Quantities

How much of this and that?

Quant Variables

Histograms

Mean

Published

November 15, 2022

Modified

October 19, 2026

Abstract

Histograms and their Siblings

“The fear of death follows from the fear of life. A man who lives fully is prepared to die at any time.”

— Mark Twain

1 Setting up R Packages

library(tidyverse) # Sine Qua Non package in R
library(mosaic) # Our Favourite Bag of Tricks
library(ggformula) # Graphing
library(skimr) # Data Inspection and Summary
##
library(crosstable) # Fast stats for multiple variables in table form
library(tinytable) # Elegant Tables for our data
library(visdat) # Mapping missing data
library(naniar) # Missing data visualization and munging
library(janitor) # Clean the data
library(tinytable) # Printing Tables for our data
library(DT) # Interactive Tables for our data

2 What graphs will we see today?

Variable #1	Variable #2	Chart Names	Chart Shape
Quant	None	Histogram

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: Golf Drive Distance over the years

4.1 What does this Chart tell us?

What do we see here? In about two-and-a-half decades, golf drive distances have increased, on the average, by 35 yards. The maximum distance has also gone up by 30 yards, and the minimum is now at 250 yards, which was close to average in 1983! What was a decent average in 1983 is just the bare minimum in 2017!!

Is it the dimples that the golf balls have? But these have been around a long time…or is it the clubs, and the swing technique invented by more recent players?

4.2 Hans Rosling’s famous Presentation

Now, let us listen to the late great Hans Rosling from the Gapminder Project, which aims at telling stories of the world with data, to remove systemic biases about poverty, income and gender related issues.

5 How do Histograms Work?

Histograms are best to show the distribution of raw Quantitative data, by displaying the number of values that fall within defined ranges, often called buckets or bins. We use a Quant variable on the x-axis and the histogram shows us how frequently different values occur for that variable by showing counts/frequencies on the y-axis. The x-axis is typically broken up into “buckets” or ranges for the x-variable. And usually you can adjust the bucket ranges to explore frequency patterns. For example, you can widen histogram buckets from 0-1, 1-2, 2-3, etc. to 0-2, 2-4, etc.

Although Bar Charts may look similar to Histograms, the two are different. Bar Charts show counts of observations with respect to a Qualitative variable. For instance, bar charts show categorical data with multiple levels, such as fruits, clothing, household products in an inventory. Each bar has a height proportional to the count per shirt-size, in this example.

Histograms do not usually show spaces between buckets because the buckets represent contiguous ranges, while bar charts show spaces to separate each (unconnected) category/level within a Qual variable.

6 Case Study-1: `diamonds` dataset

We will first look at at a dataset that is directly available in R, the diamonds dataset. As always, we will use our munging process to clean up the data for analysis.

data("diamonds", package = "ggplot2")
glimpse(diamonds)

Rows: 53,940
Columns: 10
$ carat   <dbl> 0.23, 0.21, 0.23, 0.29, 0.31, 0.24, 0.24, 0.26, 0.22, 0.23, 0.…
$ cut     <ord> Ideal, Premium, Good, Premium, Good, Very Good, Very Good, Ver…
$ color   <ord> E, E, E, I, J, J, I, H, E, H, J, J, F, J, E, E, I, J, J, J, I,…
$ clarity <ord> SI2, SI1, VS1, VS2, SI2, VVS2, VVS1, SI1, VS2, VS1, SI1, VS1, …
$ depth   <dbl> 61.5, 59.8, 56.9, 62.4, 63.3, 62.8, 62.3, 61.9, 65.1, 59.4, 64…
$ table   <dbl> 55, 61, 65, 58, 58, 57, 57, 55, 61, 61, 55, 56, 61, 54, 62, 58…
$ price   <int> 326, 326, 327, 334, 335, 336, 336, 337, 337, 338, 339, 340, 34…
$ x       <dbl> 3.95, 3.89, 4.05, 4.20, 4.34, 3.94, 3.95, 4.07, 3.87, 4.00, 4.…
$ y       <dbl> 3.98, 3.84, 4.07, 4.23, 4.35, 3.96, 3.98, 4.11, 3.78, 4.05, 4.…
$ z       <dbl> 2.43, 2.31, 2.31, 2.63, 2.75, 2.48, 2.47, 2.53, 2.49, 2.39, 2.…

We have ~54000 observations, with 10 variables. There are no missing values in any variable. It is a nice mix of Quantitative and Qualitative variables.

