.. _2-3-tools-for-numerical-quantitative-data:

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Tools for Numerical (Quantitative) Data
=================================================================

Numerical data offers richer visualization possibilities than categorical data
because it now contains information about *distance* between values.
We are now interested in the *overall shape* of the distribution,
*where the numbers are clustered*, and *how far they spread*. 
A **histogram** answers all three at a glance. 
In this chapter, we'll focus on building good histograms.

.. admonition:: Road Map 🧭
   :class: important

   * Visualize numerical variables with **histograms**.
   * Understand the impact of choosing different numbers of **classes/bins**
     for a histogram.

Building your first histogram
--------------------------------

A **histogram** divides the number line into adjacent **bins** of equal 
width and draws a bar whose height equals the count (or relative frequency) 
falling in each bin.

Example 1 – insect counts (discrete numeric)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter2/insect-count-hist.png
   :alt: Histogram of insect counts from Beall (1942)
   :align: center

This histogram displays insect counts from an agricultural study testing different insecticide sprays. Each bar represents the number of observations falling within a range of insect counts. The twin peaks reflect two different spray formulations—detail we will dig into in Chapter 3 when we learn how to *separate* a quantitative variable by group.

.. code-block:: r

   library(ggplot2)
   data(InsectSprays)

   # rule of thumb: round(sqrt(n)) + 2 bins
   n_obs  <- nrow(InsectSprays)
   n_bins <- round(sqrt(n_obs)) + 2

   ggplot(InsectSprays, aes(x = count)) +
     geom_histogram(bins = n_bins, colour = "black", fill = "skyblue", linewidth = 1.2) +
     labs(title = "Distribution of insect counts (Beall, 1942)",
          x = "Number of insects", y = "Frequency")

Choosing the number of bins
------------------------------

Bin width is a Goldilocks choice: too wide hides detail, too narrow shows only noise. For sample sizes up to a few hundred, the square-root rule works well:

.. math::
   \text{bins} \;=\; \bigl\lceil \sqrt{n} \bigr\rceil + 2.

For larger datasets (n > 400), experiment with 20-30 bins until the main pattern is clear without a saw-tooth fringe. The goal is to reveal the data's fundamental structure while filtering out noise.

Example 2 – furnace energy use (continuous numeric)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We illustrate bin width choice by plotting the same furnace energy consumption data four times with different bin counts.

.. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter2/bin-width-comparison.png
   :alt: Furnace BTU histograms: 6, 10, 15 and 30 bins
   :align: center

These four histograms display the same furnace energy consumption data with different bin counts:

1. **6 bins (top left)**: Oversmooths the data, hiding important features
2. **10 bins (top right)**: Balances detail and clarity, revealing the general right-skewed shape
3. **15 bins (bottom left)**: Shows more granular structure but begins to display some potentially random fluctuation
4. **30 bins (bottom right)**: Too detailed, resulting in a jagged appearance dominated by sampling variability

Ten to fifteen bins reveal the slight right-skew without drowning the eye in high-frequency jitter.

.. code-block:: r

   library(ggplot2)
   furnace <- read.csv("furnace.txt")   # replace with your path

   for (bins in c(6, 10, 15, 30)) {
     p <- ggplot(furnace, aes(Consumption)) +
       geom_histogram(bins = bins, colour = "black", fill = "grey70") +
       labs(title = paste(bins, "bins"), x = "BTU", y = "Frequency")
     print(p)
   }

Discrete vs. continuous data visualization
--------------------------------------------

For discrete numerical data with few unique values (like counts from 0-7), a bar graph may be more appropriate than a histogram. Bar graphs place one bar at each distinct value, with space between them to indicate their categorical nature, even if the categories are numbers.

The difference between histograms and bar graphs for quantitative data lies in how they handle the underlying value space:
* Bar graphs treat each value as a distinct category
* Histograms group ranges of values into bins with no gaps between bars

For discrete data with many possible values, histograms become the preferred choice to avoid cluttered displays.

Example – Number of children per family
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter2/children-bar.png
   :alt: Bar graph of number of children per family
   :align: center

This bar graph displays the number of children per family from a survey of 100 couples aged 30-40. Unlike a histogram, which would group values into bins, this graph shows the exact count for each possible value (0 through 7 children). Since there are only a few possible values, this bar graph displays the discrete data more effectively than a histogram would.

