Answer :
To find the lower quartile (first quartile, [tex]$Q_1$[/tex]) of the data set
[tex]$$\{66, 70, 75, 72, 67, 68, 65, 69, 68\},$$[/tex]
follow these steps:
1. Sort the data in increasing order.
The sorted data is:
[tex]$$65,\ 66,\ 67,\ 68,\ 68,\ 69,\ 70,\ 72,\ 75.$$[/tex]
2. Identify the median of the entire data set.
Since there are [tex]$9$[/tex] numbers (an odd number), the median is the middle value, which is the [tex]$5^{th}$[/tex] number in the sorted list. Here, the median is [tex]$68$[/tex]. For finding the lower quartile, we will not include this median value.
3. Determine the lower half of the data.
Exclude the median and consider the first half of the data. The lower half consists of the first [tex]$4$[/tex] numbers:
[tex]$$65,\ 66,\ 67,\ 68.$$[/tex]
4. Compute the median of this lower half.
Since there are [tex]$4$[/tex] numbers (an even number), the median is the average of the two middle numbers. The middle numbers are [tex]$66$[/tex] and [tex]$67$[/tex]. Thus, the lower quartile is calculated as:
[tex]$$Q_1 = \frac{66 + 67}{2} = \frac{133}{2} = 66.5.$$[/tex]
The lower quartile for the given data is [tex]$\boxed{66.5}$[/tex].
[tex]$$\{66, 70, 75, 72, 67, 68, 65, 69, 68\},$$[/tex]
follow these steps:
1. Sort the data in increasing order.
The sorted data is:
[tex]$$65,\ 66,\ 67,\ 68,\ 68,\ 69,\ 70,\ 72,\ 75.$$[/tex]
2. Identify the median of the entire data set.
Since there are [tex]$9$[/tex] numbers (an odd number), the median is the middle value, which is the [tex]$5^{th}$[/tex] number in the sorted list. Here, the median is [tex]$68$[/tex]. For finding the lower quartile, we will not include this median value.
3. Determine the lower half of the data.
Exclude the median and consider the first half of the data. The lower half consists of the first [tex]$4$[/tex] numbers:
[tex]$$65,\ 66,\ 67,\ 68.$$[/tex]
4. Compute the median of this lower half.
Since there are [tex]$4$[/tex] numbers (an even number), the median is the average of the two middle numbers. The middle numbers are [tex]$66$[/tex] and [tex]$67$[/tex]. Thus, the lower quartile is calculated as:
[tex]$$Q_1 = \frac{66 + 67}{2} = \frac{133}{2} = 66.5.$$[/tex]
The lower quartile for the given data is [tex]$\boxed{66.5}$[/tex].
