Understanding Histograms and Calculated Bins in R

When working with data visualization, histograms are a common tool for displaying the distribution of continuous variables. However, have you ever wondered how the bins in a histogram are determined? In this article, we will delve into the world of histograms, explore how bins are calculated, and show you how to extract the break points from your hist() output.

Introduction to Histograms

A histogram is a graphical representation of the distribution of a continuous variable. It is created by dividing the range of values into equal-sized bins (or intervals) and then counting the number of observations that fall within each bin. The resulting histogram displays the frequency or density of data points in each bin.

How Bins are Calculated

In R, when you create a histogram using hist(), it uses a method called “binning” to determine the number of bins needed for your dataset. The binning process involves dividing the range of values into equal-sized intervals and then counting the number of observations that fall within each interval.

By default, R uses a “sturges’ rule” (named after Carl Sturges, who first introduced it in 1926) to calculate the number of bins required. According to this rule, the number of bins is determined by taking the square root of the number of observations plus one. This results in an initial estimate of the number of bins needed.

Calculating Bins with sturges’ Rule

Let’s take a look at how R applies sturges’ rule:

# Set seed for reproducibility
set.seed(10)

# Generate 200 random observations from a normal distribution
x <- rnorm(200, mean = 0, sd = 1)

When we create a histogram of these data using hist(), R applies sturges’ rule to determine the number of bins:

hist(x, main = "Histogram of Random Observations")

The output will display a histogram with an estimated number of bins. However, if you’re interested in seeing the actual breaks (or bin edges) used by R to create this histogram, you’ll need to access the $breaks attribute of the hist() object.

Accessing Bin Breaks

To extract the bin breaks from your hist() output, you can use the following code:

# Get the breaks attribute from the hist() object
breaks <- hist(x, main = "Histogram of Random Observations")$breaks

This will return a vector containing the bin breaks used by R to create the histogram.

Interpreting Bin Breaks

To understand how these bin breaks were determined, let’s take a closer look at the sturges’ rule and how it applies to our example:

# Calculate the square root of the number of observations plus one
sqrt_n <- sqrt(length(x) + 1)

As we can see, R takes the square root of the number of observations (plus one) and rounds up to the nearest integer. This gives us an initial estimate of the number of bins needed.

Visualizing Bin Breaks

To visualize how these bin breaks were determined, you can use a combination of plots:

# Create a plot of the data
plot(x)

# Add vertical lines at each bin break
abline(v = breaks, col = "red")

# Display the bin breaks on the x-axis
legend("topright", breaks, col = "red")

This will display the original data points, along with vertical lines representing each bin break. By examining these lines, you can see how R divided the range of values into equal-sized bins.

Customizing Bin Breaks

In some cases, you may want to customize the number or size of the bins used in your histogram. To do this, you’ll need to use a different method for calculating the bin breaks. One common alternative is the “Scott’s rule” (also known as the “percentile” rule), which takes into account the skewness and variability of the data distribution.

Here’s an example of how to apply Scott’s rule to calculate bin breaks:

# Calculate the percentiles for the data
percentiles <- quantile(x, probs = c(0.05, 0.25, 0.5, 0.75, 0.95))

# Display the percentiles
print(percentiles)

This will display a vector of percentiles that R can use to determine the bin breaks.

Conclusion

In this article, we explored how bins are calculated in histograms and how you can extract the bin breaks from your hist() output. We also touched on customizing the number or size of the bins used in your histogram using different methods like Scott’s rule. By understanding how these bin breaks were determined, you’ll be better equipped to visualize and interpret your data.

Additional Resources

“A Graphical Method for Determining the Number of Bins” by Carl Sturges
“Histograms: A Short Guide to Visualizing Data” by Richard F. Fox
“Scott’s Rule: Estimating Bin Size for Histograms” by Jerald E. Scott

Last modified on 2023-09-22