Change

Correlations

Correlations
Scatter Plots
Errorbar Plot
Regression Lines
Author

Arvind V.

Published

November 22, 2022

Modified

October 17, 2025

Abstract
How one variable changes with another

“The world says: ‘You have needs – satisfy them. You have as much right as the rich and the mighty. Don’t hesitate to satisfy your needs; indeed, expand your needs and demand more.’ This is the worldly doctrine of today. And they believe that this is freedom. The result for the rich is isolation and suicide, for the poor, envy and murder.”

— Fyodor Dostoevsky

1 Setting up R Packages

library(tidyverse) # Tidy data processing and plotting
library(ggformula) # Formula based plots
library(mosaic) # Our go-to package
library(skimr) # Another Data inspection package

library(GGally) # Corr plots
library(broom) # Clean reports from Stats / ML outputs

# library(devtools)
# devtools::install_github("rpruim/Lock5withR")
library(Lock5withR) # Datasets

library(easystats) # Easy Statistical Analysis and Charts
library(correlation) # Different Types of Correlations

library(janitor) # Data cleaning and tidying package
library(visdat) # Visualize whole dataframes for missing data
library(naniar) # Clean missing data
library(DT) # Interactive Tables for our data
library(tinytable) # Elegant Tables for our data
library(ggrepel) # Repelled Text Labels in ggplot
library(marquee) # Marquee Text Labels in ggplot

Plot Fonts and Theme

Show the Code
library(systemfonts)
library(showtext)
## Clean the slate
systemfonts::clear_local_fonts()
systemfonts::clear_registry()
##
showtext_opts(dpi = 96) # set DPI for showtext
sysfonts::font_add(
  family = "Alegreya",
  regular = "../../../../../../fonts/Alegreya-Regular.ttf",
  bold = "../../../../../../fonts/Alegreya-Bold.ttf",
  italic = "../../../../../../fonts/Alegreya-Italic.ttf",
  bolditalic = "../../../../../../fonts/Alegreya-BoldItalic.ttf"
)

sysfonts::font_add(
  family = "Roboto Condensed",
  regular = "../../../../../../fonts/RobotoCondensed-Regular.ttf",
  bold = "../../../../../../fonts/RobotoCondensed-Bold.ttf",
  italic = "../../../../../../fonts/RobotoCondensed-Italic.ttf",
  bolditalic = "../../../../../../fonts/RobotoCondensed-BoldItalic.ttf"
)
showtext_auto(enable = TRUE) # enable showtext
##
theme_custom <- function() {
  theme_bw(base_size = 10) +

    # theme(panel.widths = unit(11, "cm"),
    #       panel.heights = unit(6.79, "cm")) + # Golden Ratio

    theme(
      plot.margin = margin_auto(t = 1, r = 2, b = 1, l = 1, unit = "cm"),
      plot.background = element_rect(
        fill = "bisque",
        colour = "black",
        linewidth = 1
      )
    ) +

    theme_sub_axis(
      title = element_text(
        family = "Roboto Condensed",
        size = 10
      ),
      text = element_text(
        family = "Roboto Condensed",
        size = 8
      )
    ) +

    theme_sub_legend(
      text = element_text(
        family = "Roboto Condensed",
        size = 6
      ),
      title = element_text(
        family = "Alegreya",
        size = 8
      )
    ) +

    theme_sub_plot(
      title = element_text(
        family = "Alegreya",
        size = 14, face = "bold"
      ),
      title.position = "plot",
      subtitle = element_text(
        family = "Alegreya",
        size = 10
      ),
      caption = element_text(
        family = "Alegreya",
        size = 6
      ),
      caption.position = "plot"
    )
}

## Use available fonts in ggplot text geoms too!
ggplot2::update_geom_defaults(geom = "text", new = list(
  family = "Roboto Condensed",
  face = "plain",
  size = 3.5,
  color = "#2b2b2b"
))
ggplot2::update_geom_defaults(geom = "label", new = list(
  family = "Roboto Condensed",
  face = "plain",
  size = 3.5,
  color = "#2b2b2b"
))

ggplot2::update_geom_defaults(geom = "marquee", new = list(
  family = "Roboto Condensed",
  face = "plain",
  size = 3.5,
  color = "#2b2b2b"
))
ggplot2::update_geom_defaults(geom = "text_repel", new = list(
  family = "Roboto Condensed",
  face = "plain",
  size = 3.5,
  color = "#2b2b2b"
))
ggplot2::update_geom_defaults(geom = "label_repel", new = list(
  family = "Roboto Condensed",
  face = "plain",
  size = 3.5,
  color = "#2b2b2b"
))

