Find the median of the following frequency distribution series: 1. Size: 20, 25, 30, 35, 40, 45, 50, 55; Frequency: 14, 18, 33, 30, 20, 15, 13, 7 2. Selling Price (paise): 45, 46,... Find the median of the following frequency distribution series: 1. Size: 20, 25, 30, 35, 40, 45, 50, 55; Frequency: 14, 18, 33, 30, 20, 15, 13, 7 2. Selling Price (paise): 45, 46, 47, 48, 49, 50, 51, 52; Frequency: 23, 20, 42, 50, 41, 12, 8, 4
Understand the Problem
The question is asking to find the median of two frequency distribution series, the first series contains Size and Frequency, the second series contains Selling Price and Frequency. The median is the middle value that separates the higher half from the lower half of a data set.
Answer
The median of the first series is 35. The median of the second series is 48.
Answer for screen readers
The median of the first series is 35. The median of the second series is 48.
Steps to Solve
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Calculate the cumulative frequencies for the first series To find the median, we first need to calculate the cumulative frequencies for the first series. This is done by adding up the frequencies as we go along.
- For size 20, the cumulative frequency is 14.
- For size 25, the cumulative frequency is $14 + 18 = 32$.
- For size 30, the cumulative frequency is $32 + 33 = 65$.
- For size 35, the cumulative frequency is $65 + 30 = 95$.
- For size 40, the cumulative frequency is $95 + 20 = 115$.
- For size 45, the cumulative frequency is $115 + 15 = 130$.
- For size 50, the cumulative frequency is $130 + 13 = 143$.
- For size 55, the cumulative frequency is $143 + 7 = 150$.
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Determine the median position for the first series The total frequency is 150. The median position is given by $\frac{N+1}{2}$, where $N$ is the total frequency. In this case, the median position is $\frac{150+1}{2} = \frac{151}{2} = 75.5$.
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Identify the median value for the first series Since the median position is 75.5, we look for the value in the cumulative frequency distribution that contains this position. The cumulative frequency of 65 corresponds to size 30, and the cumulative frequency of 95 corresponds to size 35. Therefore, the median lies in the size 35. The median is 35.
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Calculate the cumulative frequencies for the second series Now calculate the cumulative frequencies for the second series.
- For selling price 45, the cumulative frequency is 23.
- For selling price 46, the cumulative frequency is $23 + 20 = 43$.
- For selling price 47, the cumulative frequency is $43 + 42 = 85$.
- For selling price 48, the cumulative frequency is $85 + 50 = 135$.
- For selling price 49, the cumulative frequency is $135 + 41 = 176$.
- For selling price 50, the cumulative frequency is $176 + 12 = 188$.
- For selling price 51, the cumulative frequency is $188 + 8 = 196$.
- For selling price 52, the cumulative frequency is $196 + 4 = 200$.
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Determine the median position for the second series The total frequency is 200. The median position is given by $\frac{N+1}{2}$, where $N$ is the total frequency. In this case, the median position is $\frac{200+1}{2} = \frac{201}{2} = 100.5$.
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Identify the median value for the second series Since the median position is 100.5, we look for the value in the cumulative frequency distribution that contains this position. The cumulative frequency of 85 corresponds to selling price 47, and the cumulative frequency of 135 corresponds to selling price 48. Therefore, the median lies in the selling price 48.
The median of the first series is 35. The median of the second series is 48.
More Information
The median is a measure of central tendency that divides a dataset into two equal parts.
Tips
A common mistake is to forget to calculate the cumulative frequencies. Another common mistake is to not include the +1 when calculating $\frac{N+1}{2}$. A final common mistake is to simply divide $N$ by 2, without considering whether $N$ is even or odd.
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