Quantitative
Analysis

Theory

“Theory is a well-established principle or set of general principles to explain a broad range of observations.”

Theory = Explanation = Generalisation

Population

The population is the complete set of all individuals, items, or observations that share a common characteristic and are of interest to the researcher.
It has unknown parameters which we estimate.

Sample

A sample is a subset of the population that’s actually observed or measured in the study.
Samples have statistics

Statistical Inference

The goal of most research is statistical inference: using sample statistics to make educated guesses about population parameters.

Variables

“Elements in the study that are measured or manipulated.”

Independent variable
Dependent variable

Statistical Learning

“Statistical Learning refers to a vast set of tools for understanding data.”

Descriptive Statistics
Inferential Statistics

Summarizes and organizes data to describe its main features.
Limited to the observed data.

Makes predictions or inferences about a population from a sample.
Goes beyond the observed data to infer population characteristics.

Central Tendency & Spread

R Packages

library(palmerpenguins)
library(tidyverse)
library(broom)

Measures of central tendency

mean(penguins$body_mass_g, na.rm = TRUE)

[1] 4201.754

median(penguins$body_mass_g, na.rm = TRUE)

[1] 4050

na.rm = TRUE is essential here — without it, mean() returns NA if even one value is missing
Mean vs. median: mean is pulled by outliers/skew; median is more robust. Always check both, especially with real-world (messy) data

Key Insights

Mean: Best for symmetric datasets but affected by outliers.
Median: Robust to outliers, useful for skewed data.
Mode: Describes the most common value, useful for categorical data.

Skewness

Measures of spread

sd(penguins$body_mass_g, na.rm = TRUE)

[1] 801.9545

var(penguins$body_mass_g, na.rm = TRUE)

[1] 643131.1

IQR(penguins$body_mass_g, na.rm = TRUE)

[1] 1200

quantile(penguins$body_mass_g, na.rm = TRUE)

  0%  25%  50%  75% 100% 
2700 3550 4050 4750 6300

SD: average distance from the mean, same units as the variable
Variance: SD squared — used more in formulas than in reporting
IQR: range of the middle 50% — robust to outliers, pairs naturally with the median

Grouped Summaries

`group_by()` + `summarise()`

penguins |>
  group_by(species) |>
  summarise(
    mean_mass = mean(body_mass_g, na.rm = TRUE),
    sd_mass   = sd(body_mass_g, na.rm = TRUE),
    n         = n()
  )

# A tibble: 3 × 4
  species   mean_mass sd_mass     n
  <fct>         <dbl>   <dbl> <int>
1 Adelie        3701.    459.   152
2 Chinstrap     3733.    384.    68
3 Gentoo        5076.    504.   124

This is the single most-used pattern in applied quantitative work: “give me this statistic, for each level of this category.”

Grouping by more than one variable

penguins |>
  group_by(species, sex) |>
  summarise(mean_mass = mean(body_mass_g, na.rm = TRUE), .groups = "drop")

# A tibble: 8 × 3
  species   sex    mean_mass
  <fct>     <fct>      <dbl>
1 Adelie    female     3369.
2 Adelie    male       4043.
3 Adelie    <NA>       3540 
4 Chinstrap female     3527.
5 Chinstrap male       3939.
6 Gentoo    female     4680.
7 Gentoo    male       5485.
8 Gentoo    <NA>       4588.

.groups = "drop" avoids a message and returns a plain (ungrouped) tibble — good habit once you start chaining more steps after summarise().

Standard Error

penguins |>
  filter(!is.na(body_mass_g)) |>
  group_by(species) |>
  summarise(
    mean_mass = mean(body_mass_g),
    se_mass   = sd(body_mass_g) / sqrt(n())
  )

# A tibble: 3 × 3
  species   mean_mass se_mass
  <fct>         <dbl>   <dbl>
1 Adelie        3701.    37.3
2 Chinstrap     3733.    46.6
3 Gentoo        5076.    45.5

SD describes spread of individual penguins. Standard error (SE) describes uncertainty in the mean itself — how much would this mean likely shift if we resampled?

