What is the Median in a Stem Plot?

The median in a stem plot, also known as a stem-and-leaf plot, is the middle number of all the data points represented in the plot, once they are ordered from least to greatest. A significant advantage of using a stem plot is that it inherently organizes the data in ascending order, making the process of identifying the median straightforward.

Understanding the Basics of a Stem Plot

A stem plot is a simple yet effective way to display quantitative data, preserving individual data points while also showing the distribution. It separates each data value into two parts: the "stem" (the leading digit(s)) and the "leaf" (the trailing digit).

For example, if a data point is 23, the stem might be 2 and the leaf would be 3. If a data point is 105, the stem could be 10 and the leaf 5.

• Key: A crucial part of a stem plot is the key, which explains how to read the data. For instance, "1 | 2 = 12" means a stem of 1 and a leaf of 2 represents the number 12.

Because leaves are listed in increasing order from their respective stems, the entire dataset within a stem plot is naturally arranged from the smallest to the largest value. This pre-sorting is what simplifies finding the median. You can learn more about reading stem plots on resources like Wikipedia's Stem-and-leaf display page.

Step-by-Step Guide to Finding the Median

To accurately locate the median value from a stem plot, follow these clear steps:

1. Extract All Data Points: Begin by listing every individual data point from the stem plot. Read them systematically, starting from the smallest value (the first leaf of the top stem) and proceeding through to the largest value (the last leaf of the bottom stem). Since the stem plot sorts the data, your list will already be in ascending order.
2. Count the Total Number of Data Points (N): Determine the total count of observations in your dataset. Each leaf represents one data point.
3. Determine the Middle Position(s):
   • If N is Odd: The median is the single value located at the (N + 1) / 2 position.
   • If N is Even: The median is the average of the two middle values. These values are found at the N / 2 and (N / 2) + 1 positions.
4. Identify the Median Value(s): Count through your ordered list of data points (or directly on the stem plot) to find the value(s) at the position(s) determined in the previous step. If you have two middle values (for an even N), add them together and divide by two to get the median.

For a broader understanding of the median as a measure of central tendency, refer to educational materials like Investopedia's explanation of Median.

Practical Example: Calculating the Median

Let's illustrate with an example dataset represented by a stem plot:

Stem-and-Leaf Plot for Test Scores
Stem | Leaves
-----|-------
6    | 8 9
7    | 0 2 5 7
8    | 1 3 4 4 6
9    | 0 2 5
Key: 6 | 8 = 68

Example 1: Odd Number of Data Points

From the stem plot above, let's list all the data points in ascending order:
68, 69, 70, 72, 75, 77, 81, 83, 84, 84, 86, 90, 92, 95

1. Data Points (N): There are 14 leaves, so N = 14.

2. Wait, I counted wrong. Let's re-count the leaves from the example:

   • Stem 6: 2 leaves (8, 9)
   • Stem 7: 4 leaves (0, 2, 5, 7)
   • Stem 8: 5 leaves (1, 3, 4, 4, 6)
   • Stem 9: 3 leaves (0, 2, 5)
     Total N = 2 + 4 + 5 + 3 = 14.

   My previous example text was for N=11 then N=10, I should use the plot for consistency.

   Example using the provided plot with N=14:

3. Extract Data Points: 68, 69, 70, 72, 75, 77, 81, 83, 84, 84, 86, 90, 92, 95

4. Count N: N = 14 (even number)

5. Determine Middle Positions: Since N is even, we need the average of the two middle values.

   • Position 1: N / 2 = 14 / 2 = 7th position
   • Position 2: (N / 2) + 1 = 7 + 1 = 8th position

6. Identify Median Values:

   • Counting to the 7th value: 68, 69, 70, 72, 75, 77, 81 (7th value)
   • Counting to the 8th value: 68, 69, 70, 72, 75, 77, 81, 83 (8th value)
   • The two middle values are 81 and 83.

7. Calculate Median: (81 + 83) / 2 = 164 / 2 = 82

Therefore, the median test score in this dataset is 82.

Example 2: Odd Number of Data Points (Adjusted)

Let's imagine the stem plot only had these scores: 68, 69, 70, 72, 75, 77, 81, 83, 84, 84, 86, 90, 92 (N=13)

1. Extract Data Points: 68, 69, 70, 72, 75, 77, 81, 83, 84, 84, 86, 90, 92
2. Count N: N = 13 (odd number)
3. Determine Middle Position: (N + 1) / 2 = (13 + 1) / 2 = 14 / 2 = 7th position
4. Identify Median Value: Counting to the 7th value: 68, 69, 70, 72, 75, 77, 81
5. Median: 81

Why Stem Plots Simplify Median Calculation

The inherent ordering of data in a stem plot eliminates the initial step of sorting the entire dataset, which is often required when calculating the median from raw, unsorted data. This makes stem plots particularly efficient for quickly determining the central tendency, especially for smaller to moderately sized datasets.