Based on the sample data set, 27 25 39 30 11 27 6 25 54 find the lower fence and upper fence. Lower Fence = Upper Fence = Is 30 a outlier? No Yes Is 27 a outlier? Yes No Is 27 a outlier? Yes No
The Correct Answer and Explanation is:
Here are the correct answers and the explanation for the problem.
Answers:
- Lower Fence = -6.75
- Upper Fence = 59.25
- Is 30 a outlier? No
- Is 27 a outlier? No
- Is 27 a outlier? No
Explanation:
To find the lower and upper fences and determine any outliers, we first need to organize the data and calculate key statistical values. The process involves several steps, starting with ordering the data set and finding the quartiles.
The given sample data set is 27, 25, 39, 30, 11, 27, 6, 25, 54. First, we must arrange these nine numbers in ascending order: 6, 11, 25, 25, 27, 27, 30, 39, 54.
Next, we find the median, also known as the second quartile (Q2), which is the middle value of the sorted set. With nine data points, the middle value is the fifth one in the list, which is 27.
Then, we calculate the first quartile (Q1). This is the median of the lower half of the data. The lower half of the data set consists of the numbers to the left of the median: 6, 11, 25, 25. Since there is an even number of values in this half, Q1 is the average of the two middle numbers, which is (11 + 25) / 2 = 18.
Similarly, we find the third quartile (Q3), which is the median of the upper half of the data. The upper half consists of the numbers to the right of the median: 27, 30, 39, 54. The average of its two middle numbers is (30 + 39) / 2 = 34.5.
Now we can compute the interquartile range (IQR), which measures the spread of the middle 50% of the data. It is the difference between Q3 and Q1. The IQR is 34.5 – 18 = 16.5.
With the IQR, we can find the fences that define the boundaries for outliers. The lower fence is calculated using the formula Q1 – 1.5 * IQR. This gives us 18 – 1.5 * 16.5 = 18 – 24.75 = -6.75.
The upper fence is calculated using the formula Q3 + 1.5 * IQR. This gives us 34.5 + 1.5 * 16.5 = 34.5 + 24.75 = 59.25.
An outlier is any data point that falls below the lower fence or above the upper fence. The value 30 is between -6.75 and 59.25, so it is not an outlier. The value 27 is also within this range, so it is not an outlier either.