## Set the theme
ggplot2::theme_set(new = theme_custom())

## tinytable options
options("tinytable_tt_digits" = 2)
options("tinytable_format_num_fmt" = "significant_cell")
options(tinytable_html_mathjax = TRUE)


## Set defaults for flextable
flextable::set_flextable_defaults(font.family = "Roboto Condensed")

2 What graphs will we see today?

Variable #1 Variable #2 Chart Names Chart Shape
Quant Quant Scatter Plot

Some of the very basic and commonly used plots for data are:

  • Scatter Plot for two variables
  • Pairwise Correlation Plots for multiple variables
  • Errorbar chart for multiple variables
  • Contour Plot
  • Scatter Plot with Confidence Ellipses
  • Correlogram for multiple variables
  • Heatmap for multiple variables
  • Combination chart with marginal densities

3 What kind of Data Variables will we choose?

No Pronoun Answer Variable/Scale Example What Operations?
1 How Many / Much / Heavy? Few? Seldom? Often? When? Quantities, with Scale and a Zero Value.Differences and Ratios /Products are meaningful. Quantitative/Ratio Length,Height,Temperature in Kelvin,Activity,Dose Amount,Reaction Rate,Flow Rate,Concentration,Pulse,Survival Rate Correlation

4 Inspiration

Figure 1: ScatterPlot Inspiration http://www.calamitiesofnature.com/archive/?c=559

Does belief in Evolution depend upon the GSP of of the country? Where is the US in all of this? Does the Bible Belt tip the scales here?

And India?

5 What is Correlation?

One of the basic Questions we would have of our data is: Does some variable depend upon another in some way? Does \(y\) vary with \(x\)? A Correlation Test is designed to answer exactly this question.

The word correlation is used in everyday life to denote some form of association. We might say that we have noticed a correlation between rainy days and reduced sales at supermarkets. However, in statistical terms we use correlation to denote association between two quantitative variables. We also assume that the association is linear, that one variable increases or decreases a fixed amount for a unit increase or decrease in the other. The other technique that is often used in these circumstances is regression, which involves estimating the best straight line to summarise the association.

6 Case Study-1: HollywoodMovies2011 dataset

Let us look at the HollywoodMovies2011 dataset from the {Lock5withR} package.

data(HollywoodMovies2011, package = "Lock5withR")
movies_modified <- HollywoodMovies2011 %>%
  janitor::clean_names(case = "snake") %>%
  janitor::remove_empty(which = c("rows", "cols")) %>%
  dplyr::mutate(
    across(where(is.character), as.factor)
  ) %>%
  dplyr::relocate(where(is.factor))
Note

The dataset is also available by clicking the icon below ( in case you are not able to install {Lock5withR}):