Since the variables are nicely converted to factor where needed, and the names are already clean we will just use janitor to essentially rename our dataset.

Show the Code

diamonds_modified <- diamonds %>%
  janitor::clean_names(case = "snake") %>%
  janitor::remove_empty(which = c("rows", "cols")) # Empty columns and rows if any
diamonds_modified %>% glimpse()

Rows: 53,940
Columns: 10
$ carat   <dbl> 0.23, 0.21, 0.23, 0.29, 0.31, 0.24, 0.24, 0.26, 0.22, 0.23, 0.…
$ cut     <ord> Ideal, Premium, Good, Premium, Good, Very Good, Very Good, Ver…
$ color   <ord> E, E, E, I, J, J, I, H, E, H, J, J, F, J, E, E, I, J, J, J, I,…
$ clarity <ord> SI2, SI1, VS1, VS2, SI2, VVS2, VVS1, SI1, VS2, VS1, SI1, VS1, …
$ depth   <dbl> 61.5, 59.8, 56.9, 62.4, 63.3, 62.8, 62.3, 61.9, 65.1, 59.4, 64…
$ table   <dbl> 55, 61, 65, 58, 58, 57, 57, 55, 61, 61, 55, 56, 61, 54, 62, 58…
$ price   <int> 326, 326, 327, 334, 335, 336, 336, 337, 337, 338, 339, 340, 34…
$ x       <dbl> 3.95, 3.89, 4.05, 4.20, 4.34, 3.94, 3.95, 4.07, 3.87, 4.00, 4.…
$ y       <dbl> 3.98, 3.84, 4.07, 4.23, 4.35, 3.96, 3.98, 4.11, 3.78, 4.05, 4.…
$ z       <dbl> 2.43, 2.31, 2.31, 2.63, 2.75, 2.48, 2.47, 2.53, 2.49, 2.39, 2.…

6.1 Examine the Data

As per our Workflow, we will look at the data using all the three methods we have seen.

glimpse(diamonds_modified)

Rows: 53,940
Columns: 10
$ carat   <dbl> 0.23, 0.21, 0.23, 0.29, 0.31, 0.24, 0.24, 0.26, 0.22, 0.23, 0.…
$ cut     <ord> Ideal, Premium, Good, Premium, Good, Very Good, Very Good, Ver…
$ color   <ord> E, E, E, I, J, J, I, H, E, H, J, J, F, J, E, E, I, J, J, J, I,…
$ clarity <ord> SI2, SI1, VS1, VS2, SI2, VVS2, VVS1, SI1, VS2, VS1, SI1, VS1, …
$ depth   <dbl> 61.5, 59.8, 56.9, 62.4, 63.3, 62.8, 62.3, 61.9, 65.1, 59.4, 64…
$ table   <dbl> 55, 61, 65, 58, 58, 57, 57, 55, 61, 61, 55, 56, 61, 54, 62, 58…
$ price   <int> 326, 326, 327, 334, 335, 336, 336, 337, 337, 338, 339, 340, 34…
$ x       <dbl> 3.95, 3.89, 4.05, 4.20, 4.34, 3.94, 3.95, 4.07, 3.87, 4.00, 4.…
$ y       <dbl> 3.98, 3.84, 4.07, 4.23, 4.35, 3.96, 3.98, 4.11, 3.78, 4.05, 4.…
$ z       <dbl> 2.43, 2.31, 2.31, 2.63, 2.75, 2.48, 2.47, 2.53, 2.49, 2.39, 2.…

skim(diamonds_modified)

