Find the median of the following series: Items: 20-30, 30-40, 40-50, 50-60, 60-70, 70-80, 80-90 Frequency: 5, 10, 16, 18, 12, 10, 8
Understand the Problem
The question is asking to find the median of a given series of data. The data is presented in a table with 'Items' (ranges of values) and their corresponding 'Frequency'. This is a statistics problem involving finding the central value of a frequency distribution.
Answer
$Median \approx 54.72$
Answer for screen readers
$Median \approx 54.72$
Steps to Solve
- Calculate the cumulative frequencies
To find the median, we first need to calculate the cumulative frequencies for each class interval.
- Constructing the Cumulative Frequency Table
Here's the cumulative frequency table:
| Items | Frequency ($f$) | Cumulative Frequency ($cf$) |
|---|---|---|
| 20-30 | 5 | 5 |
| 30-40 | 10 | 15 |
| 40-50 | 16 | 31 |
| 50-60 | 18 | 49 |
| 60-70 | 12 | 61 |
| 70-80 | 10 | 71 |
| 80-90 | 8 | 79 |
- Determine the Median Class
The total frequency $N = 79$. The median is the value that splits the data into two halves, so we need to find the class interval that contains the $N/2 = 79/2 = 39.5^{th}$ value. From the cumulative frequency table, the cumulative frequency just greater than 39.5 is 49, which corresponds to the class interval 50-60. Therefore, the median class is 50-60.
- Apply the Median Formula
The formula for the median of a grouped data is:
$Median = L + \frac{\frac{N}{2} - cf}{f} \times h$
Where:
- $L$ is the lower boundary of the median class (50)
- $N$ is the total frequency (79)
- $cf$ is the cumulative frequency of the class preceding the median class (31)
- $f$ is the frequency of the median class (18)
- $h$ is the class width (10)
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Substitute the values into the formula $Median = 50 + \frac{39.5 - 31}{18} \times 10$
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Calculate the Median $Median = 50 + \frac{8.5}{18} \times 10$ $Median = 50 + \frac{85}{18}$ $Median = 50 + 4.72$ $Median = 54.72$
$Median \approx 54.72$
More Information
The median is approximately 54.72. This means that about half of the data points fall below 54.72 and half fall above it. This value lies within the 50-60 range, which we identified as the median class.
Tips
A common mistake is using the cumulative frequency of the median class instead of the cumulative frequency of the class preceding the median class. Also, errors can occur during the calculation, especially when dividing and multiplying the terms in the formula. Ensure to double-check these calculations.
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