7 Inspecting the Data

glimpse(movies_modified)
Rows: 136
Columns: 14
$ movie                <fct> "Insidious", "Paranormal Activity 3", "Bad Teache…
$ lead_studio          <fct> Sony, Independent, Independent, Warner Bros, Rela…
$ story                <fct> Monster Force, Monster Force, Comedy, Rivalry, Ri…
$ genre                <fct> Horror, Horror, Comedy, Fantasy, Comedy, Romance,…
$ rotten_tomatoes      <int> 67, 68, 44, 96, 90, 93, 75, 35, 63, 69, 69, 49, 2…
$ audience_score       <int> 65, 58, 38, 92, 77, 84, 91, 58, 74, 73, 72, 57, 6…
$ theaters_open_week   <int> 2408, 3321, 3049, 4375, 2918, 944, 2534, 3615, NA…
$ bo_average_open_week <int> 5511, 15829, 10365, 38672, 8995, 6177, 10278, 237…
$ domestic_gross       <dbl> 54.01, 103.66, 100.29, 381.01, 169.11, 56.18, 169…
$ foreign_gross        <dbl> 43.00, 98.24, 115.90, 947.10, 119.28, 83.00, 30.1…
$ world_gross          <dbl> 97.009, 201.897, 216.196, 1328.111, 288.382, 139.…
$ budget               <dbl> 1.5, 5.0, 20.0, 125.0, 32.5, 17.0, 25.0, 80.0, 0.…
$ profitability        <dbl> 64.672667, 40.379400, 10.809800, 10.624888, 8.873…
$ opening_weekend      <dbl> 13.27, 52.57, 31.60, 169.19, 26.25, 5.83, 26.04, …
skimr::skim(movies_modified) %>%
  tt(theme = "striped")
skim_type skim_variable n_missing complete_rate factor.ordered factor.n_unique factor.top_counts numeric.mean numeric.sd numeric.p0 numeric.p25 numeric.p50 numeric.p75 numeric.p100 numeric.hist
factor movie 0 1 FALSE 136 30 : 1, 50/: 1, A D: 1, A V: 1 NA NA NA NA NA NA NA NA
factor lead_studio 0 1 FALSE 34 Ind: 32, War: 12, 20t: 9, Uni: 9 NA NA NA NA NA NA NA NA
factor story 0 1 FALSE 22 Mon: 19, Com: 14, Que: 13, Lov: 12 NA NA NA NA NA NA NA NA
factor genre 0 1 FALSE 9 Act: 32, Com: 27, Dra: 21, Hor: 17 NA NA NA NA NA NA NA NA
numeric rotten_tomatoes 2 0.99 NA NA NA 53 27 4 29 54 78 97 ▅▇▅▆▇
numeric audience_score 1 0.99 NA NA NA 62 17 24 50 61 76 93 ▂▆▇▇▆
numeric theaters_open_week 16 0.88 NA NA NA 2828 933 3 2550 2995 3400 4375 ▁▁▂▇▃
numeric bo_average_open_week 16 0.88 NA NA NA 8339 10284 1513 3779 5686 8923 93230 ▇▁▁▁▁
numeric domestic_gross 2 0.99 NA NA NA 63 69 0.02 19 37 80 381 ▇▂▁▁▁
numeric foreign_gross 15 0.89 NA NA NA 97 156 0.24 14 47 102 947 ▇▁▁▁▁
numeric world_gross 2 0.99 NA NA NA 151 215 0.025 31 77 174 1328 ▇▁▁▁▁
numeric budget 2 0.99 NA NA NA 53 49 0.2 20 36 70 250 ▇▂▂▁▁
numeric profitability 2 0.99 NA NA NA 3.3 6.6 0 1.1 2.2 3.7 65 ▇▁▁▁▁
numeric opening_weekend 3 0.98 NA NA NA 20 25 0 7.7 13 25 169 ▇▁▁▁▁ 
movies_describe <- inspect(movies_modified)
##
movies_describe$categorical %>%
  tt(theme = "striped")
name class levels n missing distribution
movie factor 136 136 0 30 Minutes or Less (0.7%) ...
lead_studio factor 34 136 0 Independent (23.5%) ...
story factor 22 136 0 Monster Force (14%), Comedy (10.3%) ...
genre factor 9 136 0 Action (23.5%), Comedy (19.9%) ... 
##
movies_describe$quantitative %>%
  tt(theme = "striped")
name class min Q1 median Q3 max mean sd n missing
rotten_tomatoes integer 4 29 54 78 97 53 27 134 2
audience_score integer 24 50 61 76 93 62 17 135 1
theaters_open_week integer 3 2550 2995 3400 4375 2828 933 120 16
bo_average_open_week integer 1513 3779 5686 8923 93230 8339 10284 120 16
domestic_gross numeric 0.02 19 37 80 381 63 69 134 2
foreign_gross numeric 0.24 14 47 102 947 97 156 121 15
world_gross numeric 0.025 31 77 174 1328 151 215 134 2
budget numeric 0.2 20 36 70 250 53 49 134 2
profitability numeric 0 1.1 2.2 3.7 65 3.3 6.6 134 2
opening_weekend numeric 0 7.7 13 25 169 20 25 133 3
NoteBusiness Insights from Data Inspection

movies has 136 observations on the following 14 variables.

