Question
Robert and his track team practice jumping as far as they can and record their results in inches. Their results are as follows:
41, 46, 50, 48, 48, 47, 51, 44, 42, 49
Which number, if added to the dataset, would represent an outlier in the data?
41, 46, 50, 48, 48, 47, 51, 44, 42, 49
Which number, if added to the dataset, would represent an outlier in the data?
Answers
GPT-4o mini
To determine which number would represent an outlier in the dataset, we can use the interquartile range (IQR) method. Outliers are typically defined as values that are more than 1.5 times the IQR above the third quartile (Q3) or below the first quartile (Q1).
First, we need to calculate Q1, Q3, and the IQR:
1. **Order the dataset**:
The sorted results are:
41, 42, 44, 46, 47, 48, 48, 49, 50, 51
2. **Find the median (Q2)**:
The median (middle value) is the average of the 5th and 6th values in the sorted list:
\( Q2 = \frac{47 + 48}{2} = 47.5 \)
3. **Find Q1 (the median of the first half)**:
The first half of the data (lower half) is:
41, 42, 44, 46, 47
The median of this set is 44.
4. **Find Q3 (the median of the second half)**:
The second half of the data (upper half) is:
48, 48, 49, 50, 51
The median of this set is 49.
5. **Calculate the IQR**:
\( \text{IQR} = Q3 - Q1 = 49 - 44 = 5 \)
6. **Determine the outlier thresholds**:
- Lower threshold: \( Q1 - 1.5 \times \text{IQR} = 44 - 1.5 \times 5 = 44 - 7.5 = 36.5 \)
- Upper threshold: \( Q3 + 1.5 \times \text{IQR} = 49 + 1.5 \times 5 = 49 + 7.5 = 56.5 \)
Values lower than 36.5 or higher than 56.5 are considered outliers.
Thus, any number less than 36.5 or greater than 56.5 would represent an outlier in this dataset.
For example:
- A number like 57 would be an outlier since it is greater than 56.5.
- A number like 35 would also be an outlier since it is less than 36.5.
Therefore, if we added either 57 or any number above it, it would represent an outlier. Similarly, adding 35 or any number below it would also yield an outlier.
First, we need to calculate Q1, Q3, and the IQR:
1. **Order the dataset**:
The sorted results are:
41, 42, 44, 46, 47, 48, 48, 49, 50, 51
2. **Find the median (Q2)**:
The median (middle value) is the average of the 5th and 6th values in the sorted list:
\( Q2 = \frac{47 + 48}{2} = 47.5 \)
3. **Find Q1 (the median of the first half)**:
The first half of the data (lower half) is:
41, 42, 44, 46, 47
The median of this set is 44.
4. **Find Q3 (the median of the second half)**:
The second half of the data (upper half) is:
48, 48, 49, 50, 51
The median of this set is 49.
5. **Calculate the IQR**:
\( \text{IQR} = Q3 - Q1 = 49 - 44 = 5 \)
6. **Determine the outlier thresholds**:
- Lower threshold: \( Q1 - 1.5 \times \text{IQR} = 44 - 1.5 \times 5 = 44 - 7.5 = 36.5 \)
- Upper threshold: \( Q3 + 1.5 \times \text{IQR} = 49 + 1.5 \times 5 = 49 + 7.5 = 56.5 \)
Values lower than 36.5 or higher than 56.5 are considered outliers.
Thus, any number less than 36.5 or greater than 56.5 would represent an outlier in this dataset.
For example:
- A number like 57 would be an outlier since it is greater than 56.5.
- A number like 35 would also be an outlier since it is less than 36.5.
Therefore, if we added either 57 or any number above it, it would represent an outlier. Similarly, adding 35 or any number below it would also yield an outlier.