Construct A Stem And Leaf Display For The Following Data

The stem and leaf display, also known as a stemplot, is a powerful tool in descriptive statistics that allows you to visualize the distribution of a dataset in a simple yet informative way. It combines features of a histogram with the actual data values, offering a quick overview of the data's central tendency, spread, and shape. Constructing a stem and leaf display for a given dataset involves organizing the data into stems and leaves, providing a clear visual representation that is easy to interpret.

Understanding the Stem and Leaf Display

A stem and leaf display is essentially a method of presenting quantitative data in a graphical format. The "stem" typically represents the leading digit(s) of a data value, while the "leaf" represents the trailing digit. Think of it like this: you're breaking down each number into two parts, a main part (the stem) and a smaller part (the leaf).

For example, consider the number 47. In a stem and leaf display, '4' could be the stem and '7' the leaf. This simple representation allows you to see the frequency of different value ranges within your data.

Key benefits of using a stem and leaf display:

• Preserves Data: Unlike histograms, stem and leaf displays retain the original data values, making it easier to reconstruct the dataset.
• Visual Representation: Provides a clear visual representation of the data's distribution.
• Easy to Construct: Relatively simple to create by hand, especially for smaller datasets.
• Identifies Outliers: Helps in identifying potential outliers within the data.
• Reveals Distribution Shape: Shows the shape of the distribution (symmetric, skewed, etc.).

Steps to Construct a Stem and Leaf Display

Here's a step-by-step guide on how to construct a stem and leaf display for a given dataset:

Step 1: Organize the Data

Before you start constructing the display, organize your data in ascending order. This will make the process much easier and help prevent errors. Let's assume we have the following dataset:

23, 25, 28, 31, 33, 33, 35, 37, 40, 41, 42, 45, 48, 49, 52, 55, 58, 61, 63, 68

So, our ordered dataset is:

23, 25, 28, 31, 33, 33, 35, 37, 40, 41, 42, 45, 48, 49, 52, 55, 58, 61, 63, 68

Step 2: Identify the Stems

The stem usually consists of the leading digit(s) of the data values. Look at your dataset and determine the range of values. In our example, the data ranges from 23 to 68. Therefore, our stems will be 2, 3, 4, 5, and 6. These represent the tens place of our numbers.

Step 3: Identify the Leaves

The leaf is the trailing digit of the data value. For each data point, identify its corresponding leaf based on its stem. For example:

• For 23, the stem is 2 and the leaf is 3.
• For 31, the stem is 3 and the leaf is 1.
• For 40, the stem is 4 and the leaf is 0.

Step 4: Construct the Stem and Leaf Display

Draw a vertical line. On the left side of the line, write the stems in ascending order. On the right side of the line, write the leaves corresponding to each stem in ascending order.

Here's how our stem and leaf display would look:

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

Step 5: Add a Key

A key is crucial for interpreting the stem and leaf display. It explains what the stems and leaves represent. For our example, a suitable key would be:

Key: 2 | 3 = 23

This indicates that a stem of 2 and a leaf of 3 represents the data value 23.

Step 6: Add a Title

Give your stem and leaf display a descriptive title. For example:

"Stem and Leaf Display of Exam Scores"

The complete stem and leaf display would look like this:

Stem and Leaf Display of Exam Scores

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

Key: 2 | 3 = 23

Variations and Considerations

While the basic stem and leaf display is straightforward, there are variations that can be used to handle different types of data or to provide more detailed information.

1. Split Stems:

If you have a large dataset or want to show more detail, you can split the stems. This involves creating multiple stems for the same leading digit(s). For instance, you might split each stem into two, one for leaves 0-4 and another for leaves 5-9.

Let's illustrate this with a modified version of our dataset:

21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41

Here's the split stem and leaf display:

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

Key: 2 | 1 = 21

Notice how each stem (2 and 3) is split into two rows to provide a finer-grained representation of the data.

2. Back-to-Back Stem and Leaf Display:

This variation is used to compare two related datasets. You create a common stem and then display the leaves for each dataset on either side of the stem. One set of leaves extends to the left, and the other to the right.

Imagine we have two sets of test scores for two different classes:

• Class A: 52, 55, 58, 61, 63, 65, 68, 70, 72, 75
• Class B: 50, 53, 56, 60, 62, 64, 67, 71, 73, 76

The back-to-back stem and leaf display would look like this:

Class A     | Stem | Class B
----------- | ---- | -----------
8 5 2       | 5    | 0 3 6
8 5 3 1     | 6    | 0 2 4 7
5 2 0       | 7    | 1 3 6

Key: 2 | 5 | 0 = Class A: 52, Class B: 50

This allows for a direct visual comparison of the distribution of scores in the two classes.