Data summary
Name	diamonds_modified
Number of rows	53940
Number of columns	10
_______________________
Column type frequency:
factor	3
numeric	7
________________________
Group variables	None

Variable type: factor

skim_variable	complete_rate	ordered	n_unique	top_counts
cut	1	TRUE	5	Ide: 21551, Pre: 13791, Ver: 12082, Goo: 4906
color	1	TRUE	7	G: 11292, E: 9797, F: 9542, H: 8304
clarity	1	TRUE	8	SI1: 13065, VS2: 12258, SI2: 9194, VS1: 8171

Variable type: numeric

skim_variable	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
carat	1	0.80	0.47	0.2	0.40	0.70	1.04	5.01	▇▂▁▁▁
depth	1	61.75	1.43	43.0	61.00	61.80	62.50	79.00	▁▁▇▁▁
table	1	57.46	2.23	43.0	56.00	57.00	59.00	95.00	▁▇▁▁▁
price	1	3932.80	3989.44	326.0	950.00	2401.00	5324.25	18823.00	▇▂▁▁▁
x	1	5.73	1.12	0.0	4.71	5.70	6.54	10.74	▁▁▇▃▁
y	1	5.73	1.14	0.0	4.72	5.71	6.54	58.90	▇▁▁▁▁
z	1	3.54	0.71	0.0	2.91	3.53	4.04	31.80	▇▁▁▁▁

inspect(diamonds_modified)


categorical variables:  
     name   class levels     n missing
1     cut ordered      5 53940       0
2   color ordered      7 53940       0
3 clarity ordered      8 53940       0
                                   distribution
1 Ideal (40%), Premium (25.6%) ...             
2 G (20.9%), E (18.2%), F (17.7%) ...          
3 SI1 (24.2%), VS2 (22.7%), SI2 (17%) ...      

quantitative variables:  
   name   class   min     Q1  median      Q3      max         mean           sd
1 carat numeric   0.2   0.40    0.70    1.04     5.01    0.7979397    0.4740112
2 depth numeric  43.0  61.00   61.80   62.50    79.00   61.7494049    1.4326213
3 table numeric  43.0  56.00   57.00   59.00    95.00   57.4571839    2.2344906
4 price integer 326.0 950.00 2401.00 5324.25 18823.00 3932.7997219 3989.4397381
5     x numeric   0.0   4.71    5.70    6.54    10.74    5.7311572    1.1217607
6     y numeric   0.0   4.72    5.71    6.54    58.90    5.7345260    1.1421347
7     z numeric   0.0   2.91    3.53    4.04    31.80    3.5387338    0.7056988
      n missing
1 53940       0
2 53940       0
3 53940       0
4 53940       0
5 53940       0
6 53940       0
7 53940       0

6.2 Data Dictionary

Quantitative Data

carat(dbl): weight of the diamond 0.2-5.01
depth(dbl): depth total depth percentage 43-79
table(dbl): width of top of diamond relative to widest point 43-95
price(dbl): price in US dollars $326-$18,823
x(dbl): length in mm 0-10.74
y(dbl): width in mm 0-58.9
z(dbl): depth in mm 0-31.8

Qualitative Data

cut: diamond cut Fair, Good, Very Good, Premium, Ideal
color: diamond color J (worst) to D (best). (7 levels)
clarity. measurement of how clear the diamond is I1 (worst), SI2, SI1, VS2, VS1, VVS2, VVS1, IF (best).

These have 5, 7, and 8 levels respectively. The fact that the class for these is ordered suggests that these are factors and that the levels have a sequence/order.

Business Insights on Examining the diamonds dataset

This is a large dataset (54K rows).
There are several Qualitative variables:
carat, price, x, y, z, depth and table are Quantitative variables.
There are no missing values for any variable, all are complete with 54K entries.

6.3 Hypothesis and Research Questions

Let us formulate a few Questions about this dataset. At some point, we might develop a hunch or two, and these would become our hypotheses to investigate. This is an iterative process!

