The following tables gives the frequency distribution of number of peas per pea pod of 198 pods: Number of peas 1 2 3 4 5 6 7 Frequency 4 33 76 50 26 8 1
What is the median of this distribution?
3
Calculating the Median of a Frequency Distribution
The problem asks us to find the median of a given frequency distribution representing the number of peas per pea pod for 198 pods. The median is the middle value in a dataset when arranged in ascending order. For a frequency distribution, we first need to determine the total number of observations and then find the value corresponding to the middle position(s) using cumulative frequency.
Understanding the Data Table
The provided table shows the frequency of each number of peas:
| Number of peas | Frequency (Number of pods) |
|---|---|
| 1 | 4 |
| 2 | 33 |
| 3 | 76 |
| 4 | 50 |
| 5 | 26 |
| 6 | 8 |
| 7 | 1 |
Finding the Total Number of Observations
The total number of observations, denoted by \(N\), is the sum of all frequencies:
\(N = 4 + 33 + 76 + 50 + 26 + 8 + 1 = 198\)
The total number of pea pods is 198.
Determining the Median Position
Since \(N = 198\) is an even number, the median is the average of the \(\frac{N}{2}\)th term and the \(\left(\frac{N}{2} + 1\right)\)th term. The positions of the median terms are:
- The \(\frac{198}{2}\)th term = 99th term
- The \(\left(\frac{198}{2} + 1\right)\)th term = \(99 + 1\)th term = 100th term
We need to find the number of peas corresponding to the 99th and 100th pea pod when the pods are ordered by the number of peas.
Calculating Cumulative Frequency
To find the value of the 99th and 100th terms, we calculate the cumulative frequency (CF).
| Number of peas | Frequency | Cumulative Frequency (CF) |
|---|---|---|
| 1 | 4 | 4 |
| 2 | 33 | 4 + 33 = 37 |
| 3 | 76 | 37 + 76 = 113 |
| 4 | 50 | 113 + 50 = 163 |
| 5 | 26 | 163 + 26 = 189 |
| 6 | 8 | 189 + 8 = 197 |
| 7 | 1 | 197 + 1 = 198 |
Finding the Values at Median Positions
Now we use the cumulative frequency to find the values of the 99th and 100th terms:
- The first 4 pods have 1 pea (CF=4).
- The next 33 pods (up to pod 37) have 2 peas (CF=37).
- The next 76 pods (up to pod 113) have 3 peas (CF=113).
Since the 99th term falls within the range of pods having 3 peas (as 37 < 99 ≤ 113), the value of the 99th term is 3.
Similarly, the 100th term also falls within the range of pods having 3 peas (as 37 < 100 ≤ 113), so the value of the 100th term is 3.
Calculating the Median Value
The median is the average of the 99th and 100th terms:
\(\text{Median} = \frac{\text{Value of 99th term} + \text{Value of 100th term}}{2}\)
\(\text{Median} = \frac{3 + 3}{2}\)
\(\text{Median} = \frac{6}{2}\)
\(\text{Median} = 3\)
The median number of peas per pod is 3.
Revision Table: Median Calculation Steps
| Step | Action | Explanation |
|---|---|---|
| 1 | Find total frequency (N) | Sum of all frequencies. |
| 2 | Determine median position(s) | \(N/2\) and \((N/2)+1\) for even N; \((N+1)/2\) for odd N. |
| 3 | Calculate cumulative frequency (CF) | Running total of frequencies. |
| 4 | Find value at median position(s) | Use CF to locate the value where the median term(s) fall. |
| 5 | Calculate median value | Average of the values found in Step 4. |
Additional Information: Median in Statistics
The median is a measure of central tendency. It is often preferred over the mean when the data contains outliers or is skewed, as it is not affected by extreme values. For discrete data like this frequency distribution, the median is the value of the variable at the median position(s). If the median falls between two different variable values, the median is the average of those two values. In this case, both the 99th and 100th terms are 3, so the average is simply 3.
Similar Questions
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