Question

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. 53 113 68 12 113 72 72 47 22 31

Explanation:

Step1: Organize the data

First, we list the data in ascending order: \(12, 22, 31, 47, 53, 68, 72, 72, 113, 113\)
There are \(n = 10\) data points.

Step2: Calculate the mean

The mean \(\bar{x}\) is calculated by the formula \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\)
First, we find the sum of the data:
\(12+22 + 31+47+53+68+72+72+113+113\)
\(=(12 + 22)+(31+47)+(53+68)+(72+72)+(113+113)\)
\(=34+78+121+144+226\)
\(=(34 + 78)+(121+144)+226\)
\(=112+265+226\)
\(=112 + 491\)
\(=603\)
Then, the mean \(\bar{x}=\frac{603}{10}=60.3\)

Step3: Calculate the median

For \(n = 10\) (even number of data points), the median is the average of the \(\frac{n}{2}\)-th and \((\frac{n}{2}+ 1)\)-th values.
\(\frac{n}{2}=\frac{10}{2}=5\) and \(\frac{n}{2}+1 = 6\)
The 5 - th value is \(53\) and the 6 - th value is \(68\)
Median \(=\frac{53 + 68}{2}=\frac{121}{2}=60.5\)

Step4: Calculate the mode

The mode is the value that appears most frequently.
\(12\) appears 1 time, \(22\) appears 1 time, \(31\) appears 1 time, \(47\) appears 1 time, \(53\) appears 1 time, \(68\) appears 1 time, \(72\) appears 2 times, \(113\) appears 2 times.
So, the modes are \(72\) and \(113\) (bimodal)

Answer:

Mean: \(60.3\) minutes, Median: \(60.5\) minutes, Modes: \(72\) minutes and \(113\) minutes