Stem and Leaf Plots

Data analysis can feel like navigating a labyrinth of numbers. But sometimes, the most effective tools for understanding your data are also the simplest. Enter the stem and leaf plots, a powerful yet easily grasped method for organizing and visualizing data, allowing you to quickly identify patterns, distributions, and outliers.

This article dives deep into stem and leaf plots, explaining what they are, how they work, their advantages and disadvantages, and how to construct and interpret them. We’ll also provide practical examples and visuals to help you master this valuable data analysis technique.

What is a Stem and Leaf Plot?

A stem and leaf plot, also known as a stemplot, is a graphical method used to represent quantitative data in a way that allows for a quick visualization of its distribution. Unlike histograms or box plots which summarize data into categories, a stem and leaf plot retains the original data values, providing more detailed information while still presenting an overview of the overall data set.

Think of it like a simplified histogram, but with a twist. The plot splits each data value into two parts:

Stem: Usually the leading digit(s) of the number.
Leaf: The trailing digit(s) of the number.

These stems are listed down the left side, forming the “stem” of the plot. The corresponding leaves for each data point are then listed to the right of their stem, forming the “leaves.”

Why Use a Stem and Leaf Plot?

Stem and leaf plots offer several advantages over other data visualization methods:

Preserves Original Data: Unlike histograms, stem and leaf plots retain the original data values, allowing you to see each individual data point. This is particularly useful for smaller datasets where losing granular information would be detrimental.
Shows Data Distribution: The shape of the plot readily reveals the distribution of the data (e.g., symmetrical, skewed, uniform). You can easily identify clusters, gaps, and peaks.
Easy to Construct by Hand: Stem and leaf plots are relatively easy to create manually, making them a practical tool for quick data exploration, especially when you don’t have access to statistical software.
Identifies Outliers: Extreme values (outliers) are easily spotted as they will be far from the rest of the data points on the plot.
Sorted Data Representation: If the leaves are arranged in ascending order, the data is automatically sorted, facilitating calculations like the median.
Good for Smaller Datasets: Stem and leaf plots are most effective when dealing with datasets that have fewer than 50 observations.

How to Construct a Stem and Leaf Plot: A Step-by-Step Guide

Let’s illustrate the construction process with an example. Suppose we have the following set of test scores from a class of 20 students:

65, 72, 78, 81, 83, 85, 88, 89, 90, 92, 92, 93, 94, 95, 96, 96, 97, 98, 99, 100

Here’s how to create a stem and leaf plot for this data:

Step 1: Identify the Stems

Determine the leading digits for your data. In this example, the scores range from 65 to 100, so our stems will be 6, 7, 8, 9, and 10.

Step 2: List the Stems Vertically

Write the stems in a vertical column, typically from smallest to largest. Draw a vertical line to the right of the stems.

6 |
7 |
8 |
9 |
10|

Step 3: Add the Leaves

For each data point, place the trailing digit (the “leaf”) next to the corresponding stem. For example, for the score 65, the stem is 6 and the leaf is 5.

6 | 5
7 | 2 8
8 | 1 3 5 8 9
9 | 0 2 2 3 4 5 6 6 7 8 9
10| 0

Step 4: Order the Leaves (Optional but Recommended)

While not strictly necessary, arranging the leaves in ascending order from left to right within each stem makes the plot easier to interpret.

6 | 5
7 | 2 8
8 | 1 3 5 8 9
9 | 0 2 2 3 4 5 6 6 7 8 9
10| 0

Step 5: Include a Key

Always include a key to explain how to read the plot. This is crucial for avoiding misinterpretations. For example:

Key: 6 | 5 represents a score of 65

The Completed Stem and Leaf Plot:

Here is the completed stem and leaf plot:

6 | 5
7 | 2 8
8 | 1 3 5 8 9
9 | 0 2 2 3 4 5 6 6 7 8 9
10| 0
Key: 6 | 5 represents a score of 65

Interpreting the Stem and Leaf Plot

Now that we have our plot, let’s see what we can learn from it:

Distribution: The data appears to be somewhat skewed to the left (negatively skewed), meaning the scores are concentrated towards the higher end.
Range: The lowest score is 65 and the highest is 100.
Central Tendency: We can visually estimate the median (the middle value) to be around 92. To be exact you need to calculate it by locating the 10th and 11th value (as the sample size is 20) and taking the mean.
Outliers: There don’t appear to be any significant outliers in this dataset. All the data points are relatively close to each other.
Clusters: We see a cluster of scores in the 90s, suggesting that many students performed well on the test.

Another Example: Working with Decimals

Stem and leaf plots aren’t just for whole numbers. They can also be used for data containing decimals. Let’s consider the following set of heights (in inches) of 15 plants:

1.2, 1.5, 1.7, 1.8, 2.0, 2.1, 2.3, 2.3, 2.4, 2.6, 2.7, 2.7, 2.8, 2.9, 3.1

Here’s how to create a stem and leaf plot for this data:

Step 1: Choose the Stem and Leaf

In this case, the whole number part will be the stem, and the decimal portion will be the leaf.

Step 2: List the Stems and Add the Leaves (Ordered)

1 | 2 5 7 8
2 | 0 1 3 3 4 6 7 7 8 9
3 | 1

Key: 1 | 2 represents a height of 1.2 inches

Interpretation:

Distribution: The data appears relatively symmetrical, with most of the plant heights clustered around the 2-inch range.
Range: The heights range from 1.2 inches to 3.1 inches.
Central Tendency: The median is easily visible as 2.4 inches (the 8th data point when ordered).

Splitting Stems

Sometimes, when dealing with data that is very clustered, you might want to split the stems. This involves creating two or more rows for each stem value. For instance, you might have one row for leaves 0-4 and another for leaves 5-9. This can provide a more detailed view of the data’s distribution.

Example:

Let’s say we have the following data on the number of books read per year by 20 people:

12, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 30

Without splitting the stems, the plot would look like this:

1 | 2 4 5 6 7 8 9
2 | 0 1 1 2 3 4 5 6 7 8 9 9
3 | 0

Key: 1 | 2 represents 12 books

To split the stems, we’ll create two rows for each stem. The first row will contain leaves 0-4, and the second row will contain leaves 5-9.

1 | 2 4
1 | 5 6 7 8 9
2 | 0 1 1 2 3 4
2 | 5 6 7 8 9 9
3 | 0

Key: 1 | 2 represents 12 books

Splitting the stems helps to show that there are more people reading in the higher end of the teens (15-19) than in the lower end (12-14).

Disadvantages of Stem and Leaf Plots

While stem and leaf plots are a valuable tool, they have limitations:

Not Suitable for Large Datasets: Stem and leaf plots become unwieldy and difficult to interpret with large datasets. Histograms or box plots are more appropriate in these cases.
Can be Subjective: The choice of stem and leaf can be subjective, potentially influencing the visual representation of the data.
Less Common: While easy to understand, stem and leaf plots are not as widely used in professional settings as other visualizations like histograms and scatter plots.

Conclusion

Stem and leaf plots provide a simple yet powerful way to visualize and understand quantitative data. Their ability to retain original data while revealing distribution patterns makes them a valuable tool for exploratory data analysis, especially with smaller datasets. By mastering the construction and interpretation of stem and leaf plots, you can unlock valuable insights and gain a deeper understanding of the data you are working with. Remember to practice with different datasets and consider the advantages and disadvantages of this technique compared to other visualization methods. Happy analyzing! Data Science Blog