Let M denote the median of the following frequency distribution.\(x_i\)
\(f_i\)
0 - 4 2 4 - 8 4 8 - 12 7 12 - 16 8 16 - 20 6
Then 20M is equal to:
\(x_i\) | \(f_i\) |
|---|---|
| 0 - 4 | 2 |
| 4 - 8 | 4 |
| 8 - 12 | 7 |
| 12 - 16 | 8 |
| 16 - 20 | 6 |
- 416
- 104
- 52
- 208
The Correct Option is D
Solution and Explanation
Solution: First, calculate the cumulative frequency.
| Class | Frequency | Cumulative Frequency |
|---|---|---|
| 0-4 | 3 | 3 |
| 4-8 | 9 | 12 |
| 8-12 | 10 | 22 |
| 12-16 | 8 | 30 |
| 16-20 | 6 | 36 |
The total frequency \( N = 36 \), so \( \frac{N}{2} = 18 \).
The median class is 8-12, as it is the class where the cumulative frequency first exceeds 18.
Lower limit \( l = 8 \) Frequency \( f = 10 \) Cumulative frequency of the class before the median class \( C = 12 \) Class width \( h = 4 \)
Using the median formula:
\[ M = l + \left( \frac{\frac{N}{2} - C}{f} \right) \times h \]
Substitute the values:
\[ M = 8 + \left( \frac{18 - 12}{10} \right) \times 4 \] \[ = 8 + \left( \frac{6}{10} \right) \times 4 \] \[ = 8 + 0.6 \times 4 \] \[ = 8 + 2.4 = 10.4 \]
Then,
\[ 20M = 20 \times 10.4 = 208 \].
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Concepts Used:
Statistics
Statistics is a field of mathematics concerned with the study of data collection, data analysis, data interpretation, data presentation, and data organization. Statistics is mainly used to acquire a better understanding of data and to focus on specific applications. Also, Statistics is the process of gathering, assessing, and summarising data in a mathematical form.
Mathematically there are two approaches for analyzing data in statistics that are widely used:
Descriptive Statistics -
Using measures of central tendency and measures of dispersion, the descriptive technique of statistics is utilized to describe the data collected and summarise the data and its attributes.
Inferential Statistics -
This statistical strategy is utilized to produce conclusions from data. Inferential statistics rely on statistical tests on samples to make inferences, and it does so by discovering variations between the two groups. The p-value is calculated and differentiated to the probability of chance() = 0.05. If the p-value is less than or equivalent to, the p-value is considered statistically significant.