Question

The highest temperatures measured at Death Valley. California, from 1995 to 2004 are given as a dataset.
127, 125, 125, 129, 123, 126, 127, 128, 128, 125
Find the range and the interquartile range of the dataset.
The range is __, and interquartile range is __.

Answer 1

To find the range and the interquartile range (IQR) of the dataset, we will follow these steps:

1. **Calculate the range**: The range is the difference between the maximum and minimum values in the dataset.
2. **Calculate the interquartile range (IQR)**: The IQR is the difference between the third quartile (Q3) and the first quartile (Q1).

Given dataset: \(127, 125, 125, 129, 123, 126, 127, 128, 128, 125\)

**Step 1: Calculate the Range**

- Maximum value = 129
- Minimum value = 123

\[
\text{Range} = \text{Max} - \text{Min} = 129 - 123 = 6
\]

**Step 2: Calculate the Interquartile Range (IQR)**

First, we need to sort the dataset:

Sorted dataset: \(123, 125, 125, 125, 126, 127, 127, 128, 128, 129\)

- Number of observations (n) = 10

To find Q1 and Q3:

- \(Q1\) is the median of the first half of the data (the first 5 numbers).
- \(Q3\) is the median of the second half of the data (the last 5 numbers).

**Finding Q1:**
First half: \(123, 125, 125, 125, 126\)

Since there are 5 numbers:
\[
Q1 = \text{Median} = 125
\]

**Finding Q3:**
Second half: \(127, 127, 128, 128, 129\)

Since there are 5 numbers:
\[
Q3 = \text{Median} = 128
\]

**Now calculate IQR:**
\[
IQR = Q3 - Q1 = 128 - 125 = 3
\]

**Final Answer:**
- The range is **6**, and the interquartile range is **3**.