Visual Exploration

Why visualise before testing

Numbers alone can hide skew, outliers, or multi-modal distributions
A quick plot often prevents a wrong statistical choice later (e.g. running a t-test on badly skewed data)

Histogram — shape of one variable

ggplot(penguins, aes(x = body_mass_g)) +
  geom_histogram(binwidth = 250, fill = "steelblue", color = "white") +
  labs(title = "Distribution of Body Mass", x = "Body mass (g)", y = "Count")

Boxplot — spread across groups

ggplot(penguins, aes(x = species, y = body_mass_g, fill = species)) +
  geom_boxplot(alpha = 0.7) +
  labs(title = "Body Mass by Species", x = "Species", y = "Body mass (g)")

Scatterplot

ggplot(penguins, aes(x = flipper_length_mm, y = body_mass_g)) +
  geom_point(alpha = 0.6) +
  labs(title = "Flipper Length vs Body Mass")

Correlation (descriptive, not yet a test)

cor(penguins$flipper_length_mm, penguins$body_mass_g, use = "complete.obs")

[1] 0.8712018

cor() gives a single number (-1 to 1) summarising the strength and direction of a linear relationship.

From Description to Inference

What a p-value actually means

Null hypothesis (H₀): no real difference/relationship exists in the population; anything we see in our sample is due to chance
A p-value is the probability of seeing a difference this large, if H₀ were true
Small p-value (conventionally < 0.05) → the data would be unusual under H₀ → we reject H₀

What a p-value is not:

It is not the probability that H₀ is true, and it is not the probability your result is “important.” A tiny p-value can come from a trivial effect if the sample is large enough — always report the effect size alongside it.

The general workflow for every test

State H₀ (no difference/relationship) and H₁ (there is one)
Check the test’s assumptions
Run the test in R
Read the p-value and the effect size / confidence interval
Write one sentence of plain-language interpretation

Comparing Two Groups

Independent-samples t-test

Question: Does body mass differ between male and female penguins?

penguins <- na.omit(penguins)

ggplot(penguins, aes(x = sex, y = body_mass_g, fill = sex)) +
  geom_boxplot()

Running the t-test

t.test(body_mass_g ~ sex, data = penguins)


    Welch Two Sample t-test

data:  body_mass_g by sex
t = -8.5545, df = 323.9, p-value = 4.794e-16
alternative hypothesis: true difference in means between group female and group male is not equal to 0
95 percent confidence interval:
 -840.5783 -526.2453
sample estimates:
mean in group female   mean in group male 
            3862.273             4545.685

Reading the output:

t = test statistic, df = degrees of freedom
p-value — here, extremely small → strong evidence of a real difference
95% confidence interval — the likely range for the true difference in means
Reports means for each group at the bottom
“Male penguins have significantly more weight than female penguins (p < .001)”

Your Turn

Test whether flipper_length_mm differs by sex.

Boxplot first
Run t.test()
Write one sentence interpreting the result

Answer

# boxplot
ggplot(penguins, aes(x = sex, y = flipper_length_mm, fill = sex)) +
  geom_boxplot()

Answer

# Run `t.test()`
t.test(flipper_length_mm ~ sex, data = penguins)


    Welch Two Sample t-test

data:  flipper_length_mm by sex
t = -4.8079, df = 325.28, p-value = 2.336e-06
alternative hypothesis: true difference in means between group female and group male is not equal to 0
95 percent confidence interval:
 -10.064811  -4.219821
sample estimates:
mean in group female   mean in group male 
            197.3636             204.5060

“Male penguins have significantly longer flippers than female penguins (p < .001).”