Hypothesis and Research Questions

The target variable for an experiment that resulted in this data might be the price variable. Which is a numerical Quant variable.
There are also predictor variables such as carat (Quant), color(Qual), cut(Qual), and clarity(Qual).
Other predictor variables might be x, y, depth, table(all Quant)
Research Questions:
- What is the distribution of the target variable price?
- What is the distribution of the predictor variable carat?
- Does a price distribution vary based upon type of cut, clarity, and color?

These should do for now. Try and think of more Questions!

6.4 Plotting Histograms

Let’s plot some histograms to answer each of the Hypothesis questions above.

6.5 Question-1: What is the distribution of the target variable `price`?

gf_histogram(~price, data = diamonds_modified) %>%
  gf_labs(
    title = "Plot 1A: Diamond Prices",
    caption = "ggformula"
  )

## More bins
gf_histogram(~price,
  data = diamonds_modified,
  bins = 100
) %>%
  gf_labs(
    title = "Plot 1B: Diamond Prices",
    caption = "ggformula"
  )

ggplot(data = diamonds_modified) +
  geom_histogram(aes(x = price)) +
  labs(
    title = "Plot 1A: Diamond Prices",
    caption = "ggplot"
  )

## More bins
ggplot(data = diamonds_modified) +
  geom_histogram(aes(x = price), bins = 100) +
  labs(
    title = "Plot 1B: Diamond Prices",
    caption = "ggplot"
  )

The price distribution is heavily skewed to the right.
There are a great many diamonds at relatively low prices, but there are a good few diamonds at very high prices too.
Using a high number of bins does not materially change the view of the histogram.

6.6 Question-2: What is the distribution of the predictor variable `carat`?

diamonds_modified %>%
  gf_histogram(~carat) %>%
  gf_labs(
    title = "Plot 2A: Carats of Diamonds",
    caption = "ggformula"
  )

## More bins
diamonds_modified %>%
  gf_histogram(~carat,
    bins = 100
  ) %>%
  gf_labs(
    title = "Plot 2B: Carats of Diamonds",
    caption = "ggformula"
  )

diamonds_modified %>%
  ggplot() +
  geom_histogram(aes(x = carat)) +
  labs(
    title = "Plot 2A: Carats of Diamonds",
    caption = "ggplot"
  )
## More bins
diamonds_modified %>%
  ggplot() +
  geom_histogram(aes(x = carat), bins = 100) +
  labs(
    title = "Plot 2A: Carats of Diamonds",
    caption = "ggplot"
  )

carat also has a heavily right-skewed distribution.
However, there is a marked “discreteness” to the distribution. Some values of carat are far more common than others. For example, 1, 1.5, and 2 carat diamonds are large in number.
Why does the X-axis extend up to 5 carats? There must be some, very few, diamonds of very high carat value!

6.7 Question-3: Does a `price` distribution vary based upon type of `cut`, `clarity`, and `color`?

gf_histogram(~price, fill = ~cut, data = diamonds_modified) %>%
  gf_labs(title = "Plot 3A: Diamond Prices", caption = "ggformula")

diamonds_modified %>%
  gf_histogram(~price, fill = ~cut, color = "black", alpha = 0.3) %>%
  gf_labs(
    title = "Plot 3B: Prices by Cut",
    caption = "ggformula"
  )

diamonds_modified %>%
  gf_histogram(~price, fill = ~cut, color = "black", alpha = 0.3) %>%
  gf_facet_wrap(~cut) %>%
  gf_labs(
    title = "Plot 3C: Prices by Filled and Facetted by Cut",
    caption = "ggformula"
  ) %>%
  gf_theme(theme(
    axis.text.x = element_text(
      angle = 45,
      hjust = 1
    )
  ))

diamonds_modified %>%
  gf_histogram(~price, fill = ~cut, color = "black", alpha = 0.3) %>%
  gf_facet_wrap(~cut, scales = "free_y", nrow = 2) %>%
  gf_labs(
    title = "Plot 3D: Prices Filled and Facetted by Cut",
    subtitle = "Free y-scale",
    caption = "ggformula"
  ) %>%
  gf_theme(theme(
    axis.text.x =
      element_text(
        angle = 45,
        hjust = 1
      )
  ))