3. Handling Decimal Data:

If your data includes decimal values, you can adjust the stem and leaf display accordingly. Decide on the level of precision you want to represent. For example, if you have data like 12.3, 12.5, 12.7, 13.1, you might treat the whole number as the stem and the first decimal place as the leaf.

12 | 3 5 7
13 | 1

Key: 12 | 3 = 12.3

Alternatively, you could multiply all the values by a power of 10 to remove the decimal point and then create the stem and leaf display. Just remember to indicate this in your key.

4. Handling Negative Numbers:

When dealing with negative numbers, the negative sign is typically included with the leaf.

For example, consider the data: -35, -28, -22, -15, 12, 18, 23, 29.

-3 | 5
-2 | 2 8
-1 | 5
 1 | 2 8
 2 | 3 9

Key: -3 | 5 = -35

Interpreting the Stem and Leaf Display

Once you've constructed the stem and leaf display, you can use it to analyze the data. Here are some key things to look for:

• Central Tendency: The display gives you a sense of the average value. Look for where the leaves are most concentrated.
• Spread: The range of the data is easily visible from the lowest to the highest stem.
• Shape: Is the distribution symmetric, skewed to the left, or skewed to the right?
• Outliers: Are there any data points that are far away from the rest of the data? These could be potential outliers.
• Gaps: Are there any significant gaps in the data?

Advantages and Disadvantages

Like any statistical tool, the stem and leaf display has its advantages and disadvantages:

Advantages:

• Simple to create and understand.
• Preserves original data values.
• Provides a visual representation of the data's distribution.
• Helps identify outliers and gaps.

Disadvantages:

• Not suitable for very large datasets.
• Can be less effective with continuous data that has many unique values.
• Less commonly used in formal publications compared to histograms or box plots.

Example with a Larger Dataset

Let's consider a slightly larger dataset representing the scores of 50 students on a test:

62, 65, 68, 71, 73, 73, 75, 76, 77, 78, 78, 79, 80, 81, 82, 82, 83, 84, 85, 85,
86, 86, 87, 88, 88, 89, 90, 90, 91, 92, 92, 93, 94, 94, 95, 96, 96, 97, 98, 98,
99, 100, 101, 102, 103, 104, 105, 106, 107, 108

To construct a stem and leaf display for this data, we follow the same steps:

1. Organize the Data: (Already done in ascending order above)

2. Identify the Stems: The data ranges from 62 to 108, so the stems will be 6, 7, 8, 9, and 10.

3. Identify the Leaves: Extract the last digit of each number.

4. Construct the Stem and Leaf Display:

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

Key: 6 | 2 = 62

5. Add a Title:

"Stem and Leaf Display of Test Scores for 50 Students"

Final Stem and Leaf Display:

Stem and Leaf Display of Test Scores for 50 Students

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

Key: 6 | 2 = 62

From this display, we can observe:

• The scores are clustered in the 80s and 90s.
• The distribution appears to be somewhat symmetric, with a slight skew towards the higher end.
• There are no obvious outliers.

Using Software

While stem and leaf displays can be created by hand, statistical software packages like R, Python (with libraries like Pandas), SPSS, and others can also generate them automatically. This is particularly useful for larger datasets.

For example, in R, you can use the stem() function:

data <- c(62, 65, 68, 71, 73, 73, 75, 76, 77, 78, 78, 79, 80, 81, 82, 82, 83, 84, 85, 85,
          86, 86, 87, 88, 88, 89, 90, 90, 91, 92, 92, 93, 94, 94, 95, 96, 96, 97, 98, 98,
          99, 100, 101, 102, 103, 104, 105, 106, 107, 108)

stem(data)

This will output the stem and leaf display directly in the R console.

Conclusion

The stem and leaf display is a valuable tool for exploring and visualizing quantitative data. Its simplicity and ability to preserve data values make it a useful alternative to histograms, especially for smaller datasets. By understanding how to construct and interpret stem and leaf displays, you can gain insights into the distribution, central tendency, spread, and shape of your data, leading to more informed decision-making. Whether you create them by hand or using statistical software, stem and leaf displays offer a quick and effective way to summarize and present your data.