Finding the Median Length of Leaves
An Interactive Step-by-Step Guide
First, Let's Check Our Understanding!
What is a 'median'?
+The median is the **middle value** in a list of numbers that are sorted in order. It's a measure of central tendency. For grouped data like ours, we find the class where the middle value falls and then use a formula to estimate the exact median value.
What does 'continuous data' mean and why is it important here?
+Continuous data has no gaps between values. Notice our data has a gap (e.g., 118-126 is followed by 127-135). The median formula is designed for continuous data. So, our first step is to fix these gaps to make our data continuous. We do this by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class.
Breaking Down the Problem
1. What is Given?
We are given a table showing the lengths of 40 leaves, grouped into different classes.
| Length (in mm) | Number of leaves (Frequency) |
|---|---|
| 118 - 126 | 3 |
| 127 - 135 | 5 |
| 136 - 144 | 9 |
| 145 - 153 | 12 |
| 154 - 162 | 5 |
| 163 - 171 | 4 |
| 172 - 180 | 2 |
This data is discontinuous or non-overlapping.
2. What do we have to find?
We need to find the **median length** of the leaves.
How Can We Find It? (The Solution)
Step 1: Convert the data to continuous classes and calculate the cumulative frequency (cf).
| Original Classes | Continuous Classes | Frequency (f) | Cumulative Frequency (cf) |
|---|---|---|---|
| 118 - 126 | 117.5 - 126.5 | 3 | 3 |
| 127 - 135 | 126.5 - 135.5 | 5 | 3 + 5 = 8 |
| 136 - 144 | 135.5 - 144.5 | 9 | 8 + 9 = 17 |
| 145 - 153 | 144.5 - 153.5 | 12 | 17 + 12 = 29 |
| 154 - 162 | 153.5 - 162.5 | 5 | 29 + 5 = 34 |
| 163 - 171 | 162.5 - 171.5 | 4 | 34 + 4 = 38 |
| 172 - 180 | 171.5 - 180.5 | 2 | 38 + 2 = 40 |
| Total | n = 40 | ||
Step 2: Find the median class. First, find n/2.
Total frequency (n) = 40. So, n/2 = 40/2 = 20.
Now, we look for the class whose cumulative frequency is just greater than 20. That is 29. The corresponding class, 144.5 - 153.5, is our Median Class (highlighted above).
Step 3: Identify the values needed for the median formula.
Median = l + n2 - cf f × h
- l = Lower limit of the median class = 144.5
- n = Total frequency = 40
- cf = Cumulative frequency of the class *before* the median class = 17
- f = Frequency of the median class = 12
- h = Class size (Upper limit - Lower limit) = 153.5 - 144.5 = 9
Step 4: Substitute the values and calculate.
Median = 144.5 + 20 - 17 12 × 9
Median = 144.5 + 3 12 × 9
Median = 144.5 + (0.25) × 9
Median = 144.5 + 2.25
Conclusion
The median length of the leaves is:
146.75 mm
This means that if we were to arrange all 40 leaves by length, the middle leaf would have a length of 146.75 mm. Half of the leaves are shorter than this length, and the other half are longer.