diamonds_modified %>% ggplot() +
  geom_histogram(aes(x = price, fill = cut), alpha = 0.3) +
  labs(title = "Plot 3A: Prices by Cut", caption = "ggplot")
##
diamonds_modified %>%
  ggplot() +
  geom_histogram(aes(x = price, fill = cut),
    colour = "black", alpha = 0.3
  ) +
  labs(title = "Plot 3B: Prices filled by Cut", caption = "ggplot")
##
diamonds_modified %>% ggplot() +
  geom_histogram(aes(price, fill = cut),
    colour = "black", alpha = 0.3
  ) +
  facet_wrap(facets = vars(cut)) +
  labs(
    title = "Plot 3C: Prices by Filled and Facetted by Cut",
    caption = "ggplot"
  ) +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))
##
diamonds_modified %>% ggplot() +
  geom_histogram(aes(price, fill = cut),
    colour = "black", alpha = 0.3
  ) +
  facet_wrap(facets = vars(cut), scales = "free_y") +
  labs(
    title = "Plot D: Prices by Filled and Facetted by Cut",
    subtitle = "Free y-scale",
    caption = "ggplot"
  ) +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

The price distribution is heavily skewed to the right AND This long-tailed nature of the histogram holds true regardless of the cut of the diamond.
See the x-axis range for each plot in Plot D! Price ranges are the same regardless of cut !! Very surprising! So cut is perhaps not the only thing that determines price…
Facetting the plot into small multiples helps look at patterns better: overlapping histograms are hard to decipher. Adding color defines the bars in the histogram very well.

6.8 A Hypothesis

The surprise insight above should lead you to make a Hypothesis!
You should decide whether you want to investigate this question further, making more graphs, as we will see. Here, we are making a Hypothesis that more than just cut determines the price of a diamond.

6.9 An Interactive App for Histograms

Type in your Console:

```{r}
#| eval: false
install.packages("shiny")
library(shiny)
runExample("01_hello") # an interactive histogram
```

7 Distributions and Densities in the Wild

Before we conclude, let us look at a real world dataset: populations of countries. This dataset was taken from Kaggle https://www.kaggle.com/datasets/ulrikthygepedersen/populations. Click on the icon below to save the file into a subfolder called data in your project folder.

Show the Code

pop <- read_csv("data/populations.csv")
pop
glimpse(pop)

Rows: 16,400
Columns: 4
$ country_code <chr> "ABW", "ABW", "ABW", "ABW", "ABW", "ABW", "ABW", "ABW", "…
$ country_name <chr> "Aruba", "Aruba", "Aruba", "Aruba", "Aruba", "Aruba", "Ar…
$ year         <dbl> 1960, 1961, 1962, 1963, 1964, 1965, 1966, 1967, 1968, 196…
$ value        <dbl> 54608, 55811, 56682, 57475, 58178, 58782, 59291, 59522, 5…

7.1 Real World Histograms

The value variable in this dataset is the population of a country. Let us plot densities/histograms for value:

Show the Code

gf_histogram(~value, data = pop, title = "Long Tailed Histogram")
gf_density(~value, data = pop, title = "Long Tailed Density")

These graphs convey very little to us: the data is very heavily skewed to the right and much of the chart is empty. There are many countries with small populations and a few countries with very large populations. Such distributions are also called “long tailed” distributions.

7.2 Transforming the Variable

To develop better insights with this data, we should transform the variable concerned, using say a “log” transformation:

Show the Code

gf_histogram(~ log10(value), data = pop, title = "Histogram with Log transformed x-variable")
gf_density(~ log10(value), data = pop, title = "Density with Log transformed x-variable")

Be prepared to transform your data with log or sqrt transformations when you see skewed distributions!

