Find The Mean, Median, And Mode:$\[ \begin{tabular}{c|c|c|c|c|c|c|c|c} Score, $x$ & 18 & 19 & 22 & 25 & 31 & 36 & 40 & 43 \\ \hline Frequency & 1 & 2 & 3 & 1 & 3 & 4 & 2 & 3 \\ \end{tabular} \\]Calculate The
Introduction
In statistics, a frequency distribution is a representation of the number of observations that fall within a given range or category. When dealing with discrete frequency distributions, it's essential to calculate the mean, median, and mode to understand the central tendency of the data. In this article, we will explore how to find the mean, median, and mode of a discrete frequency distribution using a given dataset.
Understanding the Dataset
The given dataset consists of scores and their corresponding frequencies. The scores range from 18 to 43, and the frequencies range from 1 to 4. To calculate the mean, median, and mode, we need to understand the concept of frequency and how it affects the calculation.
Calculating the Mean
The mean is the average value of a dataset. To calculate the mean, we need to multiply each score by its frequency, add up the products, and then divide by the total frequency.
Step 1: Multiply each score by its frequency
| Score, | Frequency | Frequency |
|---|---|---|
| 18 | 1 | 18 |
| 19 | 2 | 38 |
| 22 | 3 | 66 |
| 25 | 1 | 25 |
| 31 | 3 | 93 |
| 36 | 4 | 144 |
| 40 | 2 | 80 |
| 43 | 3 | 129 |
Step 2: Add up the products
Total product = 18 + 38 + 66 + 25 + 93 + 144 + 80 + 129 = 533
Step 3: Divide by the total frequency
Total frequency = 1 + 2 + 3 + 1 + 3 + 4 + 2 + 3 = 19
Mean = Total product / Total frequency = 533 / 19 = 28.05
Calculating the Median
The median is the middle value of a dataset when it is arranged in order. Since the dataset is already arranged in order, we can find the median by locating the middle value.
Step 1: Arrange the dataset in order
| Score, | Frequency |
|---|---|
| 18 | 1 |
| 19 | 2 |
| 22 | 3 |
| 25 | 1 |
| 31 | 3 |
| 36 | 4 |
| 40 | 2 |
| 43 | 3 |
Step 2: Find the middle value
Since there are 19 values in the dataset, the middle value is the 10th value. The 10th value is 36.
Median = 36
Calculating the Mode
The mode is the value that appears most frequently in a dataset. To find the mode, we need to identify the score with the highest frequency.
Step 1: Identify the score with the highest frequency
The score with the highest frequency is 36, which appears 4 times.
Mode = 36
Conclusion
In conclusion, the mean, median, and mode of the given discrete frequency distribution are 28.05, 36, and 36, respectively. The mean is the average value of the dataset, the median is the middle value, and the mode is the value that appears most frequently. Understanding the central tendency of a dataset is essential in statistics, and calculating the mean, median, and mode is a crucial step in data analysis.
Frequently Asked Questions
Q: What is the difference between the mean and the median?
A: The mean is the average value of a dataset, while the median is the middle value. The mean is sensitive to extreme values, while the median is more robust.
Q: How do I calculate the mode?
A: To calculate the mode, you need to identify the score with the highest frequency.
Q: What is the significance of the median in statistics?
A: The median is the middle value of a dataset, which makes it a good measure of central tendency when the data is skewed or has outliers.
Q: Can the mode be more than one value?
A: Yes, the mode can be more than one value if there are multiple scores with the same highest frequency.
Q: How do I calculate the mean of a discrete frequency distribution?
A: To calculate the mean, you need to multiply each score by its frequency, add up the products, and then divide by the total frequency.
References
- [1] Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
- [2] Larson, R. E., & Farber, B. A. (2018). Elementary statistics: Picturing the world. Cengage Learning.
Glossary
- Mean: The average value of a dataset.
- Median: The middle value of a dataset.
- Mode: The value that appears most frequently in a dataset.
- Frequency: The number of times a value appears in a dataset.
- Discrete frequency distribution: A representation of the number of observations that fall within a given range or category.