To determine the lower fence and upper fence for identifying outliers in the data set (-14, 32, 22, 21, 33, 7, 56, 40), we can use the IQR (Interquartile Range) method. The IQR is the range between the first quartile (Q1) and the third quartile (Q3).
First, let's find Q1 and Q3:
1. Arrange the data in ascending order: -14, 7, 21, 22, 32, 33, 40, 56.
2. Calculate Q1, which is the median of the lower half of the data set (the first four numbers):
Q1 = (7 + 21) / 2 = 28 / 2 = 14.
3. Calculate Q3, which is the median of the upper half of the data set (the last four numbers):
Q3 = (32 + 33) / 2 = 65 / 2 = 32.5.
Next, calculate the IQR:
IQR = Q3 - Q1 = 32.5 - 14 = 18.5.
Now, find the lower fence and upper fence:
Lower Fence = Q1 - 1.5 * IQR
Lower Fence = 14 - 1.5 * 18.5
Lower Fence = 14 - 27.75
Lower Fence ≈ -13.75 (rounded to two decimal places)
Upper Fence = Q3 + 1.5 * IQR
Upper Fence = 32.5 + 1.5 * 18.5
Upper Fence = 32.5 + 27.75
Upper Fence ≈ 60.25 (rounded to two decimal places)
Now, let's determine if any of the values in the data set are outliers:
- -14 is less than the lower fence (-13.75), so it is an outlier.
- 56 is greater than the upper fence (60.25), so it is an outlier.
- 40 is not greater than the upper fence (60.25), so it is not an outlier.
So, to answer your questions:
- Is -14 an outlier? Yes.
- Is 56 an outlier? Yes.
- Is 40 an outlier? No.