What is the outlier in the following set:
29,2,28,30,26,31
The Correct Answer and Explanation is :
To identify the outlier in the given set of numbers—29, 2, 28, 30, 26, 31—we need to analyze the values and determine which one is significantly different from the others.
Identifying the Outlier:
The set consists of the following numbers: 29, 2, 28, 30, 26, and 31. By examining these values, we notice that all of them fall within a similar range except for one: 2. The other numbers (29, 28, 30, 26, and 31) are all in the mid-to-high 20s and 30s, while 2 is significantly lower.
Statistical Explanation:
To provide a more systematic identification of the outlier, we can calculate the mean and standard deviation of the dataset.
- Mean Calculation:
- Add the numbers together: 29 + 2 + 28 + 30 + 26 + 31 = 146.
- Divide by the number of values: 146 / 6 = 24.33.
- Standard Deviation:
- Calculate the differences from the mean:
- (29 – 24.33)² = 21.78
- (2 – 24.33)² = 508.78
- (28 – 24.33)² = 13.49
- (30 – 24.33)² = 31.11
- (26 – 24.33)² = 2.78
- (31 – 24.33)² = 44.49
- The sum of the squared differences: 21.78 + 508.78 + 13.49 + 31.11 + 2.78 + 44.49 = 622.43.
- Divide by the number of values: 622.43 / 6 = 103.74.
- Take the square root to find the standard deviation: √103.74 ≈ 10.18.
- Outlier Identification:
- An outlier is typically defined as a value that is more than 1.5 times the interquartile range (IQR) above the third quartile or below the first quartile or more than two standard deviations from the mean.
- The mean is 24.33, and two standard deviations would be 24.33 ± 20.36 (2 × 10.18). Therefore, any number below 3.97 or above 44.69 could be considered an outlier.
- The number 2 falls well below the lower threshold, confirming it as the outlier.
Conclusion:
Thus, the outlier in the set of numbers 29, 2, 28, 30, 26, and 31 is 2, as it is significantly different from the other values, which cluster around the mid-to-high 20s and 30s.