Sean counted the number of stuffed animals available for prizes in each of the booths at a county fair. The list shows the results.
2, 23, 27, 29, 30, 32, 32, 34, 35, 96
Select all the data values that are outliers.
A.
2
B.
27
C.
34
D.
96
The value 2 is an outlier.
Therefore, the correct answer is:
A. 2
To determine if a data value is an outlier, we can use the Tukey's Fences method. According to this method, any data value that falls below Q1 - 1.5 * IQR or above Q3 + 1.5 * IQR is considered an outlier.
First, let's find the quartiles (Q1 and Q3) and the interquartile range (IQR) for the data set:
The data set in ascending order: 2, 23, 27, 29, 30, 32, 32, 34, 35, 96
Q1: The median of the lower half of the data set = (27 + 29) / 2 = 28
Q3: The median of the upper half of the data set = (34 + 35) / 2 = 34.5
IQR: Q3 - Q1 = 34.5 - 28 = 6.5
Next, let's calculate the fences:
Lower fence = Q1 - 1.5 * IQR = 28 - 1.5 * 6.5 = 18.25
Upper fence = Q3 + 1.5 * IQR = 34.5 + 1.5 * 6.5 = 44.75
The outlier data values are the ones that fall below the lower fence (18.25) or above the upper fence (44.75).
From the given data set, the only value that falls outside these fences is 96.
Therefore, the correct answer is:
D. 96
To identify outliers in a dataset, we can use the following steps:
1. Arrange the data values in ascending order: 2, 23, 27, 29, 30, 32, 32, 34, 35, 96.
2. Calculate the interquartile range (IQR), which is the difference between the third quartile (Q3) and the first quartile (Q1). To find Q1 and Q3:
a. Find the median, which is the middle value of the dataset. In this case, the median is the average of the fifth and sixth values, which is (30 + 32)/2 = 31.
b. Find Q1, which is the median of the lower half of the dataset. In this case, the lower half is 2, 23, 27, and 29. The median of these values is (23 + 27)/2 = 25.
c. Find Q3, which is the median of the upper half of the dataset. In this case, the upper half is 32, 32, 34, 35, and 96. The median of these values is (34 + 35)/2 = 34.5.
Therefore, Q1 = 25, Q3 = 34.5.
The IQR = Q3 - Q1 = 34.5 - 25 = 9.5.
3. Identify any data values that are more than 1.5 times the IQR above Q3 or below Q1. In this case, any values below Q1 - 1.5 * IQR or above Q3 + 1.5 * IQR can be considered outliers.
- Q1 - 1.5 * IQR = 25 - 1.5 * 9.5 = 11.75
- Q3 + 1.5 * IQR = 34.5 + 1.5 * 9.5 = 48.25
We can see that the data value 2 is below Q1 - 1.5 * IQR and is therefore an outlier.
Therefore, the correct answer is:
A. 2