Question

The following are the ages of physics teachers in a school district.
25,28 ,33 ,33 ,37 ,38 ,39 ,42 ,42 ,46 ,54 ,54 ,57
Notice that the ages are ordered from least to greatest.

Give the five-number summary and the interquartile range for the data set.
Five-number summary
Minimum:

Lower quartile:
Median:
Upper quartile:
Maximum:
Interquartile range:

Answer 1

To calculate the five-number summary and the interquartile range (IQR) for the given data set, we follow these steps:

First, let's summarize the ages in a list:

25, 28, 33, 33, 37, 38, 39, 42, 42, 46, 54, 54, 57

### Step 1: Five-number summary
1. **Minimum**: The smallest value in the data set.
- Minimum = 25

2. **Maximum**: The largest value in the data set.
- Maximum = 57

3. **Median**: The middle value of the ordered data set. Since there are 13 values (an odd number), the median is the 7th value.
- Median = 39

4. **Lower Quartile (Q1)**: The median of the first half of the data (below the overall median). The first half is:
- 25, 28, 33, 33, 37, 38 (6 values)
- Q1 is the average of the 3rd and 4th values: (33 + 33) / 2 = 33

5. **Upper Quartile (Q3)**: The median of the second half of the data (above the overall median). The second half is:
- 42, 42, 46, 54, 54, 57 (6 values)
- Q3 is the average of the 3rd and 4th values: (46 + 54) / 2 = 50

### Step 2: Interquartile Range (IQR)
The IQR is calculated as:
\[ \text{IQR} = Q3 - Q1 \]

- IQR = 50 - 33 = 17

### Summary
Now, we can compile the five-number summary and the interquartile range:

Five-number summary:
- Minimum: **25**
- Lower quartile (Q1): **33**
- Median: **39**
- Upper quartile (Q3): **50**
- Maximum: **57**

Interquartile range (IQR): **17**

### Final Summary
- Minimum: **25**
- Lower quartile: **33**
- Median: **39**
- Upper quartile: **50**
- Maximum: **57**
- Interquartile range: **17**