1. Welcome to Stem and Leaf Diagrams
Welcome to one of the most useful tools in Statistics! Have you ever had a huge list of numbers and found it messy and confusing to work with? A stem and leaf diagram (sometimes called a stem-and-leaf plot) is a clever way to organize your raw data into an ordered display that looks like a horizontal bar chart, while still keeping every single original number visible.
Did you know? Unlike a histogram or bar chart where individual values get grouped and "lost" inside bars, a stem and leaf diagram lets you see the exact data values at any time!
2. The Anatomy of a Stem and Leaf Diagram
Every number in your dataset is split into two distinct parts:
• The Stem: This represents the leading digit or digits (such as the tens, hundreds, or whole numbers). The stems are written vertically in a column on the left.
• The Leaf: This is always a single digit (usually the units or the first decimal place). The leaves are written in horizontal rows to the right of their corresponding stem, in ascending numerical order.
• The Key: This is the most crucial part of your diagram! Without a key, nobody knows what your numbers represent. For example, does \(3 \mid 4\) mean \(34\), \(3.4\), or \(340\)? The key tells the reader exactly how to interpret the diagram.
Memory Aid: The Tree Analogy
Think of a tree branch: one main stem can hold many small individual leaves. Just like in nature, each leaf must be attached to its correct stem!
Key Takeaway: A stem and leaf diagram consists of a vertical stem, horizontal leaves (which must always be single digits), and a key that explains the place value.
3. How to Construct an Ordered Stem and Leaf Diagram
Don't worry if this seems tricky at first—following these simple steps will ensure you get full marks every time in your CCEA exam.
Step-by-Step Guide with an Example
Let's organize the test scores of \(12\) students:
\(23, 45, 31, 28, 19, 34, 42, 23, 37, 15, 34, 50\)
Step 1: Identify the lowest and highest values.
The lowest value is \(15\) and the highest is \(50\). This tells us our stems must run from \(1\) to \(5\) without skipping any numbers in between.
Step 2: Draw an unordered (rough) diagram.
Put the stems in a vertical line with a vertical dividing line to the right. Place each leaf next to its stem as you read through the list:
Stem \(1 \mid 9, 5\)
Stem \(2 \mid 3, 8, 3\)
Stem \(3 \mid 1, 4, 7, 4\)
Stem \(4 \mid 5, 2\)
Stem \(5 \mid 0\)
Step 3: Create the final ordered diagram.
Rewrite the leaves in order from smallest to largest. Make sure the numbers are equally spaced in neat columns so the visual shape is accurate:
\(1 \mid 5 \quad 9\)
\(2 \mid 3 \quad 3 \quad 8\)
\(3 \mid 1 \quad 4 \quad 4 \quad 7\)
\(4 \mid 2 \quad 5\)
\(5 \mid 0\)
Step 4: Add the Key!
Key: \(2 \mid 3 = 23\text{ marks}\)
Crucial Rules for Drawing:
• Never use commas between the leaves in your final diagram.
• Keep vertical spacing uniform so rows with more data visibly look longer.
• Never skip a stem, even if it has no leaves. If there were no values in the \(40\)s, you would still write the stem \(4\) and leave the row blank.
Key Takeaway: Always draw a rough unordered diagram first, then rewrite it in ascending order with evenly spaced columns, and finish by writing a clear key.
4. Finding Statistical Measures from a Stem and Leaf Diagram
Once your data is neatly ordered, finding averages and spread is straightforward!
1. The Mode (Modal Value)
The mode is the most frequently occurring value. Look for identical leaves sitting next to each other on the same stem.
In our example, on stem \(2\) we have two \(3\)s (\(23\)) and on stem \(3\) we have two \(4\)s (\(34\)).
Therefore, this dataset is bimodal with modes \(23\) and \(34\).
2. The Median
The median is the middle value. For \(n\) pieces of data, the position of the median is given by the formula:
\(\text{Position of median} = \frac{n + 1}{2}\)
For our \(n = 12\) students:
\(\text{Position} = \frac{12 + 1}{2} = 6.5\text{th value}\)
This means the median lies halfway between the \(6\)th and \(7\)th values.
