Question

1. what do they measure? for each of the following summary statistics, decide (i) whether it could be used to measure center or variability and (ii) whether it is resistant. (a) \\(\frac{q_1 + q_3}{2}\\) (b) \\(\frac{\text{max} - \text{min}}{2}\\)

Explanation:

Part (a): $\boldsymbol{\frac{Q_1 + Q_3}{2}}$

Step 1: Determine Center/Variability

The first quartile ($Q_1$) is the median of the lower half of the data, and the third quartile ($Q_3$) is the median of the upper half. Their average, $\frac{Q_1 + Q_3}{2}$, is a measure of the center (a type of mid - spread or a central tendency measure related to the middle 50% of the data). It focuses on the middle portion of the data, so it is a measure of center.

Step 2: Determine Resistance

A resistant measure is not strongly affected by outliers. $Q_1$ and $Q_3$ are both resistant to outliers (since they are based on the median of subsets of the data, and the median is resistant). When we take the average of two resistant measures ($Q_1$ and $Q_3$), the resulting measure $\frac{Q_1+Q_3}{2}$ is also resistant. Outliers, which are extreme values in the data set, do not have a large impact on $Q_1$ and $Q_3$, so their average is also not strongly affected by outliers.

Part (b): $\boldsymbol{\frac{\text{Max}-\text{Min}}{2}}$

Step 1: Determine Center/Variability

The range of a data set is defined as $\text{Max}-\text{Min}$, which is a measure of variability (it tells us how spread out the data is from the minimum to the maximum value). When we take $\frac{\text{Max}-\text{Min}}{2}$, we are essentially taking half of the range. Since the range is a measure of variability, half - range is also a measure of variability. It still describes how spread out the data is (just scaled by a factor of $\frac{1}{2}$).

Step 2: Determine Resistance

A resistant measure should not be strongly influenced by outliers. The maximum ($\text{Max}$) and minimum ($\text{Min}$) values of a data set are very sensitive to outliers. If there is an outlier (either a very large or a very small value), it will directly affect the value of $\text{Max}$ or $\text{Min}$, and thus affect $\frac{\text{Max}-\text{Min}}{2}$. So, $\frac{\text{Max}-\text{Min}}{2}$ is not resistant.

Final Answers

(a)

(i) It measures the center.
(ii) It is resistant.

(b)

(i) It measures variability.
(ii) It is not resistant.

Answer:

Part (a): $\boldsymbol{\frac{Q_1 + Q_3}{2}}$

Step 1: Determine Center/Variability

The first quartile ($Q_1$) is the median of the lower half of the data, and the third quartile ($Q_3$) is the median of the upper half. Their average, $\frac{Q_1 + Q_3}{2}$, is a measure of the center (a type of mid - spread or a central tendency measure related to the middle 50% of the data). It focuses on the middle portion of the data, so it is a measure of center.

Step 2: Determine Resistance

A resistant measure is not strongly affected by outliers. $Q_1$ and $Q_3$ are both resistant to outliers (since they are based on the median of subsets of the data, and the median is resistant). When we take the average of two resistant measures ($Q_1$ and $Q_3$), the resulting measure $\frac{Q_1+Q_3}{2}$ is also resistant. Outliers, which are extreme values in the data set, do not have a large impact on $Q_1$ and $Q_3$, so their average is also not strongly affected by outliers.

Part (b): $\boldsymbol{\frac{\text{Max}-\text{Min}}{2}}$

Step 1: Determine Center/Variability

The range of a data set is defined as $\text{Max}-\text{Min}$, which is a measure of variability (it tells us how spread out the data is from the minimum to the maximum value). When we take $\frac{\text{Max}-\text{Min}}{2}$, we are essentially taking half of the range. Since the range is a measure of variability, half - range is also a measure of variability. It still describes how spread out the data is (just scaled by a factor of $\frac{1}{2}$).

Step 2: Determine Resistance

A resistant measure should not be strongly influenced by outliers. The maximum ($\text{Max}$) and minimum ($\text{Min}$) values of a data set are very sensitive to outliers. If there is an outlier (either a very large or a very small value), it will directly affect the value of $\text{Max}$ or $\text{Min}$, and thus affect $\frac{\text{Max}-\text{Min}}{2}$. So, $\frac{\text{Max}-\text{Min}}{2}$ is not resistant.

Final Answers

(a)

(i) It measures the center.
(ii) It is resistant.

(b)

(i) It measures variability.
(ii) It is not resistant.