Understanding Quartile Deviation Calculation

The question asks us to calculate the quartile deviation for a given data set, where the third quartile is already provided. Quartile deviation is a measure of dispersion that describes the spread of the middle 50% of a data set. It is half the difference between the third quartile ($Q_3$) and the first quartile ($Q_1$).

Steps to Calculate Quartile Deviation

To find the quartile deviation, we need to determine the values of the first quartile ($Q_1$) and the third quartile ($Q_3$). The formula for quartile deviation (QD) is:

$$QD = \frac{Q_3 - Q_1}{2}$$

We are given the data set: 7, 10, 7, 8, 9 and the third quartile, $Q_3 = 9.5$. We need to find the first quartile, $Q_1$, from the data set.

Ordering the Data Set

First, let's arrange the data set in ascending order:

• 7
• 7
• 8
• 9
• 10

The ordered data set is 7, 7, 8, 9, 10. There are $n=5$ data points.

Calculating the First Quartile ($Q_1$)

The first quartile ($Q_1$) is the value at the 25th percentile. For an ordered data set with $n$ observations, the position of $Q_1$ can be found using the formula: Position of $Q_1 = \frac{n+1}{4}$.

For this data set, $n=5$.

Position of $Q_1 = \frac{5+1}{4} = \frac{6}{4} = 1.5$$

This means $Q_1$ is located halfway between the 1st and 2nd values in the ordered data set.

The 1st value is 7.

The 2nd value is 7.

$$Q_1 = \frac{\text{1st value} + \text{2nd value}}{2} = \frac{7+7}{2} = \frac{14}{2} = 7$$

So, the first quartile is $Q_1 = 7$.

Calculating the Third Quartile ($Q_3$)

The third quartile ($Q_3$) is the value at the 75th percentile. The position of $Q_3$ can be found using the formula: Position of $Q_3 = \frac{3(n+1)}{4}$.

For this data set, $n=5$.

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.

The 4th value is 9.

The 5th value is 10.

$$Q_3 = \frac{\text{4th value} + \text{5th value}}{2} = \frac{9+10}{2} = \frac{19}{2} = 9.5$$

The calculated $Q_3$ is 9.5, which matches the value given in the question.

Calculating the Quartile Deviation

Now that we have $Q_1 = 7$ and $Q_3 = 9.5$, we can calculate the quartile deviation using the formula:

$$QD = \frac{Q_3 - Q_1}{2}$$

$$QD = \frac{9.5 - 7}{2} = \frac{2.5}{2} = 1.25$$

Thus, the quartile deviation is 1.25.

Conclusion

Based on the calculations, the first quartile ($Q_1$) is 7 and the third quartile ($Q_3$) is 9.5. Using these values, the quartile deviation is 1.25.

Revision Table: Key Quartile Concepts

Additional Information on Measures of Dispersion

Quartile deviation is a type of measure of dispersion, which tells us how spread out the data points are. Other common measures include:

• Range: The difference between the maximum and minimum values in a data set. It is simple to calculate but is highly affected by outliers.
• Variance: The average of the squared differences from the mean. It measures how far each number in the set is from the mean and thus from every other number in the set.
• Standard Deviation: The square root of the variance. It is a widely used measure of dispersion as it is in the same units as the data. A low standard deviation indicates that the data points tend to be close to the mean (and to each other), while a high standard deviation indicates that the data points are spread out over a wider range of values.

Quartile deviation is particularly useful for data sets that are skewed or have outliers, as it is not affected by the extreme values.