Suppose a given data set has the following characteristics:

Minimum value: 120
[tex]Q_1: 160[/tex]
[tex]Q_2: 190[/tex]
[tex]Q_3: 200[/tex]
Maximum value: 220

Which of the following is true about the distribution of the data?

A. The distribution is negatively skewed.
B. The distribution is symmetrical.
C. The distribution is positively skewed.
D. Nothing can be said about the skew of the distribution.



Answer :

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.