7.3 Pareto, Power Laws, and Fat-Tailed Distributions

City Populations, Sales across product categories, Salaries, Instagram connections, number of customers vs Companies, net worth / valuation of Companies, extreme events on stock markets….all of these could have highly skewed distributions. In such a case, the standard statistics of mean/median/sd may not convey too much information. With such distributions, one additional observation on say net worth, like say Mr Gates’, will change these measures completely. (More when we discuss Sampling)

Since very large observations are indeed possible, if not highly probable, one needs to look at the result of such an observation and its impact on a situation rather than its (mere) probability. Classical statistical measures and analysis cannot apply with long-tailed distributions. More on this later in the Module on Statistical Inference, but for now, here is a video that talks in detail about fat-tailed distributions, and how one should use them and get used to them:

7.4 Pareto, Power Laws, and Fat-Tailed Distributions

7.5 Types of Distribution Shapes

Show the Code

# options(ragg.max_dim = 7000) # to avoid error in ragg device

# Build dataset with different distributions
library(hrbrthemes)

data <- data.frame(
  type = c(rep("edge peak", 1000), rep("comb", 1000), rep("normal", 1000), rep("uniform", 1000), rep("bimodal", 1000), rep("skewed", 1000)),
  value = c(rnorm(900), rep(3, 100), rnorm(360, sd = 0.5), rep(c(-1, -0.75, -0.5, -0.25, 0, 0.25, 0.5, 0.75), 80), rnorm(1000), runif(1000), rnorm(500, mean = -2), rnorm(500, mean = 2), abs(log(rnorm(1000))))
)

# Represent it
data %>%
  ggplot(aes(x = value)) +
  geom_histogram(fill = "#69b3a2", color = "#e9ecef", alpha = 0.9) +
  facet_wrap(~type, scale = "free_x") +
  theme_custom()

What insights could you develop based on these distribution shapes?

Bimodal: Maybe two different systems or phenomena or regimes under which the data unfolds. Like the geyser dataset (. Or a machine that works differently when cold and when hot. Intermittent faulty behaviour…
Comb: Some specific Observations occur predominantly, in an otherwise even spread or observations. In a survey many respondents round off numbers to nearest 100 or 1000. Check the distribution of the diamonds dataset for carat values which are suspiciously integer numbers in too many cases.
Edge Peak: Could even be a data entry artifact!! All unknown / unrecorded observations are recorded as 999 !!🙀
Normal: Just what it says! Course Marks in a Univ cohort…
Skewed: Income, or friends count in a set of people. Do UI/UX peasants have more followers on Insta than say CAP people?
Uniform: The World is ~~not~~ flat. Anything can happen within a range. But not much happens outside! Sharp limits…

8 Z-scores

Look at the 4 graphs below:

Show the Code

TeachHistDens(Mean = 60, Sd = 5, VLine1 = 70, AxisFontSize = 14)

TeachHistDens(Mean = 60, Sd = 15, VLine1 = 70, AxisFontSize = 14)

xpnorm(
  mean = 60, sd = 5, q = 70, return = "plot", alpha = 0.5,
  method = "gg"
) %>%
  gf_vline(xintercept = 60, colour = "red", linewidth = 1) %>%
  gf_annotate("label", x = 75, y = 0.05, label = "area = probability = 0.02275") %>%
  gf_annotate("curve",
    x = 76, y = 0.045, xend = 72.5, yend = 0.005,
    curvature = -0.3, arrow = arrow(length = unit(0.2, "cm"))
  ) %>%
  gf_labs(title = "Z-score = 2", subtitle = "mean = 60, sd = 5") %>%
  gf_refine(
    scale_x_continuous(
      limits = c(35, 85),
      breaks = seq(35, 85, by = 5),
      expand = c(0, 0)
    )
  )

xpnorm(
  mean = 60, sd = 15, q = 70, return = "plot", alpha = 0.5,
  method = "gg"
) %>%
  gf_vline(xintercept = 60, colour = "red", linewidth = 1) %>%
  gf_annotate("label", x = 100, y = 0.02, label = "area = probability = 0.2525") %>%
  gf_annotate("curve",
    x = 100, y = 0.018, xend = 75, yend = 0.005,
    curvature = -0.3, arrow = arrow(length = unit(0.2, "cm"))
  ) %>%
  gf_labs(title = "Z-score = 0.6667", subtitle = "mean = 60, sd = 15") %>%
  gf_refine(scale_x_continuous(
    limits = c(-15, 135),
    breaks = seq(-15, 135, by = 15), expand = c(0, 0)
  ))