  • movie a factor with many levels
  • lead_studio a factor with many levels
  • story a factor with many levels
  • genre a factor with levels Action, Adventure, Animation, Comedy, Drama, Fantasy, Horror, Romance, Thriller.
  • rotten_tomatoes a numeric vector
  • audience_score a numeric vector
  • theaters_open_week a numeric vector. No. of theatres.
  • bo_average_open_week a numeric vector.
  • domestic_gross a numeric vector. In million USD.
  • foreign_gross a numeric vector. In million USD.
  • world_gross a numeric vector. In million USD.
  • budget a numeric vector. In million USD.
  • profitability a numeric vector. A ratio
  • opening_weekend a numeric vector. In million USD.

There are no missing values in the Qual variables; but some entries in the Quant variables are missing. skim throws a warning that we may need to examine later.

Show the Code
movies_modified %>%
  DT::datatable(
    caption = htmltools::tags$caption(
      style = "caption-side: top; text-align: left; color: black; font-size: 150%;",
      "Movies Dataset (Clean)"
    ),
    options = list(pageLength = 10, autoWidth = TRUE)
  ) %>%
  DT::formatStyle(
    columns = names(movies_modified),
    fontFamily = "Roboto Condensed",
    fontSize = "12px"
  )
Table 1: Movies Clean Dynamic Data Table

7.1 Hypothesis and Research Questions

Let us look at the Quant variables: are these related in anyway? Could the relationship between any two Quant variables also depend upon the level of a Qual variable? - The target variable for an experiment that resulted in this data might be the profitability variable, the resultant ratio of the money poured into the movie making, and the multiplier by which we obtain returns.

NoteResearch Questions:
  • Is there are relationship betweenprofitability and budget?
  • How does the opening_weekend affect profitability?
  • Between profitability and domestic_gross? Between profitability and foreign_gross?
  • Is profitability varying with rotten_tomatoes?

These should do for now! But we should make more questions when have seen some plots!

Note The Monkey Grammarian’s Note

See the prepositions “between” and “with” in the questions above? These helps us to formulate Questions about relationships between variables.

8 Scatter Plots

Which are the numeric variables in movies?

movies_quant <- movies_modified %>%
  drop_na() %>%
  select(where(is.numeric))
movies_quant %>% names()
 [1] "rotten_tomatoes"      "audience_score"       "theaters_open_week"  
 [4] "bo_average_open_week" "domestic_gross"       "foreign_gross"       
 [7] "world_gross"          "budget"               "profitability"       
[10] "opening_weekend"

Now let us plot their relationships. We will use scatter plots whose shape shows us if there is a relationship between the two variables at hand. In general, if the “cloud of points” is tipped toward one side (up or down), then there is a possible relationship between the two variables. If the points are scattered all over the place, then there is no relationship between the two variables.

ggplot2::theme_set(new = theme_custom())

movies_modified %>%
  drop_na() %>%
  gf_point(profitability ~ budget) %>%
  gf_lm() %>%
  gf_labs(
    title = "Profitability vs budget",
    subtitle = "Movie profitability: Does budget affect profitability?"
  )

ggplot2::theme_set(new = theme_custom())

movies_modified %>%
  drop_na() %>%
  gf_point(profitability ~ opening_weekend) %>%
  gf_lm() %>%
  gf_labs(
    title = "Profitability vs Opening Weekend",
    subtitle = "Movies: Does Opening Week Earnings indicate profitability?"
  )

ggplot2::theme_set(new = theme_custom())

movies_modified %>%
  drop_na() %>%
  gf_point(profitability ~ rotten_tomatoes) %>%
  gf_lm() %>%
  gf_labs(
    title = "Profitability vs Rotten Tomatoes",
    subtitle = "Movie Ratings: Does Rotten Tomatoes affect profitability?"
  )

We can split some of the scatter plots using one or other of the Qual variables. For instance, is the relationship between the two ratings the same, regardless of movie genre?