Enhancing histograms – density & normal overlay
-------------------------------------------------

A quick visual test for *normality* is to overlay a theoretical bell curve and compare. We can also add a density curve that smooths the histogram data to get a better sense of the underlying distribution.

Example 1 – Residential furnace energy use
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter2/furnace-density.png
   :alt: Histogram with kernel density (red) and normal curve (blue)
   :align: center

This enhanced histogram of furnace energy consumption includes:
* Blue bars showing the distribution of BTU measurements
* A red density curve that smooths the data 
* A blue normal distribution curve for comparison

The gap between these curves reveals that the data is slightly right-skewed compared to a normal distribution.

.. code-block:: r

   library(ggplot2)
   furnace <- read.csv("furnace.txt")   # replace with your path
   xbar <- mean(furnace$Consumption)
   s    <- sd(furnace$Consumption)
   bins <- round(sqrt(nrow(furnace))) + 2

   ggplot(furnace, aes(Consumption)) +
     geom_histogram(aes(y = after_stat(density)), bins = bins,
                   fill = "lightblue", colour = "black") +
     geom_density(colour = "red",  linewidth = 1.2) +
     stat_function(fun = dnorm, args = list(mean = xbar, sd = s),
                   colour = "blue", linewidth = 1.2) +
     labs(title = "Residential furnace energy consumption",
          y = "Density", x = "BTU")

..
   Example 2 – Old Faithful eruption lengths
   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   .. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter2/faithful-hist.png
      :alt: Histogram of Old Faithful eruption times with overlays
      :align: center

   This histogram of eruption times from Old Faithful geyser clearly shows a bimodal distribution—two distinct peaks indicating two different types of eruptions. Note how dramatically the actual distribution (red density curve) differs from the normal curve (blue), which assumes a single central peak.


   Additional examples – Black Cherry tree data
   -----------------------------------------------

   The `trees` dataset contains measurements of Black Cherry trees, including their diameter (Girth), Height, and Volume. Below are histograms for two of these variables.

   .. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter2/tree-girth-hist.png
      :alt: Histogram of Black Cherry tree diameters with overlays
      :align: center

   .. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter2/tree-height-hist.png
      :alt: Histogram of Black Cherry tree heights with overlays
      :align: center

   These histograms reveal different distribution patterns:
   * Tree diameter (girth) follows a roughly normal distribution with a slight right skew
   * Tree height shows more pronounced deviation from normality with multiple smaller peaks


   Histograms by group – insect spray effectiveness
   ---------------------------------------------------

   When we have both categorical and quantitative variables, we can create separate histograms for each category to compare distributions.

   .. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter2/insect-spray-facets.png
      :alt: Panel of histograms for six insect sprays
      :align: center

   These faceted histograms compare the effectiveness of six different insect spray formulations (labeled A through F). Each panel shows the distribution of insect counts after applying that specific spray type.

   The visualization reveals several patterns:
   * Sprays C, D, and E compress the distribution near zero (highly effective)
   * Sprays A and B leave many insects alive (less effective)
   * Spray F shows intermediate effectiveness

   This qualitative insight sets the stage for formal statistical tests later in the course.

.. code-block:: r

   library(ggplot2)

   ggplot(InsectSprays, aes(x = count, fill = spray)) +
     geom_histogram(bins = 5, colour = "black", linewidth = 0.8) +
     facet_wrap(~ spray, scales = "free_y") +
     theme_minimal() + theme(legend.position = "none") +
     labs(title = "Insect count distribution by spray type",
          x = "Number of insects", y = "Frequency")



Bringing It All Together
----------------------------

Visualizing numerical data effectively requires choosing the right technique for your 
specific data type and research question. Histograms serve as the foundation for 
understanding numerical distributions, revealing patterns that guide further analysis. 
As we progress, we'll build on these visual insights with formal numerical summaries 
and statistical tests.

.. admonition:: Key Takeaways 📝
   :class: important

   1. Histograms turn walls of numbers into shape, center, spread, and possible outliers.
   2. Use :math:`\lceil \sqrt{n} \rceil + 2` bins as a starting point, then adjust by eye.
   3. Overlay normal curves to assess skewness and normality.