Comparing Three or More Groups

One-way ANOVA

Question: does body mass differ across the three species?

model_aov <- aov(body_mass_g ~ species, data = penguins)
summary(model_aov)

             Df    Sum Sq  Mean Sq F value Pr(>F)    
species       2 145190219 72595110   341.9 <2e-16 ***
Residuals   330  70069447   212332                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

ANOVA tests whether at least one group differs from the others — it does not tell you which pair(s) differ. The significant Pr(>F) here says species matters, but we need a follow-up test to know which species differ from which.

Post-hoc test: Tukey HSD

TukeyHSD(model_aov)

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = body_mass_g ~ species, data = penguins)

$species
                       diff       lwr       upr    p adj
Chinstrap-Adelie   26.92385 -132.3528  186.2005 0.916431
Gentoo-Adelie    1386.27259 1252.2897 1520.2554 0.000000
Gentoo-Chinstrap 1359.34874 1194.4304 1524.2671 0.000000

Every pairwise comparison, with an adjusted p-value that corrects for multiple testing. Reading the output: adelie-Gentoo, Adelie-Chinstrap, Chinstrap-Gentoo — check the p adj column for each pair.

Checking ANOVA assumptions visually

ggplot(penguins, aes(x = species, y = body_mass_g, fill = species)) +
  geom_boxplot()

Your Turn

Run an ANOVA testing whether flipper_length_mm differs by species
Run TukeyHSD() on it
Which species pairs differ significantly?

Answer

model2 <- aov(flipper_length_mm ~ species, data = penguins)
summary(model2)

             Df Sum Sq Mean Sq F value Pr(>F)    
species       2  50526   25263   567.4 <2e-16 ***
Residuals   330  14693      45                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

TukeyHSD(model2)

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = flipper_length_mm ~ species, data = penguins)

$species
                     diff       lwr       upr p adj
Chinstrap-Adelie  5.72079  3.414364  8.027215     0
Gentoo-Adelie    27.13255 25.192399 29.072709     0
Gentoo-Chinstrap 21.41176 19.023644 23.799885     0

All three pairwise comparisons are significant — every species differs from every other in flipper length.

Testing Relationships

Correlation test

cor.test(penguins$bill_length_mm, penguins$flipper_length_mm)


    Pearson's product-moment correlation

data:  penguins$bill_length_mm and penguins$flipper_length_mm
t = 15.691, df = 331, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.5868092 0.7106869
sample estimates:
      cor 
0.6530956

plain cor() — now we get a p-value and confidence interval alongside the correlation coefficient (cor in the output), telling us whether this relationship is distinguishable from zero.

Chi-square test of independence

Question: is species associated with island?

tbl <- table(penguins$species, penguins$island)
tbl

           
            Biscoe Dream Torgersen
  Adelie        44    55        47
  Chinstrap      0    68         0
  Gentoo       119     0         0

chisq.test(tbl)


    Pearson's Chi-squared test

data:  tbl
X-squared = 284.59, df = 4, p-value < 2.2e-16

Chi-square tests whether two categorical variables are independent. Here, the tiny p-value confirms — species and island are strongly associated

Your Turn

Test the correlation between bill_depth_mm and body_mass_g with cor.test()
Build a table of species by sex and run a chi-square test — are they associated?

Answer

cor.test(penguins$bill_depth_mm, penguins$body_mass_g)


    Pearson's product-moment correlation

data:  penguins$bill_depth_mm and penguins$body_mass_g
t = -9.741, df = 331, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.5515130 -0.3840214
sample estimates:
       cor 
-0.4720157

tbl2 <- table(penguins$species, penguins$sex)
tbl2

           
            female male
  Adelie        73   73
  Chinstrap     34   34
  Gentoo        58   61

chisq.test(tbl2)


    Pearson's Chi-squared test

data:  tbl2
X-squared = 0.048607, df = 2, p-value = 0.976

Species and sex are not significantly associated (roughly equal male/female split within each species) — a good example of a non-significant, and equally informative, result.

