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 4 years are listed below. \( 54 \quad 46 \quad 28 \quad 129 \quad 65 \quad 39 \quad 39 \quad 78 \)

To find the mean, median, and mode of the given data, we first need to arrange the data in ascending order: \( 28, 39, 39, 46, 54, 65, 78, 129 \) 1. **Mean**: The mean is calculated by adding all the numbers in the data set and then dividing by the total number of values. Mean = \( \frac{28 + 39 + 39 + 46 + 54 + 65 + 78 + 129}{8} \) 2. **Median**: The median is the middle value of the data set when it is arranged in ascending order. If the data set has an odd number of values, the median is the middle value. If the data set has an even number of values, the median is the average of the two middle values. In this case, the data set has 8 values, so the median is the average of the 4th and 5th values. Median = \( \frac{46 + 54}{2} \) 3. **Mode**: The mode is the value that appears most frequently in the data set. In this case, the mode is the value that appears more than once. Now, let's calculate the mean, median, and mode of the given data. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(28+39+39+46+54+65+78+129\right)}{8}\) - step1: Remove the parentheses: \(\frac{28+39+39+46+54+65+78+129}{8}\) - step2: Add the numbers: \(\frac{478}{8}\) - step3: Reduce the fraction: \(\frac{239}{4}\) The mean of the data is \( 59.75 \) minutes. Next, let's calculate the median. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(46+54\right)}{2}\) - step1: Remove the parentheses: \(\frac{46+54}{2}\) - step2: Add the numbers: \(\frac{100}{2}\) - step3: Divide the terms: \(50\) The median of the data is \( 50 \) minutes. Finally, let's determine the mode of the data. Numbers by following steps: - step0: Solution: \(39\) The mode of the data is \( 39 \) minutes. In summary: - Mean: \( 59.75 \) minutes - Median: \( 50 \) minutes - Mode: \( 39 \) minutes All three measures of central tendency (mean, median, and mode) have been calculated for the given data set.