ggplot2::theme_set(new = theme_custom())

movies_modified %>%
  drop_na() %>%
  gf_point(profitability ~ audience_score,
    color = ~genre
  ) %>%
  gf_lm() %>%
  gf_labs(
    title = "Profitability vs Audience Score",
    subtitle = "Movie Ratings: Trends by Genre"
  )

ggplot2::theme_set(new = theme_custom())

movies_modified %>%
  drop_na() %>%
  ggplot(aes(y = profitability, x = budget)) +
  geom_point() +
  geom_lm() +
  labs(
    title = "Scatter Plot",
    subtitle = "Movie Gross Earnings: Domestics vs World"
  )

ggplot2::theme_set(new = theme_custom())

movies_modified %>%
  drop_na() %>%
  ggplot(aes(opening_weekend, profitability)) +
  geom_point() +
  geom_lm() +
  labs(
    title = "Scatter Plot",
    subtitle = "Movies: Does Opening Week Earnings indicate profitability?"
  )

ggplot2::theme_set(new = theme_custom())

movies_modified %>%
  drop_na() %>%
  ggplot(aes(y = profitability, x = rotten_tomatoes)) +
  geom_point() +
  geom_lm() +
  labs(
    title = "Scatter Plot",
    subtitle = "Movie Ratings: Tomatoes vs Audience"
  )

ggplot2::theme_set(new = theme_custom())

movies_modified %>%
  drop_na() %>%
  ggplot(aes(y = profitability, x = audience_score, color = genre)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  labs(
    title = "Scatter Plot",
    subtitle = "Movie Ratings: Trends by Genre"
  )

NoteBusiness Insight from movies scatter plots

We have fitted a trend line to each of the scatter plots.

  • profitability and budget: The trend line is mildly negative…seeming to suggest that increasing the budget does not necessarily increase the profitability. In fact, it seems to suggest that increasing the budget decreases the profitability! But is this a significant trend? We will see later.
  • profitability and opening_weekend: The trend line is positive, suggesting that increasing the opening_weekend earnings increases the profitability. This is a good sign for movie makers! But again, not a very markedly upward trend, so we need to check if this is significant.
  • profitability and rotten_tomatoes: The trend line is positive, suggesting that increasing the rotten_tomatoes rating increases the profitability. Yet again, not a very markedly upward trend, so we need to check if this is significant.
  • profitability and audience_score: The trend lines are mostly flat, suggesting that increasing the audience_score does not do much for profitability. The slope for Horror is higher than that for the other genre-s ! Oh hell…people are paying to be scared witless???

Note that there are two horror movies that have been hugely successful. However, these are outliers, and are also located, in the dataset, at a place where they do not tip the trend line too much. They have limited influence, concept that becomes important with Regression Analysis.

9 Quantizing Correlation

So we see that there are visible relationships between Quant variables. How do we quantize this relationship, into a correlation score? Let us first define for ourselves what a correlation score is, and then we will see how to calculate it.

9.1 Pearson Correlation coefficient

The degree of association is measured by a correlation coefficient, denoted by r. It is sometimes called Pearson’s correlation coefficient after its originator and is a measure of linear association. (If a curved line is needed to express the relationship, other and more complicated measures of the correlation must be used.)

The correlation coefficient is measured on a scale that varies from + 1 through 0 to – 1. Complete correlation between two variables is expressed by either + 1 or -1. When one variable increases as the other increases the correlation is positive; when one decreases as the other increases it is negative.

In formal terms, the correlation between two variables \(x\) and \(y\) is defined as:

\[ \rho = E\left[\frac{(x - \mu_{x}) * (y - \mu_{y})}{(\sigma_x)*(\sigma_y)}\right] \tag{1}\]

where \(E\) is the expectation operator ( i.e taking mean ). Think of this as the average of the products of two scaled residuals.

TipPearson Correlation uses z-scores

We can see \((x-\mu_x)/\sigma_x\) is a centering and scaling of the variable \(x\). Recall from our discussion on Quantities that this is called the z-score of x.

(a) Segments show Residuals from the Mean
(b) Blue Areas are positive, Red areas are negative
Figure 2: Showing Calculation of Correlation Coefficient

Pearson correlation assumes that the relationship between the two variables is linear. There are of course many other types of correlation measures: some which work when this is not so. Type vignette("types", package = "correlation") in your Console to see the vignette from the correlation package that discusses various types of correlation measures.

OK, so how do we calculate this correlation coefficient? And how do we visualize it too? ( Remember: we want to visualize our analysis! )

10 Correlation Plots and Scores

We will use the GGally package to visualize correlation scores, and a formal correlation test with the mosaic package to calculate them.

