Question

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) =

Answers

Answered by GPT-4o mini
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:

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

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