A Random Sample Of 66 Students Was Selected, And They Were Asked The Number Of Pairs Of Shoes They Have. The Results Are As Follows: # Of Pairs Of Shoes 4 5 6 7 8 9 10 11 12 13 14 15 Frequency 5 8 5 4 8 2 8 4 4 6 6 6 Round Answers To 4 Decimal Places.
Introduction
In statistics, the central tendency of a discrete random variable is a measure that describes the middle or typical value of the variable. It is a crucial concept in understanding the distribution of a variable and making inferences about the population. In this article, we will explore the central tendency of a discrete random variable using a real-world example. We will analyze the number of pairs of shoes owned by a random sample of 66 students and calculate the mean, median, and mode of the variable.
Data Collection and Analysis
A random sample of 66 students was selected, and they were asked to report the number of pairs of shoes they own. The results are presented in the following table:
| # of Pairs of Shoes | Frequency |
|---|---|
| 4 | 5 |
| 5 | 8 |
| 6 | 5 |
| 7 | 4 |
| 8 | 8 |
| 9 | 2 |
| 10 | 8 |
| 11 | 4 |
| 12 | 4 |
| 13 | 6 |
| 14 | 6 |
| 15 | 6 |
To calculate the central tendency of the variable, we need to round the frequencies to 4 decimal places. The rounded frequencies are:
| # of Pairs of Shoes | Frequency |
|---|---|
| 4 | 5.0 |
| 5 | 8.0 |
| 6 | 5.0 |
| 7 | 4.0 |
| 8 | 8.0 |
| 9 | 2.0 |
| 10 | 8.0 |
| 11 | 4.0 |
| 12 | 4.0 |
| 13 | 6.0 |
| 14 | 6.0 |
| 15 | 6.0 |
Calculating the Mean
The mean of a discrete random variable is calculated by multiplying each value of the variable by its frequency, summing the products, and then dividing by the total frequency. In this case, the mean is calculated as follows:
Mean = (4 × 5.0) + (5 × 8.0) + (6 × 5.0) + (7 × 4.0) + (8 × 8.0) + (9 × 2.0) + (10 × 8.0) + (11 × 4.0) + (12 × 4.0) + (13 × 6.0) + (14 × 6.0) + (15 × 6.0) / 66
Mean = 20 + 40 + 30 + 28 + 64 + 18 + 80 + 44 + 48 + 78 + 84 + 90 / 66
Mean = 585 / 66
Mean = 8.864
Calculating the Median
The median of a discrete random variable is the middle value of the variable when it is arranged in ascending order. To calculate the median, we need to arrange the values of the variable in ascending order and then find the middle value. In this case, the values of the variable are already arranged in ascending order. The median is the 33rd value of the variable, which is 8.
Calculating the Mode
The mode of a discrete random variable is the value that appears most frequently. In this case, the value 8 appears most frequently, with a frequency of 8. Therefore, the mode of the variable is 8.
Conclusion
In this article, we analyzed the number of pairs of shoes owned by a random sample of 66 students and calculated the mean, median, and mode of the variable. The mean of the variable is 8.864, the median is 8, and the mode is 8. The results of this study demonstrate the importance of understanding the central tendency of a discrete random variable in making inferences about the population.
Implications of the Study
The results of this study have several implications for understanding the behavior of discrete random variables. Firstly, the study demonstrates the importance of considering the central tendency of a variable when making inferences about the population. Secondly, the study highlights the need to consider the distribution of the variable when making inferences about the population. Finally, the study demonstrates the importance of using statistical methods to analyze and interpret data.
Limitations of the Study
The study has several limitations. Firstly, the sample size is relatively small, which may affect the accuracy of the results. Secondly, the study only considers the number of pairs of shoes owned by the students, which may not be representative of the entire population. Finally, the study does not consider other factors that may affect the behavior of the variable.
Future Research Directions
The study suggests several future research directions. Firstly, the study could be replicated with a larger sample size to improve the accuracy of the results. Secondly, the study could be extended to consider other factors that may affect the behavior of the variable. Finally, the study could be used as a basis for further research on the behavior of discrete random variables.
References
- [1] Kendall, M. G. (1975). Multivariate analysis. New York: Hafner Press.
- [2] Johnson, R. A. (1998). Applied multivariate statistical analysis. New York: Prentice Hall.
- [3] Hogg, R. V., & Tanis, E. A. (2001). Probability and statistical inference. Upper Saddle River, NJ: Prentice Hall.