Find the median of the following series: Size: 20, 25, 30, 35, 40, 45, 50, 55 Frequency: 14, 18, 33, 30, 20, 15, 13, 7
Understand the Problem
The question asks us to find the median of the given frequency distribution. To do this, we need to calculate the cumulative frequencies and then identify the value corresponding to the middle observation.
Answer
$M = 35$
Answer for screen readers
$M = 35$
Steps to Solve
- Calculate the cumulative frequencies
To find the median, we first need to calculate the cumulative frequencies for each size.
- Cumulative frequency for size 20
The cumulative frequency for size 20 is the same as its frequency, which is 14.
- Cumulative frequency for size 25
The cumulative frequency for size 25 is the sum of the frequencies of sizes 20 and 25: $14 + 18 = 32$.
- Cumulative frequency for size 30
The cumulative frequency for size 30 is the sum of the cumulative frequency of size 25 and the frequency of size 30: $32 + 33 = 65$.
- Cumulative frequency for size 35
The cumulative frequency for size 35 is the sum of the cumulative frequency of size 30 and the frequency of size 35: $65 + 30 = 95$.
- Cumulative frequency for size 40
The cumulative frequency for size 40 is the sum of the cumulative frequency of size 35 and the frequency of size 40: $95 + 20 = 115$.
- Cumulative frequency for size 45
The cumulative frequency for size 45 is the sum of the cumulative frequency of size 40 and the frequency of size 45: $115 + 15 = 130$.
- Cumulative frequency for size 50
The cumulative frequency for size 50 is the sum of the cumulative frequency of size 45 and the frequency of size 50: $130 + 13 = 143$.
- Cumulative frequency for size 55
The cumulative frequency for size 55 is the sum of the cumulative frequency of size 50 and the frequency of size 55: $143 + 7 = 150$.
- Find the median position
The total frequency is 150. The median position is the middle value, which is $\frac{150}{2} = 75$.
- Identify the median
The median is the size corresponding to the cumulative frequency that is just greater than or equal to 75. The cumulative frequency for size 30 is 65, and the cumulative frequency for size 35 is 95. Since 95 is the first cumulative frequency greater than 75, the median size is 35.
$M = 35$
More Information
The median is the middle value in a dataset. In this case, since we have a frequency distribution, we found the cumulative frequencies and then identified the size corresponding to the middle observation.
Tips
A common mistake is to take the average of the sizes instead of considering the frequencies. Another mistake is to calculate the cumulative frequencies incorrectly. It's crucial to add the frequencies cumulatively to find the correct cumulative frequencies.
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