Count through the ordered leaves from the smallest value:
\(1\)st: \(15\), \(2\)nd: \(19\), \(3\)rd: \(23\), \(4\)th: \(23\), \(5\)th: \(28\), \(6\)th: \(31\), \(7\)th: \(34\).
\(\text{Median} = \frac{31 + 34}{2} = 32.5\)
3. The Range
The range measures the total spread of the data:
\(\text{Range} = \text{Highest value} - \text{Lowest value}\)
\(\text{Range} = 50 - 15 = 35\)
4. Quartiles and the Interquartile Range (IQR)
• Lower Quartile (\(Q_1\)): The value one-quarter of the way into the ordered data.
\(\text{Position of } Q_1 = \frac{n + 1}{4} = \frac{13}{4} = 3.25\text{th value}\) (or the middle of the lower half: between \(23\) and \(23\), so \(Q_1 = 23\)).
• Upper Quartile (\(Q_3\)): The value three-quarters of the way into the ordered data.
\(\text{Position of } Q_3 = \frac{3(n + 1)}{4} = \frac{39}{4} = 9.75\text{th value}\) (or the middle of the upper half: between \(42\) and \(45\), so \(Q_3 = 43.5\)).
• Interquartile Range (\(\text{IQR}\)): Measures the spread of the middle \(50\%\) of data and is not affected by extreme outliers:
\(\text{IQR} = Q_3 - Q_1 = 43.5 - 23 = 20.5\)
Key Takeaway: Because the stem and leaf diagram orders data from lowest to highest, you can easily count through the leaves to locate the median, quartiles, and extreme values.
5. Back-to-Back Stem and Leaf Diagrams
When you want to compare two related sets of data (for example, scores of Class A vs. Class B, or heights of Boys vs. Girls), you can share a single central stem. This is called a back-to-back stem and leaf diagram.
How to Read and Construct:
• The shared stem is placed in the center column.
• One group's leaves go to the right (ordered smallest to largest, reading left to right as normal).
• The other group's leaves go to the left (ordered smallest to largest, reading from the stem outwards to the left).
Example: Ages of Visitors at Two Museum Exhibits
Exhibit A (Left) \(\mid\) Stem \(\mid\) Exhibit B (Right)
\(8 \quad 2 \mid 1 \mid 4 \quad 7 \quad 9\)
\(5 \quad 3 \quad 0 \mid 2 \mid 1 \quad 6\)
\(9 \quad 4 \mid 3 \mid 2 \quad 5 \quad 8\)
\(1 \mid 4 \mid 0 \quad 3\)
Important Key for Back-to-Back Diagrams: You must include a two-sided key!
Key:
\(2 \mid 1\) on left means \(12\text{ years old}\) (Exhibit A)
\(1 \mid 4\) on right means \(14\text{ years old}\) (Exhibit B)
Watch out: On the left side, notice that the leaves for stem \(2\) are written as \(5 \quad 3 \quad 0\). As you move away from the central stem towards the left, the numbers increase: \(20, 23, 25\).
Key Takeaway: In back-to-back diagrams, the left-hand leaves must increase in value as you move leftwards away from the stem.
6. Advantages and Limitations
Advantages:
• Retains all original raw data values.
• Simultaneously displays the shape of the distribution (like a histogram turned on its side).
• Makes finding order-based statistics (median, quartiles, range) very quick.
• Back-to-back plots allow direct comparison of two distributions.
Limitations:
• Unsuitable for very large datasets (e.g., \(1,000\) values would make the diagram too cluttered and time-consuming to draw).
• Only suitable for discrete or continuous numerical data with a limited range of digits.
7. Quick Review: Common Mistakes to Avoid
• Forgetting the Key: This is the most common reason students lose an easy mark in exams. Always write down your key!
• Leaving leaves unordered: Stems and leaves must both be in numerical order.
• Omitting repeated numbers: If the number \(24\) appears three times, write the leaf \(4\) three times.
• Putting multiple digits in a leaf: A leaf must contain only one digit. For example, to represent \(125\), the stem is \(12\) and the leaf is \(5\).
• Reading the left side of a back-to-back diagram backwards: Remember that numbers start from the stem and read outwards.