To determine if there is an outlier in the data set and which measure of central tendency best represents the data, we can follow these steps:
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Identify the Outlier: A common method to detect outliers is to calculate the interquartile range (IQR) and find points that lie outside of \(1.5 \times IQR\) from the first and third quartiles.
- Sorting the data: 104, 104, 104, 109, 113, 117, 136, 189
- Finding Quartiles:
- Q1 (the median of the first half) = 104
- Q3 (the median of the second half) = 117
- Calculating IQR: \[ IQR = Q3 - Q1 = 117 - 104 = 13 \]
- Calculating outlier boundaries:
- Lower bound = \(Q1 - 1.5 \times IQR = 104 - 19.5 = 84.5\)
- Upper bound = \(Q3 + 1.5 \times IQR = 117 + 19.5 = 136.5\)
The only value that lies outside these bounds in the dataset is 189. Therefore, it is an outlier.
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Choosing the Measure of Central Tendency:
- Mean: \[ \text{Mean} = \frac{104 + 104 + 104 + 109 + 113 + 117 + 136 + 189}{8} = \frac{976}{8} = 122 \]
- Median:
- The dataset (after sorting) has an even number of observations (8), so the median will be the average of the two middle numbers: \[ \text{Median} = \frac{109 + 113}{2} = 111 \]
Given that the mean can be affected by the outlier (189), and the median is a better measure for skewed distributions or data with outliers, the median is often the preferred measure of central tendency in this case.
Conclusion:
- Yes, there is an outlier at 189. The median is the best measure.
Thus, the appropriate response is: Yes outlier at 189. Median is the best measure.