Question

How many outliers are in the following dataset?

{22,22,23,23,24,26,27,56,80}
{
22
,
22
,
23
,
23
,
24
,
26
,
27
,
56
,
80
}

(1 point)

Answer 1

To identify outliers in the dataset {22, 22, 23, 23, 24, 26, 27, 56, 80}, we can use the Interquartile Range (IQR) method.

1. **Order the dataset** (already ordered in this case):
- {22, 22, 23, 23, 24, 26, 27, 56, 80}

2. **Calculate Q1 (the first quartile)**:
- Q1 is the median of the first half of the data.
- The first half of the data is {22, 22, 23, 23}. The median is the average of the two middle values (22 and 23):
\[
Q1 = \frac{22 + 23}{2} = 22.5
\]

3. **Calculate Q3 (the third quartile)**:
- Q3 is the median of the second half of the data.
- The second half of the data is {24, 26, 27, 56, 80}. The median is 27.
\[
Q3 = 27
\]

4. **Calculate the IQR**:
\[
IQR = Q3 - Q1 = 27 - 22.5 = 4.5
\]

5. **Determine the outlier thresholds**:
- Lower bound:
\[
Q1 - 1.5 \times IQR = 22.5 - 1.5 \times 4.5 = 22.5 - 6.75 = 15.75
\]

- Upper bound:
\[
Q3 + 1.5 \times IQR = 27 + 1.5 \times 4.5 = 27 + 6.75 = 33.75
\]

6. **Identify the outliers**:
- Values below 15.75 or above 33.75 are considered outliers.
- In the dataset {22, 22, 23, 23, 24, 26, 27, 56, 80}, the values 56 and 80 are greater than 33.75 and thus are outliers.

**Conclusion**: There are **2 outliers** in the dataset: 56 and 80.