8.1 Understanding Z-scores

Often when we compute wish to compare distributions with different values for means and standard deviations, we resort to a scaling of the variables that are plotted in the respective distributions.

Although the densities all look the same, they are are quite different! The x-axis in each case has two scales: one is the actual value of the x-variable, and the other is the z-score which is calculated as the scaled residual from the mean:

z_x = \frac{x - \mu_{x}}{\sigma_x} \tag{1}

where \mu_x and \sigma_x are the mean and standard deviation of the x-variable.

Important

The z-score is the distance from the mean (i.e. residual) scaled by the sd.

In the figures above, the absolute value of the random variables x_i is always 10 from the (identical) means \mu_i. However since the \sigma_i are different, the z-score is different in each case. In the left column, the z_{score} = 2; in the right column it is z_{score} = 0.6667.

We note that the variation in density is the same at the same values of z-score.

When we make comparisons of two random variables, our comparisons are done most easily when we compare z-scores to calculate probabilities, or differences in z-scores at identical probabilities.

9 Wait, But Why?

Histograms are used to study the distribution of one or a few Quant variables.
Checking the distribution of your variables one by one is probably the first task you should do when you get a new dataset.
It delivers a good quantity of information about spread, how frequent the observations are, and if there are some outlandish ones.
Comparing histograms side-by-side helps to provide insight about whether a Quant measurement varies with situation (a Qual variable). We will see this properly in a statistical way soon.

10 Conclusion

To complicate matters: Having said all that, the histogram is really a bar chart in disguise! You probably suspect that the “bucketing” of the Quant variable is tantamount to creating a Qual variable! Each bucket is a level in this fictitious bucketed Quant variable.

Histograms, Frequency Distributions, and Box Plots are used for Quantitative data variables
Histograms “dwell upon” counts, ranges, means and standard deviations
We can split histograms on the basis of another Qualitative variable.
Long tailed distributions need care in visualization and in inference making!

11 Your Turn

Old Faithful Data in R (Find it!)
Wage and Education Data from Canada
Time taken to Open or Close Packages

Some Design Students/HCD peasants tested Elderly people, some with and some without hand pain, and observed how long they took to open or close typical packages for milk, cheese, bottles etc.

Tip

Note: reading xlsx files into R may need the the readxl package. Install it, peasants!!

12 AI Generated Summary and Podcast

The author illustrates these concepts through real-world examples using datasets such as diamond prices, ultramarathon race times, and global population figures. By analyzing these datasets with histograms, the author explores various aspects of data distributions, including skewness, bimodality, and the presence of outliers. The guide also introduces additional tools like the crosstable package and z-scores to enhance data analysis. Finally, the author encourages readers to apply these concepts to real-world datasets, developing questions and insights through the use of histograms and statistical measures.

What patterns emerge from the distributions of quantitative variables in each dataset, and what insights can we gain about the relationships between these variables?
How do different qualitative variables impact the distribution of quantitative variables in the datasets, and what are the implications of these findings for understanding the underlying phenomena?
Based on the distributions and relationships between variables, what are the most relevant questions to ask about the datasets, and what further analyses could be conducted

13 References

Winston Chang (2024). R Graphics Cookbook. https://r-graphics.org
See the scrolly animation for a histogram at this website: Exploring Histograms, an essay by Aran Lunzer and Amelia McNamara https://tinlizzie.org/histograms/?s=09
Minimal R using mosaic.https://cran.r-project.org/web/packages/mosaic/vignettes/MinimalRgg.pdf
Sebastian Sauer, Plotting multiple plots using purrr::map and ggplot