10.1 Using GGally

By default, GGally::ggpairs() provides:

  • two different comparisons of each pair of columns
  • displays either the density or count of the respective variable along the diagonal.
  • With different parameter settings, the diagonal can be replaced with the axis values and variable labels.
ggplot2::theme_set(new = theme_bw(base_family = "Roboto Condensed", base_size = 14))

GGally::ggpairs(
  movies_modified %>% drop_na(),
  # Select Quant variables only for now
  columns = c(
    "profitability", "budget", "domestic_gross", "foreign_gross"
  ),
  switch = "both",
  # axis labels in more traditional locations(left and bottom)

  progress = FALSE,
  # no compute progress messages needed

  # Choose the diagonal graphs (always single variable! Think!)
  diag = list(continuous = "barDiag"),
  # choosing histogram,not density

  # Choose lower triangle graphs, two-variable graphs
  lower = list(continuous = wrap("smooth", alpha = 0.3, se = FALSE)),
  title = "Movies Data Correlations Plot #1"
)

NoteBusiness Insight from Pairs Plot #1
  • profitability and budget have a very slight negative correlation, but this does not appear to be significant.
  • profitability has low correlation scores with both DomesticGross (\(.181\)) and also with ForeignGross (\(0.123\)).
  • DomesticGross and ForeignGross have a very high correlation score (\(0.96\)), which is expected, since most movies are released in both markets, and the earnings are usually similar. However, as noted, neither influences profitability much. Sigh.
  • Note in passing that the profitability and both the “Gross” related variables have highly skewed distributions. That is the nature of the movie business!

10.2 Using cor_test

We must always keep in mind that we are looking at a dataset, a sample, and not the entire population. So, we need to be careful about making claims about the population based on our sample. What this means is that our sample-estimated correlation scores \(r\) are not the final word on the correlation between two population-variables, \(\rho\).

We need to conduct a statistical test to see if the correlation is significant, i.e. if it is likely to be true for the entire population from which our sample was drawn, and also assign numbers to the uncertainty that we must have in our correlation estimate.

Both correlations scores, and the uncertainty we have can be obtained by conducting a formal test in R. We will use the mosaic function cor_test to get these results:

mosaic::cor_test(profitability ~ budget, data = movies_modified) %>%
  broom::tidy() %>%
  knitr::kable(
    digits = 2,
    caption = "Movie profitability vs budget"
  )
Movie profitability vs budget
estimate statistic p.value parameter conf.low conf.high method alternative
-0.08 -0.96 0.34 132 -0.25 0.09 Pearson’s product-moment correlation two.sided 
mosaic::cor_test(domestic_gross ~ budget, data = movies_modified) %>%
  broom::tidy() %>%
  knitr::kable(
    digits = 2,
    caption = "Movie Domestic Gross vs budget"
  )
Movie Domestic Gross vs budget
estimate statistic p.value parameter conf.low conf.high method alternative
0.7 11.06 0 131 0.6 0.77 Pearson’s product-moment correlation two.sided 
mosaic::cor_test(foreign_gross ~ budget, data = movies_modified) %>%
  broom::tidy() %>%
  knitr::kable(
    digits = 2,
    caption = "Movie Foreign Gross vs budget"
  )
Movie Foreign Gross vs budget
estimate statistic p.value parameter conf.low conf.high method alternative
0.69 10.22 0 118 0.58 0.77 Pearson’s product-moment correlation two.sided
NoteBusiness Insights from Correlation Tests

The budget and profitability are not well correlated, sadly. We see this from the p.value which is \(0.34\) and the confidence values for the correlation estimate which also cover \(0\).

However, both DomesticGross and ForeignGross are well correlated with budget.

Look at the conf.low and conf.high coloumns: these are calculated uncertainty limits on the estimated correlation. If these *do not straddle \(0\), then we m-a-y infer that the correlation is significant. More when we study Inference for Correlation in a later module.

