If the first quartile of data set 8, 10, 8, 7, 9 is 7.5, then the value of quartile deviation is
1.0
Calculating Quartile Deviation for a Data Set
The question asks us to find the value of the quartile deviation for a given data set, provided that the first quartile (Q1) is 7.5. The data set is 8, 10, 8, 7, 9.
Quartile deviation is a measure of dispersion that is half of the difference between the third quartile (Q3) and the first quartile (Q1). The formula for quartile deviation (QD) is:
\( QD = \frac{Q_3 - Q_1}{2} \)
We are given \( Q_1 = 7.5 \). To calculate the quartile deviation, we first need to find the value of the third quartile (\( Q_3 \)).
Steps to Find the Third Quartile (Q3)
- Order the data set: Arrange the data points in ascending order.
The data set {8, 10, 8, 7, 9} sorted in ascending order is {7, 8, 8, 9, 10}.
There are \( n = 5 \) data points in this set.
- Find the position of the third quartile: The position of the third quartile in a sorted data set with \( n \) observations is typically calculated using the formula:
\( \text{Position of } Q_3 = \frac{3(n+1)}{4} \)
For our data set, \( n = 5 \):
\( \text{Position of } Q_3 = \frac{3(5+1)}{4} = \frac{3 \times 6}{4} = \frac{18}{4} = 4.5 \)
This means \( Q_3 \) is located halfway between the 4th and 5th values in the ordered data set.
- Calculate the value of the third quartile: Find the values at the 4th and 5th positions in the ordered data set {7, 8, 8, 9, 10}.
- The 4th value is 9.
- The 5th value is 10.
To find \( Q_3 \), we take the average of these two values:
\( Q_3 = \frac{\text{4th value} + \text{5th value}}{2} = \frac{9 + 10}{2} = \frac{19}{2} = 9.5 \)
So, the third quartile \( Q_3 = 9.5 \).
Calculate the Quartile Deviation
Now that we have \( Q_1 = 7.5 \) (given) and we calculated \( Q_3 = 9.5 \), we can find the quartile deviation using the formula:
\( QD = \frac{Q_3 - Q_1}{2} \)
Substitute the values of \( Q_1 \) and \( Q_3 \):
\( QD = \frac{9.5 - 7.5}{2} \)
\( QD = \frac{2.0}{2} \)
\( QD = 1.0 \)
The quartile deviation for the given data set is 1.0.
Summary of Quartiles and Quartile Deviation
| Measure | Value |
|---|---|
| Data Set (Sorted) | {7, 8, 8, 9, 10} |
| First Quartile (Q1) | 7.5 (Given) |
| Third Quartile (Q3) | 9.5 (Calculated) |
| Quartile Deviation (QD) | 1.0 (Calculated) |
The quartile deviation provides a measure of the spread of the middle 50% of the data. A smaller quartile deviation indicates less variability in the central part of the data set.
Revision Table: Key Concepts
| Term | Definition | Calculation Method |
|---|---|---|
| Quartile (Qi) | Values that divide a data set into four equal parts. Q1 is the 25th percentile, Q2 (Median) is the 50th percentile, and Q3 is the 75th percentile. | Sort data. Use position formula \( \frac{i(n+1)}{4} \) and interpolate if needed. |
| First Quartile (Q1) | The value below which 25% of the data falls. | Position \( \frac{(n+1)}{4} \) in sorted data. |
| Third Quartile (Q3) | The value below which 75% of the data falls. | Position \( \frac{3(n+1)}{4} \) in sorted data. |
| Interquartile Range (IQR) | The difference between the third and first quartiles. | \( IQR = Q_3 - Q_1 \) |
| Quartile Deviation (QD) | Half of the interquartile range. | \( QD = \frac{IQR}{2} = \frac{Q_3 - Q_1}{2} \) |
Additional Information on Quartiles and Dispersion
Quartiles are a type of quantile. Quantiles divide a dataset into equal parts. Quartiles divide it into four parts. Percentiles divide it into 100 parts, and Deciles divide it into 10 parts.
The quartile deviation is a measure of dispersion or variability. Unlike measures like range which use only the extreme values, or standard deviation which uses all values, quartile deviation focuses on the spread of the central portion of the data. This makes it particularly useful when the data contains outliers or is skewed, as outliers do not significantly affect the quartiles.
Other measures of dispersion include:
- Range: Maximum value - Minimum value.
- Mean Deviation: Average of absolute deviations from the mean or median.
- Standard Deviation: Square root of the variance. Measures the typical distance of data points from the mean.
- Variance: Average of the squared differences from the mean.
Understanding these different measures helps in choosing the most appropriate one to describe the spread of a specific data set based on its characteristics.
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