To identify an outlier in a set of numbers, we first need to determine the central tendency of the data, such as the mean or median. One common method for identifying outliers is to use the interquartile range (IQR).
To do this, we first need to find the median of the data set:
45, 52, 17, 63, 57, 42, 54, 58
Arranging them in ascending order:
17, 42, 45, 52, 54, 57, 58, 63
The median is the middle value, which is 54.
Next, we need to find the IQR. The IQR is the range between the first and third quartiles of the data. The first quartile (Q1) is the median of the lower half of the data, and the third quartile (Q3) is the median of the upper half of the data.
To find Q1 and Q3, we split the data into two halves:
Lower half: 17, 42, 45, 52
Upper half: 54, 57, 58, 63
Q1 is the median of the lower half, which is (42 + 45)/2 = 43.5.
Q3 is the median of the upper half, which is (57 + 58)/2 = 57.5.
Therefore, the IQR is 57.5 - 43.5 = 14.
Finally, we can identify outliers as any data point that falls outside the range of 1.5 times the IQR above Q3 or below Q1.
The upper limit is Q3 + 1.5(IQR) = 57.5 + 1.5(14) = 78.5.
The lower limit is Q1 - 1.5(IQR) = 43.5 - 1.5(14) = 22.5.
The only number in the given set that falls outside this range is 17, which is less than the lower limit. Therefore, 17 is the outlier in this data set.