Question
Outliers _ the range and interquatile range in different ways.
Data with No outlier
4, 5, 7, 11, 12, 15
15 - 4 = _
Data with one outlier
4, 5, 7, 11, 12, 15, 58
58 - 4 = _
The range is much larger with this outlier included in the data set.
Outliers have a significant impact on the range.
Data with No outlier
4, 5, 7, 11, 12, 15
15 - 4 = _
Data with one outlier
4, 5, 7, 11, 12, 15, 58
58 - 4 = _
The range is much larger with this outlier included in the data set.
Outliers have a significant impact on the range.
Answers
Answered by
GPT-4o mini
You are correct that outliers can significantly impact the range of a dataset.
Let's break down your examples and calculations:
1. **Data with No Outlier:**
- Data: 4, 5, 7, 11, 12, 15
- Range Calculation: The range is calculated by subtracting the smallest value from the largest value.
- Range = 15 - 4 = **11**
2. **Data with One Outlier:**
- Data: 4, 5, 7, 11, 12, 15, 58
- Range Calculation: Again, the range is calculated by subtracting the smallest value from the largest value.
- Range = 58 - 4 = **54**
From these examples, you can see that the presence of the outlier (58) greatly increases the range from 11 to 54.
### Impact on the Interquartile Range (IQR):
The **interquartile range (IQR)** is less affected by outliers, as it focuses on the central portion of the data. It is calculated as the difference between the first quartile (Q1) and the third quartile (Q3).
1. **Data with No Outlier:**
- Quartiles:
- Q1 (25th percentile) = (5 + 7) / 2 = 6
- Q3 (75th percentile) = (12 + 15) / 2 = 13.5
- IQR = Q3 - Q1 = 13.5 - 6 = **7.5**
2. **Data with One Outlier:**
- Quartiles:
- The 1st quartile (Q1) remains the same: 6
- Q3 can be calculated based on the dataset: the middle value of the upper half (11, 12, 15, 58), which becomes 12.5.
- IQR = Q3 - Q1 = 12.5 - 6 = **6.5**
### Summary:
- The range significantly increases with outliers, while the IQR remains more stable and reliable for understanding the spread of the central data points.
- In your examples:
- No outlier range: 11, IQR: 7.5
- One outlier range: 54, IQR: 6.5
This illustrates how outliers can skew measures of spread like the range, while measures like the IQR provide a more reliable understanding of variability in the presence of outliers.
Let's break down your examples and calculations:
1. **Data with No Outlier:**
- Data: 4, 5, 7, 11, 12, 15
- Range Calculation: The range is calculated by subtracting the smallest value from the largest value.
- Range = 15 - 4 = **11**
2. **Data with One Outlier:**
- Data: 4, 5, 7, 11, 12, 15, 58
- Range Calculation: Again, the range is calculated by subtracting the smallest value from the largest value.
- Range = 58 - 4 = **54**
From these examples, you can see that the presence of the outlier (58) greatly increases the range from 11 to 54.
### Impact on the Interquartile Range (IQR):
The **interquartile range (IQR)** is less affected by outliers, as it focuses on the central portion of the data. It is calculated as the difference between the first quartile (Q1) and the third quartile (Q3).
1. **Data with No Outlier:**
- Quartiles:
- Q1 (25th percentile) = (5 + 7) / 2 = 6
- Q3 (75th percentile) = (12 + 15) / 2 = 13.5
- IQR = Q3 - Q1 = 13.5 - 6 = **7.5**
2. **Data with One Outlier:**
- Quartiles:
- The 1st quartile (Q1) remains the same: 6
- Q3 can be calculated based on the dataset: the middle value of the upper half (11, 12, 15, 58), which becomes 12.5.
- IQR = Q3 - Q1 = 12.5 - 6 = **6.5**
### Summary:
- The range significantly increases with outliers, while the IQR remains more stable and reliable for understanding the spread of the central data points.
- In your examples:
- No outlier range: 11, IQR: 7.5
- One outlier range: 54, IQR: 6.5
This illustrates how outliers can skew measures of spread like the range, while measures like the IQR provide a more reliable understanding of variability in the presence of outliers.