Find the mean, median, and mode of the data, if possible. If any of these measures cannot be found or a measure does not represent the center of the data, explain why. The durations (in minutes) of power failures at a residence in the last 6 years are listed below. \( 56 \quad 67 \quad 113 \quad 35 \quad 113 \quad 29 \quad 67 \quad 17 \quad 72 \)

To find the mean, median, and mode of the given data set, we will first organize the data and then calculate each measure. ### Given Data: The durations (in minutes) of power failures are: \[ 56, 67, 113, 35, 113, 29, 67, 17, 72 \] ### Step 1: Organize the Data Let's arrange the data in ascending order: \[ 17, 29, 35, 56, 67, 67, 72, 113, 113 \] ### Step 2: Calculate the Mean The mean is calculated by summing all the values and dividing by the number of values. **Sum of the values:** \[ 17 + 29 + 35 + 56 + 67 + 67 + 72 + 113 + 113 = 499 \] **Number of values:** There are 9 values. **Mean:** \[ \text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}} = \frac{499}{9} \approx 55.44 \] ### Step 3: Calculate the Median The median is the middle value when the data is ordered. Since there are 9 values (an odd number), the median is the value at position \((n + 1) / 2\), where \(n\) is the number of values. **Position of the median:** \[ \text{Position} = \frac{9 + 1}{2} = 5 \] The 5th value in the ordered list is: \[ 67 \] **Median:** \[ \text{Median} = 67 \] ### Step 4: Calculate the Mode The mode is the value that appears most frequently in the data set. From the ordered data: - 17 appears 1 time - 29 appears 1 time - 35 appears 1 time - 56 appears 1 time - 67 appears 2 times - 72 appears 1 time - 113 appears 2 times The modes are 67 and 113, as both appear twice. **Mode:** \[ \text{Mode} = 67 \text{ and } 113 \] ### Summary of Results - **Mean:** Approximately \( 55.44 \) - **Median:** \( 67 \) - **Mode:** \( 67 \) and \( 113 \) ### Conclusion All three measures of central tendency (mean, median, and mode) can be calculated for this data set. The mean may not represent the center of the data well because it is influenced by the higher values (like 113), while the median (67) and modes (67 and 113) provide a better representation of the center of the data. The presence of two modes indicates that the data is bimodal, which can also affect the interpretation of the central tendency.