R Package Citations

Package	Version	Citation
crosstable	0.9.0	Chaltiel (2026)
ggridges	0.5.7	Wilke (2025)
janitor	2.2.1	Firke (2024)
naniar	1.1.0	Tierney and Cook (2023)
NHANES	2.1.4	Pruim (2026)
TeachHist	0.2.1	Lange (2023)
TeachingDemos	2.13	Snow (2024)
tinytable	0.19.0	Arel-Bundock (2026)
visdat	0.6.1	Tierney (2017)
visualize	4.5.0	Balamuta (2023)

Arel-Bundock, Vincent. 2026. tinytable: Simple and Configurable Tables in “HTML,” “LaTeX,” “Markdown,” “Word,” “PNG,” “PDF,” and “Typst” Formats. https://vincentarelbundock.github.io/tinytable/.

Balamuta, James. 2023. visualize: Graph Probability Distributions with User Supplied Parameters and Statistics. https://doi.org/10.32614/CRAN.package.visualize.

Chaltiel, Dan. 2026. crosstable: Crosstables for Descriptive Analyses. https://doi.org/10.32614/CRAN.package.crosstable.

Firke, Sam. 2024. janitor: Simple Tools for Examining and Cleaning Dirty Data. https://doi.org/10.32614/CRAN.package.janitor.

Lange, Carsten. 2023. TeachHist: A Collection of Amended Histograms Designed for Teaching Statistics. https://doi.org/10.32614/CRAN.package.TeachHist.

Pruim, Randall. 2026. NHANES: Data from the US National Health and Nutrition Examination Study. https://doi.org/10.32614/CRAN.package.NHANES.

Snow, Greg. 2024. TeachingDemos: Demonstrations for Teaching and Learning. https://doi.org/10.32614/CRAN.package.TeachingDemos.

Tierney, Nicholas. 2017. “visdat: Visualising Whole Data Frames.” JOSS 2 (16): 355. https://doi.org/10.21105/joss.00355.

Tierney, Nicholas, and Dianne Cook. 2023. “Expanding Tidy Data Principles to Facilitate Missing Data Exploration, Visualization and Assessment of Imputations.” Journal of Statistical Software 105 (7): 1–31. https://doi.org/10.18637/jss.v105.i07.

Wilke, Claus O. 2025. ggridges: Ridgeline Plots in “ggplot2”. https://doi.org/10.32614/CRAN.package.ggridges.

1 Setting up R Packages

2 What graphs will we see today?

3 What kind of Data Variables will we choose?

4 Inspiration

4.1 What does this Chart tell us?

4.2 Hans Rosling’s famous Presentation

5 How do Histograms Work?

6 Case Study-1: diamonds dataset

6.1 Examine the Data

6.2 Data Dictionary

6.3 Hypothesis and Research Questions

6.4 Plotting Histograms

6.5 Question-1: What is the distribution of the target variable price?

6.6 Question-2: What is the distribution of the predictor variable carat?

6.7 Question-3: Does a price distribution vary based upon type of cut, clarity, and color?

6.8 A Hypothesis

6.9 An Interactive App for Histograms

7 Distributions and Densities in the Wild

7.1 Real World Histograms

7.2 Transforming the Variable

7.3 Pareto, Power Laws, and Fat-Tailed Distributions

7.4 Pareto, Power Laws, and Fat-Tailed Distributions

7.5 Types of Distribution Shapes

8 Z-scores

8.1 Understanding Z-scores

9 Wait, But Why?

10 Conclusion

11 Your Turn

12 AI Generated Summary and Podcast

13 References

R Package Citations

6 Case Study-1: `diamonds` dataset

6.5 Question-1: What is the distribution of the target variable `price`?

6.6 Question-2: What is the distribution of the predictor variable `carat`?

6.7 Question-3: Does a `price` distribution vary based upon type of `cut`, `clarity`, and `color`?