Question

The following data was collected:
38 32 27 37 33 28 27 38 35 28 34 21
34 29 37 36 25 20 28 41 28 35 36 40
31 25 31 33 34 39 20 29 24 30 31 26
62 23 33 35 27 22 36 39 42 38 29 36
What is the 5 number summary for this data set?

Answer 1

To find the five-number summary (minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum) for the data set, we first need to organize the data in ascending order.

**Step 1: Organize the data.**

The data set is:
```
38, 32, 27, 37, 33, 28, 27, 38, 35, 28, 34, 21,
34, 29, 37, 36, 25, 20, 28, 41, 28, 35, 36, 40,
31, 25, 31, 33, 34, 39, 20, 29, 24, 30, 31, 26,
62, 23, 33, 35, 27, 22, 36, 39, 42, 38, 29, 36
```

Ordering the data gives us:
```
20, 20, 21, 22, 23, 24, 25, 25, 26, 27,
27, 27, 28, 28, 28, 28, 29, 29, 29, 30,
31, 31, 31, 31, 32, 33, 33, 33, 34, 34,
34, 34, 35, 35, 35, 36, 36, 36, 36, 37,
37, 38, 38, 38, 38, 39, 39, 40, 41, 42,
62
```

**Step 2: Identify the five-number summary.**

1. **Minimum:**
- The smallest number in the dataset is **20**.

2. **Maximum:**
- The largest number in the dataset is **62**.

3. **Median (Q2):**
- To find the median, we need the middle number.
- There are 48 values in the dataset (even number), so the median will be the average of the 24th and 25th values.
- The 24th and 25th values are both **31**.
- Median (Q2) = (31 + 31) / 2 = **31**.

4. **First Quartile (Q1):**
- Q1 is the median of the first half of the data set (the first 24 values).
- The first half of the data set (sorted) is:
```
20, 20, 21, 22, 23, 24, 25, 25, 26, 27,
27, 27, 28, 28, 28, 28, 29, 29, 29, 30,
31, 31, 31, 31
```
- There are 24 values, so Q1 is the average of the 12th and 13th values (both **27**).
- Q1 = (27 + 27) / 2 = **27**.

5. **Third Quartile (Q3):**
- Q3 is the median of the second half of the data set (the last 24 values).
- The second half of the data set (sorted) is:
```
31, 31, 32, 33, 33, 33, 34, 34, 34, 34,
35, 35, 35, 36, 36, 36, 36, 37, 37, 38,
38, 38, 38, 39, 39, 40, 41, 42, 62
```
- Again, there are 24 values, so Q3 is the average of the 12th and 13th values (both **36**).
- Q3 = (36 + 36) / 2 = **36**.

**Final Five-Number Summary:**
- Minimum: 20
- Q1: 27
- Median (Q2): 31
- Q3: 36
- Maximum: 62

Thus, the five-number summary is:
**20, 27, 31, 36, 62.**