Describing Distributions

August 24, 2026

Measuring Spread

Why Measure and Visualize Spread?

IQR

What is a Range?

Range (min and max values)
Not ideal b/c does not tell us much about where most of the values are located

vdem2022 |>
  summarize(min = min(polyarchy),
            max = max(polyarchy))

    min   max
1 0.015 0.915

Interquartile Range

IQR: 25th percentile - 75th percentile

Interquartile Range

The middle 50 percent of the countries in the data lie between 0.262 and 0.747
The IQR (0.485) is the difference between the Q3 and Q1 values

vdem2022 %>% 
  summarize(IQRlow =  quantile(polyarchy, .25),
            IQRhigh = quantile(polyarchy, .75),
            IQRlength = IQR(polyarchy)
          )

  IQRlow IQRhigh IQRlength
1 0.2585   0.734    0.4755

Box Plot

A box plot is a graphical representation of the distribution based on the median and quartiles
It is a standardized way of displaying the distribution of data based on a five number summary: minimum, first quartile, median, third quartile, and maximum

Box Plot

Code

ggplot(vdem2022, aes(x = "", y = polyarchy)) +
  geom_boxplot(fill = "steelblue") + 
   labs(
    x = "", 
    y = "Electoral Democracy", 
    title = "Distribution of Electoral Democracy in 2022", 
    caption = "Source: V-Dem Institute"
  ) +
  theme_minimal()

Standard Deviation

Measure of Spead: Standard Deviation

Can think of it as something like the “average distance” of each data point from the mean

vdem2022 |>
  summarize(mean = mean(polyarchy),
            stdDev = sd(polyarchy))

       mean   stdDev
1 0.4955978 0.262777

Standard Deviation

A low standard deviation indicates that the values tend to be close to the mean
A high standard deviation indicates that the values are spread out over a wider range

Starting with Variance

Variance is a step towards calculating the standard deviation.
It quantifies the average squared deviation of each number from the mean of the data set.

Calculating Deviation from the Mean

First, calculate the mean (\(\bar{X}\)) of the dataset.
For each data point (\(X_i\)), calculate its deviation from the mean: \[e_i = X_i - \bar{X}\]
- Example with a mean of 5:
  - For a data point where \((X_i = 0): (0 - 5 = -5)\)
  - For a data point where \((X_i = 10): (10 - 5 = 5)\)

Squaring the Deviations

Squaring each deviation (\(e_i\)) to eliminate negative values: \[e_i^2 = (X_i - \bar{X})^2\]
Summing up all squared deviations: \[\sum_{i=1}^{n} (X_i - \bar{X})^2\]
This sum represents the total squared deviation from the mean.

Calculating the Variance

Divide the total squared deviation by \((n-1)\) (to account for the sample variance): \[\text{Variance} = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2\]
Using \((n-1)\) ensures an unbiased estimate of the population variance when calculating from a sample.

Deriving the Standard Deviation

The standard deviation is the square root of the variance: \[s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2}\]
Taking the square root converts the variance back to the units of the original data.

Standard Deviation Simple Example

x = c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
e <- x - mean(x)
e

 [1] -5 -4 -3 -2 -1  0  1  2  3  4  5

Standard Deviation Simple Example

x = c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
e_squared <- e^2
e_squared

 [1] 25 16  9  4  1  0  1  4  9 16 25

Standard Deviation Simple Example

x = c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
sum_e_squared <- sum(e_squared)
sum_e_squared

[1] 110

Standard Deviation Simple Example

variance <- sum_e_squared/(length(x)-1)
variance

[1] 11

Standard Deviation Simple Example

standard_dev <- sqrt(variance)
standard_dev

[1] 3.316625

sd(x)

[1] 3.316625

Your Turn!

Calculate measures of spread for the polyarchy variable in the V-Dem data (mean, median, IQR, standard deviation)
How would you interpret these measures?
Try a box plot for the polyarchy variable
Try another variable in the V-Dem data
How does it compare to polyarchy?

Groups

Calculating Statistics by groups

What if we want to describe electoral democracy and see how it differs by some different variable? For example, by world region, or by year?
In this case we want to combine numerical summaries with categorical variables
This brings us back to bar chart

Calculating Statistics by Groups

Let’s calculate the mean and median of electoral democracy in each world region
For this, we add the group_by() to our previous code

vdem2022 |>
  group_by(region) |>
  summarize(mean_dem = mean(polyarchy),
            median_dem = median(polyarchy))

# A tibble: 6 × 3
  region         mean_dem median_dem
  <chr>             <dbl>      <dbl>
1 Africa            0.398      0.377
2 Asia              0.425      0.404
3 Eastern Europe    0.534      0.566
4 Latin America     0.606      0.692
5 Middle East       0.229      0.206
6 The West          0.854      0.853

Calculating Statistics by Groups

Let’s store our statistics as a new data object, democracy_region

democracy_region <- vdem2022 |> 
  group_by(region) |>
  summarize(mean_dem = mean(polyarchy),
            median_dem = median(polyarchy))

democracy_region

# A tibble: 6 × 3
  region         mean_dem median_dem
  <chr>             <dbl>      <dbl>
1 Africa            0.398      0.377
2 Asia              0.425      0.404
3 Eastern Europe    0.534      0.566
4 Latin America     0.606      0.692
5 Middle East       0.229      0.206
6 The West          0.854      0.853

Visualize using our Bar Chart Skills

Code

ggplot(democracy_region, aes(x = reorder(region, -mean_dem), y = mean_dem)) +
  geom_col(fill = "steelblue") + 
  labs(
    x = "Region", 
    y = "Mean Polyarchy Score", 
    title = "Democracy by region, 1990 - present", 
    caption = "Source: V-Dem Institute"
    ) + 
  theme_minimal()

Numerical Variable by Group

How should we interpret this plot?

Code

library(ggridges)
#library(forcats)
  ggplot(vdem2022, aes(x = polyarchy, y = region, fill = region)) +
    geom_density_ridges() +
  labs(
    x = "Electoral Democracy",
    y = "Region",
    title = "A Ridge Plot",
    caption = "Source: V-Dem Institute",
  ) +
  scale_fill_viridis_d() +
  theme_minimal()

Your Turn!

Make a bar chart summarizing polyarchy or some other V-Dem variable
Now try your hand at a ridge plot