To identify an outlier from the given data, we can use the interquartile range (IQR) method. Typically, an outlier is considered to be a data point that is more than 1.5 times the IQR above the third quartile (Q3) or below the first quartile (Q1).
- First, we need to organize the exam scores:
- (1.5, 65)
- (2, 68)
- (3.5, 71)
- (4.5, 82)
- (4.5, 98)
- (6, 84)
- (6.5, 88)
- (7, 85)
- (7, 80)
- Next, we can extract the scores (y-values):
65, 68, 71, 82, 98, 84, 88, 85, 80
- Now, we sort the scores in ascending order:
65, 68, 71, 80, 82, 84, 85, 88, 98
- Identify Q1 and Q3:
-
Q1 (the first quartile) is the median of the first half of the data:
- First half: 65, 68, 71, 80
- Median of first half: (68 + 71) / 2 = 69.5
-
Q3 (the third quartile) is the median of the second half of the data:
- Second half: 82, 84, 85, 88, 98
- Median of the second half: (84 + 85) / 2 = 84.5
- Now, we calculate the IQR:
IQR = Q3 - Q1 = 84.5 - 69.5 = 15
- Now we find the lower and upper bounds for outliers:
- Lower bound: Q1 - 1.5 * IQR = 69.5 - 1.5 * 15 = 69.5 - 22.5 = 47
- Upper bound: Q3 + 1.5 * IQR = 84.5 + 1.5 * 15 = 84.5 + 22.5 = 107
- Finally, we look for any exam scores outside the bounds of 47 and 107:
The scores are: 65, 68, 71, 80, 82, 84, 85, 88, 98
All the scores fall within the range of 47 to 107, but now we should consider the points individually:
Among the highlighted points for potential outliers:
- (1.5, 65) - within bounds
- (3.5, 71) - within bounds
- (4.5, 98) - within bounds
- (6.5, 88) - within bounds
The score of 98 from the point (4.5, 98) stands out compared to the others. While it is within the calculated bounds based on IQR, it could still be considered an outlier in the context of this dataset, as it is significantly higher than most scores.
Thus, the outlier is (4.5, 98) in the context of this dataset as it is far apart from the rest.
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