Choosing the Right Test

A quick decision guide

Question	Variable types	Test
Do 2 groups differ?	1 numeric + 1 categorical (2 levels)	`t.test()`
Do 3+ groups differ?	1 numeric + 1 categorical (3+ levels)	`aov()` + `TukeyHSD()`
Are two numeric variables related?	2 numeric	`cor.test()`
Are two categorical variables associated?	2 categorical	`chisq.test()`

Simple Linear Regression

From correlation to regression

cor.test() told us whether flipper_length_mm and body_mass_g are related, and how strongly
Regression goes further: it gives us an equation to predict one from the other

model1 <- lm(body_mass_g ~ flipper_length_mm, data = penguins)
summary(model1)


Call:
lm(formula = body_mass_g ~ flipper_length_mm, data = penguins)

Residuals:
     Min       1Q   Median       3Q      Max 
-1057.33  -259.79   -12.24   242.97  1293.89 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)       -5872.09     310.29  -18.93   <2e-16 ***
flipper_length_mm    50.15       1.54   32.56   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 393.3 on 331 degrees of freedom
Multiple R-squared:  0.7621,    Adjusted R-squared:  0.7614 
F-statistic:  1060 on 1 and 331 DF,  p-value: < 2.2e-16

Reading the output

summary(model1)

Intercept: predicted body_mass_g when flipper_length_mm = 0 (often not meaningful on its own)
Slope (flipper_length_mm coefficient): for every 1mm increase in flipper length, body mass increases by this many grams
p-value (per coefficient): is this slope distinguishable from zero?
R-squared: proportion of variance in body_mass_g explained by flipper_length_mm (0 to 1)

Visualising the fitted line

ggplot(penguins, aes(x = flipper_length_mm, y = body_mass_g)) +
  geom_point(alpha = 0.5) +
  geom_smooth(method = "lm", se = TRUE) +
  labs(title = "Body Mass Predicted by Flipper Length")

Your Turn 1

Fit lm(bill_length_mm ~ bill_depth_mm, data = penguins)
What is the slope? Is it significant?
What is the R²? How much variance is explained?

Answer

model_yt1 <- lm(bill_length_mm ~ bill_depth_mm, data = penguins)
summary(model_yt1)


Call:
lm(formula = bill_length_mm ~ bill_depth_mm, data = penguins)

Residuals:
     Min       1Q   Median       3Q      Max 
-12.9498  -3.9530  -0.3657   3.7327  15.5025 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)    54.8909     2.5673  21.380  < 2e-16 ***
bill_depth_mm  -0.6349     0.1486  -4.273 2.53e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.332 on 331 degrees of freedom
Multiple R-squared:  0.05227,   Adjusted R-squared:  0.04941 
F-statistic: 18.26 on 1 and 331 DF,  p-value: 2.528e-05

The slope is negative and significant here

Thank you!

Module 4 complete 🎉

SARA Institute of Data Science - 3rd SARA Summer School 22-26 July 2026 “R for Researchers”

QuantitativeAnalysis

Theory

Population

Sample

Statistical Inference

Variables

Statistical Learning

Central Tendency & Spread

R Packages

Measures of central tendency

Key Insights

Skewness

Measures of spread

Grouped Summaries

group_by() + summarise()

Grouping by more than one variable

Standard Error

Visual Exploration

Why visualise before testing

Histogram — shape of one variable

Boxplot — spread across groups

Scatterplot

Correlation (descriptive, not yet a test)

From Description to Inference

What a p-value actually means

What a p-value is not:

The general workflow for every test

Comparing Two Groups

Independent-samples t-test

Running the t-test

Your Turn

Answer

Answer

Comparing Three or More Groups

One-way ANOVA

Post-hoc test: Tukey HSD

Checking ANOVA assumptions visually

Your Turn

Answer

Testing Relationships

Correlation test

Chi-square test of independence

Your Turn

Answer

Choosing the Right Test

A quick decision guide

Simple Linear Regression

From correlation to regression

Reading the output

Visualising the fitted line

Your Turn 1

Answer

Thank you!

Quantitative
Analysis

`group_by()` + `summarise()`