Loren’s friends jump as far as they can and record their results in inches. Their results include the observations 42, 47, 50, 42, 45, 41, 49, 51, and 44. Which number, if added to the dataset, would represent an outlier in the data?(1 point)
Responses
40
40
44
44
72
72
52
52
The number 72 would represent an outlier in the data.
To determine which number would represent an outlier in the dataset, we can use the concept of the interquartile range (IQR).
1. First, let's calculate the IQR.
IQR = Q3 - Q1
Q1 represents the first quartile and Q3 represents the third quartile.
2. To find Q1, we need to calculate the 25th percentile.
25th percentile = (25/100) * (N + 1)
N is the total number of observations.
In this case, N = 9. Therefore, the 25th percentile is:
25th percentile = (25/100) * (9 + 1) = 0.25 * 10 = 2.5
Since we have a small number of observations, we need to find the average of the 2nd and 3rd smallest values, which are 41 and 42.
Q1 = (41 + 42) / 2 = 83 / 2 = 41.5
3. Similarly, we calculate the 75th percentile to find Q3.
75th percentile = (75/100) * (N + 1) = (75/100) * (9 + 1) = 0.75 * 10 = 7.5
The 7th and 8th numbers in the dataset are 49 and 50. We take their average to find Q3.
Q3 = (49 + 50) / 2 = 99 / 2 = 49.5
4. Finally, we calculate the IQR.
IQR = Q3 - Q1 = 49.5 - 41.5 = 8
5. Any value that falls more than 1.5 times the IQR below Q1 or above Q3 is considered an outlier.
Lower outlier threshold = Q1 - 1.5 * IQR = 41.5 - 1.5 * 8 = 41.5 - 12 = 29.5
Upper outlier threshold = Q3 + 1.5 * IQR = 49.5 + 1.5 * 8 = 49.5 + 12 = 61.5
Now, let's check which number is outside the range from 29.5 to 61.5:
- 40 is within the range.
- 44 is within the range.
- 72 is outside the range.
- 52 is outside the range.
Therefore, 72 is the number that, if added to the dataset, would represent an outlier in the data.