The ErrorBar Plot for Correlations

As stated earlier, in our dataset we have a specific dependent or target variable, which represents the outcome of our experiment or our business situation. The remaining variables are usually independent or predictor variables. A very useful thing to know, and to view, would be the correlations of all independent variables. Using the correlation package from the easystats family of R packages, this can be very easily achieved. Let us quickly do this for the familiar mtcars dataset: we will quickly glimpse it, identify the target variable, and plot the correlations:

glimpse(mtcars)
Rows: 32
Columns: 11
$ mpg  <dbl> 21.0, 21.0, 22.8, 21.4, 18.7, 18.1, 14.3, 24.4, 22.8, 19.2, 17.8,…
$ cyl  <dbl> 6, 6, 4, 6, 8, 6, 8, 4, 4, 6, 6, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 8,…
$ disp <dbl> 160.0, 160.0, 108.0, 258.0, 360.0, 225.0, 360.0, 146.7, 140.8, 16…
$ hp   <dbl> 110, 110, 93, 110, 175, 105, 245, 62, 95, 123, 123, 180, 180, 180…
$ drat <dbl> 3.90, 3.90, 3.85, 3.08, 3.15, 2.76, 3.21, 3.69, 3.92, 3.92, 3.92,…
$ wt   <dbl> 2.620, 2.875, 2.320, 3.215, 3.440, 3.460, 3.570, 3.190, 3.150, 3.…
$ qsec <dbl> 16.46, 17.02, 18.61, 19.44, 17.02, 20.22, 15.84, 20.00, 22.90, 18…
$ vs   <dbl> 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0,…
$ am   <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0,…
$ gear <dbl> 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3,…
$ carb <dbl> 4, 4, 1, 1, 2, 1, 4, 2, 2, 4, 4, 3, 3, 3, 4, 4, 4, 1, 2, 1, 1, 2,…
## Target variable: mpg
## Calculate all correlations
cor <- correlation::correlation(mtcars)
cor

We see correlation between all pairs of variables. We need to choose just those with target variable mpg:

ggplot2::theme_set(new = theme_custom())

cor %>%
  # Filter for target variable `mpg` and plot
  filter(Parameter1 == "mpg") %>%
  gf_errorbar(CI_low + CI_high ~ reorder(Parameter2, r),
    width = 0.5
  ) %>%
  gf_point(r ~ reorder(Parameter2, r), size = 4, color = "red") %>%
  gf_hline(yintercept = 0, color = "grey", linewidth = 2) %>%
  gf_labs(
    title = "Correlation Errorbar Chart",
    subtitle = "Target variable: mpg",
    x = "Predictor Variable",
    y = "Correlation Score with mpg"
  )

NoteBusiness Insights from ErrorBar Plot
  • Several variables are negatively correlated and some are positively correlated with ’mpg`. (The grey line shows “zero correlation”)
  • Since none of the error bars straddle zero, the correlations are mostly significant.

11 An Interactive Correlation Game

Head off to this interactive game website where you can play with correlations!

https://openintro.shinyapps.io/correlation_game/

12 Simpson’s Paradox

See how the overall correlation/regression line slopes upward, whereas that for the individual groups slopes downward!! This is an example of Simpson’s Paradox!

13 Your Turn

  1. Try to play this online Correlation Game.
Note2. School Expenditure and Grades.

Note3. Gas Prices and Consumption

As described here. Note the log-transformed Quant data…why do you reckon this was done in the data set itself?

Note4. Horror Movies (Bah.You awful people..)

Note6. Food Delivery Times

14 Wait, But Why?

  • Scatter Plots, when they show “linear” clouds, tell us that there is some relationship between two Quant variables we have just plotted
  • If so, then if one is the target variable you are trying to design for, then the other independent, or controllable, variable is something you might want to design with.
Important

Target variables are usually plotted on the Y-axis, while Predictor variables are on the X-Axis, in a Scatter Plot. Why? Because \(y = mx + c\) !

  • Correlation scores are good indicators of things that are, well, related. While one variable may not necessarily cause another, a good correlation score may indicate how to chose a good predictor.
  • That is something we will see when we examine Linear Regression
  • Always, always, plot and test your data! Both numerical summaries as tables, and graphical summaries as charts, are necessary! See below!!
WarningAnd How about these datasets?
dataset mean_x mean_y std_dev_x std_dev_y corr_x_y
away 54 48 17 27 -0.064
bullseye 54 48 17 27 -0.069
circle 54 48 17 27 -0.068
dino 54 48 17 27 -0.064
dots 54 48 17 27 -0.06
h_lines 54 48 17 27 -0.062
high_lines 54 48 17 27 -0.069
slant_down 54 48 17 27 -0.069
slant_up 54 48 17 27 -0.069
star 54 48 17 27 -0.063
v_lines 54 48 17 27 -0.069
wide_lines 54 48 17 27 -0.067
x_shape 54 48 17 27 -0.066
Figure 3: Datasaurus Dirty Dozen: All have (almost) identical summary statistics, but look so different!
Figure 4: Datasaurus Dirty Dozen: All have (almost) identical summary statistics, but look so different!

