Find the following measurements for the data set: 8, 9, 10, 12, 14, 14, 15 median = lower extreme = upper extreme = lower quartile (Q1) = upper quartile (Q3) = range = interquartile range (IQR) =

Answer 1

To find the required measurements for the dataset \( 8, 9, 10, 12, 14, 14, 15 \), we will compute each value step by step.

1. Median:

   • To find the median, we need to arrange the numbers in ascending order (which they already are): \( 8, 9, 10, 12, 14, 14, 15 \).
   • The median is the middle number. Since there are 7 numbers (an odd set), the median is the 4th number: \[ \text{Median} = 12 \]

2. Lower Extreme:

   • The lower extreme (minimum) is the smallest number in the dataset. \[ \text{Lower Extreme} = 8 \]

3. Upper Extreme:

   • The upper extreme (maximum) is the largest number in the dataset. \[ \text{Upper Extreme} = 15 \]

4. Lower Quartile (Q1):

   • To find Q1, we look at the lower half of the dataset (not including the median). The lower half is \( 8, 9, 10 \).
   • The lower quartile is the median of this subset. There are 3 numbers, so the median (2nd number) is: \[ Q1 = 9 \]

5. Upper Quartile (Q3):

   • To find Q3, we look at the upper half of the dataset (not including the median). The upper half is \( 14, 14, 15 \).
   • The upper quartile is the median of this subset. There are 3 numbers, so the median (2nd number) is: \[ Q3 = 14 \]

6. Range:

   • The range is calculated by subtracting the lower extreme from the upper extreme. \[ \text{Range} = 15 - 8 = 7 \]

7. Interquartile Range (IQR):

   • The IQR is calculated by subtracting Q1 from Q3. \[ \text{IQR} = Q3 - Q1 = 14 - 9 = 5 \]

Now, summarizing all the measurements:

\[ \text{Median} = 12, \quad \text{Lower Extreme} = 8, \quad \text{Upper Extreme} = 15, \quad Q1 = 9, \quad Q3 = 14, \quad \text{Range} = 7, \quad \text{IQR} = 5 \]