Question

Which of the following sets of data does not contain an outlier?

25, 33, 29, 31, 33, 33, 78, 28

6, 7, 5, 8, 9, 1

45, 45, 45, 45, 35, 45

101, 99, 98, 111, 101, 101, 103

Answer 1

To determine which set of data does not contain an outlier, we can use the Interquartile Range (IQR) method. An outlier is typically defined as a value lower than \( Q1 - 1.5 \times IQR \) or higher than \( Q3 + 1.5 \times IQR \), where \( Q1 \) is the first quartile and \( Q3 \) is the third quartile.

Let's examine each set:

1. **Set 1: 25, 33, 29, 31, 33, 33, 78, 28**
- Sorted: 25, 28, 29, 31, 33, 33, 33, 78
- Q1 = 29, Q3 = 33
- IQR = Q3 - Q1 = 33 - 29 = 4
- Lower Bound = Q1 - 1.5 * IQR = 29 - 6 = 23
- Upper Bound = Q3 + 1.5 * IQR = 33 + 6 = 39
- 78 is above 39, so this set contains an outlier.

2. **Set 2: 6, 7, 5, 8, 9, 1**
- Sorted: 1, 5, 6, 7, 8, 9
- Q1 = 5, Q3 = 7.5 (average of 7 and 8)
- IQR = Q3 - Q1 = 7.5 - 5 = 2.5
- Lower Bound = 5 - 3.75 = 1.25
- Upper Bound = 7.5 + 3.75 = 11.25
- 1 is below 1.25, so this set contains an outlier.

3. **Set 3: 45, 45, 45, 45, 35, 45**
- Sorted: 35, 45, 45, 45, 45, 45
- Q1 = 45, Q3 = 45
- IQR = Q3 - Q1 = 0
- Lower Bound = 45 - 0 = 45
- Upper Bound = 45 + 0 = 45
- All numbers are 45 or 35, and 35 is below 45. This set contains an outlier.

4. **Set 4: 101, 99, 98, 111, 101, 101, 103**
- Sorted: 98, 99, 101, 101, 101, 103, 111
- Q1 = 99, Q3 = 101
- IQR = Q3 - Q1 = 101 - 99 = 2
- Lower Bound = 99 - 3 = 96
- Upper Bound = 101 + 3 = 104
- All numbers lie within the bounds, so this set has no outliers.

Based on the analysis, the **4th set (101, 99, 98, 111, 101, 101, 103)** does not contain an outlier.