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Are there any outliers, or at least one value that is much higher or lower than the others, in the data set shown?
A table titled Test Scores contains the following entries. 95 comma 92 comma 98 comma 91 comma 84 comma 52 comma 88 comma 86.
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Are there any outliers, or at least one value that is much higher or lower than the others, in the data set shown?
A table titled Test Scores contains the following entries. 95 comma 92 comma 98 comma 91 comma 84 comma 52 comma 88 comma 86.
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Answers
GPT-4o mini
To determine whether there are any outliers in the given data set of test scores (95, 92, 98, 91, 84, 52, 88, 86), we can use the interquartile range (IQR) method.
Here is how to find outliers step by step:
1. **Order the data** from least to greatest:
- 52, 84, 86, 88, 91, 92, 95, 98
2. **Find the median (Q2)**:
- There are 8 scores, so the median is the average of the 4th and 5th scores:
- Median = (88 + 91) / 2 = 89.5
3. **Calculate Q1 (first quartile)** and **Q3 (third quartile)**:
- Q1: The median of the first half (52, 84, 86, 88) is (84 + 86) / 2 = 85
- Q3: The median of the second half (91, 92, 95, 98) is (92 + 95) / 2 = 93.5
4. **Calculate the IQR**:
- IQR = Q3 - Q1 = 93.5 - 85 = 8.5
5. **Determine the lower and upper bounds for outliers**:
- Lower bound = Q1 - 1.5 * IQR = 85 - 1.5 * 8.5 = 85 - 12.75 = 72.25
- Upper bound = Q3 + 1.5 * IQR = 93.5 + 1.5 * 8.5 = 93.5 + 12.75 = 106.25
6. **Identify outliers**:
- Any score below 72.25 or above 106.25 would be considered an outlier.
- In our data set, the lowest score is 52, and the highest is 98.
Since 52 is below the lower bound of 72.25, it **is an outlier**. There are no outliers on the upper end since 98 is below 106.25.
**Conclusion**: Yes, there is one outlier in this data set, and it is the score of 52.
Here is how to find outliers step by step:
1. **Order the data** from least to greatest:
- 52, 84, 86, 88, 91, 92, 95, 98
2. **Find the median (Q2)**:
- There are 8 scores, so the median is the average of the 4th and 5th scores:
- Median = (88 + 91) / 2 = 89.5
3. **Calculate Q1 (first quartile)** and **Q3 (third quartile)**:
- Q1: The median of the first half (52, 84, 86, 88) is (84 + 86) / 2 = 85
- Q3: The median of the second half (91, 92, 95, 98) is (92 + 95) / 2 = 93.5
4. **Calculate the IQR**:
- IQR = Q3 - Q1 = 93.5 - 85 = 8.5
5. **Determine the lower and upper bounds for outliers**:
- Lower bound = Q1 - 1.5 * IQR = 85 - 1.5 * 8.5 = 85 - 12.75 = 72.25
- Upper bound = Q3 + 1.5 * IQR = 93.5 + 1.5 * 8.5 = 93.5 + 12.75 = 106.25
6. **Identify outliers**:
- Any score below 72.25 or above 106.25 would be considered an outlier.
- In our data set, the lowest score is 52, and the highest is 98.
Since 52 is below the lower bound of 72.25, it **is an outlier**. There are no outliers on the upper end since 98 is below 106.25.
**Conclusion**: Yes, there is one outlier in this data set, and it is the score of 52.