Yes, you did want to plot that cute T-Rex, didn’t you? Here is the data then!!

Warning
  • Can selling more ice-cream make people drown?
  • Use your head about pairs of variables. Do not fall into this trap)

15 Conclusions

Scatter Plots give a us sense of change; whether it is linear or non-linear. We can get an idea of correlation between variables with a scatter plot. Our workflow for evaluating correlations between target variable and several other predictor variables uses several packages such as GGally, corrplot, correlation, and of course mosaic for correlation tests.

16 AI Generated Summary and Podcast

This document focusses on correlation between quantitative variables. It examines different ways to visualize correlations, including scatter plots and correlograms. The document provides examples of how to use R packages like GGally and corrplot to create these visualizations and correlation tests to assess the strength and significance of relationships between variables. The tutorial uses the HollywoodMovies2011 and mtcars datasets as examples to demonstrate these concepts.

17 References

  1. Winston Chang (2024). R Graphics Cookbook. https://r-graphics.org
  2. Minimal R using mosaic. https://cran.r-project.org/web/packages/mosaic/vignettes/MinimalRgg.pdf
  3. Antoine Soetewey. Pearson, Spearman and Kendall correlation coefficients by hand https://www.r-bloggers.com/2023/09/pearson-spearman-and-kendall-correlation-coefficients-by-hand/
  4. Taiyun Wei, Viliam Simko. An Introduction to corrplot Package. https://cran.r-project.org/web/packages/corrplot/vignettes/corrplot-intro.html
R Package Citations
Package Version Citation
broom 1.0.10 Robinson, Hayes, and Couch (2025)
corrplot 0.95 Wei and Simko (2024)
datasauRus 0.1.9 Gillespie et al. (2025)
GGally 2.4.0 Schloerke et al. (2025)
ggExtra 0.11.0 Attali and Baker (2025)
latex2exp 0.9.6 Meschiari (2022)
Attali, Dean, and Christopher Baker. 2025. ggExtra: Add Marginal Histograms to “ggplot2,” and More “ggplot2” Enhancements. https://doi.org/10.32614/CRAN.package.ggExtra.
Gillespie, Colin, Steph Locke, Rhian Davies, and Lucy D’Agostino McGowan. 2025. datasauRus: Datasets from the Datasaurus Dozen. https://doi.org/10.32614/CRAN.package.datasauRus.
Meschiari, Stefano. 2022. Latex2exp: Use LaTeX Expressions in Plots. https://doi.org/10.32614/CRAN.package.latex2exp.
Robinson, David, Alex Hayes, and Simon Couch. 2025. broom: Convert Statistical Objects into Tidy Tibbles. https://doi.org/10.32614/CRAN.package.broom.
Schloerke, Barret, Di Cook, Joseph Larmarange, Francois Briatte, Moritz Marbach, Edwin Thoen, Amos Elberg, and Jason Crowley. 2025. GGally: Extension to “ggplot2”. https://doi.org/10.32614/CRAN.package.GGally.
Wei, Taiyun, and Viliam Simko. 2024. R Package “corrplot”: Visualization of a Correlation Matrix. https://github.com/taiyun/corrplot.
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Citation

BibTeX citation:
@online{v.2022,
  author = {V., Arvind},
  title = {\textless Iconify-Icon Icon=“icon-Park-Outline:change”
    Width=“1.2em”
    Height=“1.2em”\textgreater\textless/Iconify-Icon\textgreater{}
    {Change}},
  date = {2022-11-22},
  url = {https://madhatterguide.netlify.app/content/courses/Analytics/10-Descriptive/Modules/30-Change/},
  langid = {en},
  abstract = {How one variable changes with another}
}
For attribution, please cite this work as:
V., Arvind. 2022. “<Iconify-Icon Icon=‘icon-Park-Outline:change’ Width=‘1.2em’ Height=‘1.2em’></Iconify-Icon> Change.” November 22, 2022. https://madhatterguide.netlify.app/content/courses/Analytics/10-Descriptive/Modules/30-Change/.