Find The Quartile Deviation And Its Coefficient From The Data Given Below.$[ \begin{tabular}{|c|c|c|c|c|c|c|} \hline Obtained Marks & 48 & 52 & 57 & 60 & 64 & 70 \ \hline Number Of Students & 5 & 7 & 2 & 6 & 2 & 3
Introduction to Quartile Deviation
Quartile deviation is a measure of dispersion that is used to describe the spread of a dataset. It is calculated as half of the difference between the third quartile (Q3) and the first quartile (Q1). The quartile deviation is a useful measure of dispersion because it is not affected by extreme values in the dataset. In this article, we will learn how to calculate the quartile deviation and its coefficient from a given dataset.
Calculating Quartile Deviation
To calculate the quartile deviation, we need to first arrange the data in ascending order. The given dataset is:
| Obtained Marks | Number of Students |
|---|---|
| 48 | 5 |
| 52 | 7 |
| 57 | 2 |
| 60 | 6 |
| 64 | 2 |
| 70 | 3 |
First, we need to find the first quartile (Q1) and the third quartile (Q3). The first quartile is the value below which 25% of the data falls, and the third quartile is the value below which 75% of the data falls.
Calculating the First Quartile (Q1)
To calculate the first quartile, we need to find the value below which 25% of the data falls. Since there are 22 students in the dataset, 25% of 22 is 5.5. This means that the first quartile is the value below which 5.5 students fall.
| Obtained Marks | Number of Students |
|---|---|
| 48 | 5 |
| 52 | 7 |
| 57 | 2 |
| 60 | 6 |
| 64 | 2 |
| 70 | 3 |
Since 5.5 students fall below the first quartile, we need to find the value that 5.5 students fall below. To do this, we can use the cumulative frequency table.
| Obtained Marks | Cumulative Frequency |
|---|---|
| 48 | 5 |
| 52 | 12 |
| 57 | 14 |
| 60 | 20 |
| 64 | 22 |
| 70 | 22 |
From the cumulative frequency table, we can see that 5.5 students fall below 52. Therefore, the first quartile (Q1) is 52.
Calculating the Third Quartile (Q3)
To calculate the third quartile, we need to find the value below which 75% of the data falls. Since there are 22 students in the dataset, 75% of 22 is 16.5. This means that the third quartile is the value below which 16.5 students fall.
| Obtained Marks | Number of Students |
|---|---|
| 48 | 5 |
| 52 | 7 |
| 57 | 2 |
| 60 | 6 |
| 64 | 2 |
| 70 | 3 |
Since 16.5 students fall below the third quartile, we need to find the value that 16.5 students fall below. To do this, we can use the cumulative frequency table.
| Obtained Marks | Cumulative Frequency |
|---|---|
| 48 | 5 |
| 52 | 12 |
| 57 | 14 |
| 60 | 20 |
| 64 | 22 |
| 70 | 22 |
From the cumulative frequency table, we can see that 16.5 students fall below 60. Therefore, the third quartile (Q3) is 60.
Calculating the Quartile Deviation
Now that we have found the first quartile (Q1) and the third quartile (Q3), we can calculate the quartile deviation. The quartile deviation is calculated as half of the difference between the third quartile (Q3) and the first quartile (Q1).
Quartile Deviation = (Q3 - Q1) / 2 Quartile Deviation = (60 - 52) / 2 Quartile Deviation = 8 / 2 Quartile Deviation = 4
Calculating the Coefficient of Quartile Deviation
The coefficient of quartile deviation is a measure of the relative dispersion of a dataset. It is calculated as the ratio of the quartile deviation to the third quartile (Q3).
Coefficient of Quartile Deviation = Quartile Deviation / Q3 Coefficient of Quartile Deviation = 4 / 60 Coefficient of Quartile Deviation = 0.0667
Conclusion
In this article, we learned how to calculate the quartile deviation and its coefficient from a given dataset. We used the given dataset to calculate the first quartile (Q1) and the third quartile (Q3), and then used these values to calculate the quartile deviation and its coefficient. The quartile deviation is a useful measure of dispersion because it is not affected by extreme values in the dataset. The coefficient of quartile deviation is a measure of the relative dispersion of a dataset.
Q: What is quartile deviation?
A: Quartile deviation is a measure of dispersion that is used to describe the spread of a dataset. It is calculated as half of the difference between the third quartile (Q3) and the first quartile (Q1).
Q: How is the first quartile (Q1) calculated?
A: The first quartile (Q1) is the value below which 25% of the data falls. To calculate Q1, we need to find the value that 25% of the data falls below. This can be done by using the cumulative frequency table.
Q: How is the third quartile (Q3) calculated?
A: The third quartile (Q3) is the value below which 75% of the data falls. To calculate Q3, we need to find the value that 75% of the data falls below. This can be done by using the cumulative frequency table.
Q: What is the formula for calculating the quartile deviation?
A: The formula for calculating the quartile deviation is:
Quartile Deviation = (Q3 - Q1) / 2
Q: What is the formula for calculating the coefficient of quartile deviation?
A: The formula for calculating the coefficient of quartile deviation is:
Coefficient of Quartile Deviation = Quartile Deviation / Q3
Q: Why is the quartile deviation a useful measure of dispersion?
A: The quartile deviation is a useful measure of dispersion because it is not affected by extreme values in the dataset. This makes it a good choice for datasets that have outliers.
Q: What is the difference between the quartile deviation and the standard deviation?
A: The quartile deviation and the standard deviation are both measures of dispersion, but they are calculated differently. The standard deviation is calculated as the square root of the variance, while the quartile deviation is calculated as half of the difference between the third quartile (Q3) and the first quartile (Q1).
Q: Can the quartile deviation be used to compare the dispersion of different datasets?
A: Yes, the quartile deviation can be used to compare the dispersion of different datasets. However, it is generally more useful to compare the coefficient of quartile deviation, which is a relative measure of dispersion.
Q: What are some common applications of the quartile deviation?
A: The quartile deviation is commonly used in statistics and data analysis to describe the spread of a dataset. It is also used in finance and economics to measure the dispersion of stock prices and other financial data.
Q: How can the quartile deviation be used in real-world scenarios?
A: The quartile deviation can be used in a variety of real-world scenarios, such as:
- Describing the spread of a dataset in a report or presentation
- Comparing the dispersion of different datasets
- Measuring the effectiveness of a treatment or intervention
- Identifying outliers in a dataset
Q: What are some common mistakes to avoid when calculating the quartile deviation?
A: Some common mistakes to avoid when calculating the quartile deviation include:
- Failing to arrange the data in ascending order
- Failing to calculate the correct values for Q1 and Q3
- Failing to use the correct formula for calculating the quartile deviation
- Failing to check for outliers in the dataset
Q: How can the quartile deviation be used in conjunction with other statistical measures?
A: The quartile deviation can be used in conjunction with other statistical measures, such as the mean, median, and standard deviation, to provide a more complete picture of the dataset.