To determine the skewness of the given data set, we should analyze the positions of the quartiles [tex]\( Q_1 \)[/tex], [tex]\( Q_2 \)[/tex] (the median), and [tex]\( Q_3 \)[/tex] relative to each other.
Given data points:
- Minimum value: 120
- [tex]\( Q_1 \)[/tex]: 160
- [tex]\( Q_2 \)[/tex]: 190
- [tex]\( Q_3 \)[/tex]: 200
- Maximum value: 220
Here is the procedure to determine the skewness:
1. Calculate the distances between [tex]\( Q_1 \)[/tex] and [tex]\( Q_2 \)[/tex], and between [tex]\( Q_2 \)[/tex] and [tex]\( Q_3 \)[/tex]:
- Distance between [tex]\( Q_1 \)[/tex] and [tex]\( Q_2 \)[/tex]:
[tex]\[
Q_2 - Q_1 = 190 - 160 = 30
\][/tex]
- Distance between [tex]\( Q_2 \)[/tex] and [tex]\( Q_3 \)[/tex]:
[tex]\[
Q_3 - Q_2 = 200 - 190 = 10
\][/tex]
2. Compare these distances:
- If [tex]\( Q_2 - Q_1 \)[/tex] is greater than [tex]\( Q_3 - Q_2 \)[/tex], the data is negatively skewed.
- If [tex]\( Q_2 - Q_1 \)[/tex] is less than [tex]\( Q_3 - Q_2 \)[/tex], the data is positively skewed.
- If [tex]\( Q_2 - Q_1 \)[/tex] is equal to [tex]\( Q_3 - Q_2 \)[/tex], the data is symmetrical.
From our calculations:
- [tex]\( Q_2 - Q_1 = 30 \)[/tex]
- [tex]\( Q_3 - Q_2 = 10 \)[/tex]
Since [tex]\( Q_2 - Q_1 \)[/tex] (30) is greater than [tex]\( Q_3 - Q_2 \)[/tex] (10), the data distribution is negatively skewed.
Therefore, the correct answer is:
- The distribution is negatively skewed.
