data into equal-width "bins”

gf_densitiy ()

Creates a smooth curve showing where data points are most concentrated

gf_bar()

Bar charts used for counting distinct categories

Modality

the number of peaks

Unimodal

one peak

Bimodal

two peaks

Uniform

flat

Skewness

the direction of the long tail

right-skewed (positive)

the tail stretching out to the right (higher values. The mean id typically pulled higher than the median.

Left-skewed (negative)

The tail stretches out to the left (lower values), The mean is typically pulled lower than the median

Symmetry

Mirror-image apparence on both sides of the center

faceting

using gf_facet_grid(Outcome ~ Predictor) to split a single plot into multiple panels based on a categorical grouping variable

Explaining vs. Describing

Moving from looking at one variable (univariate to looking at the relaitonship between two variables (bivariate)

Outcome Variable (Y)

The dependent variable; the thing you are trying to predict or explain (plotted on the y-axis

Explanatory Variable (X)

The independent variable; the thing that you suspect causes or predicts changesin the outcome (plotted on the x-axis)

Observational Studies

Reseachers observe variables without intervening. The studies can show correlation/association, but cannot prove causation due ot confounding variables

confounding variables

variables that influence both X and Y

Experimental Studies

Researchers actively manipulate the explanatory variable

Random Assignment

It distributes confounding variables equally across treatment groups, isolating X as the sole cause of changes in Y, allowing for causal conclusions.

Induced Variation

Variation created intentionally by a researcher’s experimental manipulation

Error/Real-World Variation

Natural difference among individuals, or measurement errors

Statisitical Model Definition

A simpliflied mathematical representation of the Data Generating Process

The empty Model Equation

Yi = b0 + ei

Why is the empty model empty?

contains zero explanatory variables

The Mean as the Ultimate Empty Model

For quantitative

What does the sum of all deviations of all data points from the mean equal?

Zero

Deviation

Distance

What does the mean do for the data?

Balances it

What does the mean do to the sum of squared deviations

minimises the SS deviations

R-code for fitting the empty model

lm(Outcome ~ NULL, data = Dataset)

Residuals

The mathematical definition for error for an individual case

Residuals equation

ei = Yi - Y-bar

Data point above the model’s prediction

residual = positive

Data point below the prediction

Residual = negative

Sum of Squares

SS = Sum(Yi-Y-bar)²

What does SS quantify?

total amount of error

Degrees of Freedom (df)

The number of indepdent pieces of information that went into calculating a statistic

Degree of freedom for the empty model

n - 1, n = sample size

Variance (s²)

The mean squared error —- quantifies the average squared distance from the model

Variance (s²) Equation

s² = SS/df = Sum(Yi-Y-bar)²/n-1

Standard Deviation (s)

The square root of variance

Standard Deviation Equation

s = Sqrt of s²

Standard Deviation Equation Purpose

Takes the square root of the variance unit and puts it into the original units of the data. This then tells us the typical distance a data point falls from the mean

Z-Scores (Standardized Scores)

Measures how many standard deviations a specific data point is away from the mean

Z-Score EQ

z = Yi-Y-bar/s

z-score exactly at the mean equals?

zero

Purpose of z-score

allows you to compare values from completely different distributions

Empirical Rule (“68-95-99.7)

In a bell-shaped distribution: ~68% of observations fall within ± 1 standard deviation of the mean; ~96% of observations fall within ± 2 standard deviations of the mean;and ~99.7% of observations fall within +/_ 3 standard deviations of the mean

Extracts the individual error values (ei) for every row

reside(EmptyModel)

Shows the predicted value (Y-bar) for each row

predict(EmptyModel)

Gives the area under the normal curve to the left of a given z-score (percentile)

pnorm(z-score)

The inverse of pnorm. Input a percentile/probability and it returns the corresponding z-score

qnorm(probability)

Visualizations for Quantitative variables

Histogram, Boxplot

VIsualizations for Categorical Variables

Frequency tables, Bar chart

Qualitative outcome variable visualizations

jitter plot, Boxplot, faceted histogram

Quantitaive outcome variable visualizations

scatterplot, faceted hsitogram

The select command — subsets columns

select(dataframe, column1, column2…). Returns a data frame that contains the columns that you specified as arguments.

The filter command — subsets rows

filter(dataframe, criteria). Returns a data frame that contains the rows that fit your criteria

Understand structure of data

str(dataframe)

R-code for arranging dataframe based on a specific variable, in descending order

datafram$variable

R-code for going from lowest to highest in a list

arrange(dataframe, (desc(variable))

R-code for going from highest to lowest in a list

arrange(dataframe, variable)

R-code for listing numbers’ frequency and numbers in order from lowest to highest

tally(dataframe$variable) or tally(~variable, data = dataframe)

R-code for displaying the last 6 rows

tail(dataframe)

R-Code for displaying the first 6 rows

head(dataframe)