document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197349>Question 1197349</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The duration (minutes) of 26 electric power outages in the community of Sonando Heights over the past five years are shown below.<BR>\n" );
document.write( "35 48 25 65 25 9 57 11 51 5 15 46 41 <BR>\n" );
document.write( "101 32 17 13 50 23 5 18 26 89 28 13 19<BR>\n" );
document.write( "a) Find the mean, median, and mode. (Round your answer to 2 decimal places.)<BR>\n" );
document.write( "b) Are the mean and median about the same?<BR>\n" );
document.write( "c) Is the mode a good measure of center for this data set?<BR>\n" );
document.write( "d) is the distribution skewed? </SPAN>\n" );
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document.write( "**1. Calculate the Mean**\r<BR>\n" );
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document.write( "* Sum of all durations: 35 + 48 + 25 + 65 + 25 + 9 + 57 + 11 + 51 + 5 + 15 + 46 + 41 + 101 + 32 + 17 + 13 + 50 + 23 + 5 + 18 + 26 + 89 + 28 + 13 + 19 = 744<BR>\n" );
document.write( "* Mean = Sum of durations / Number of observations = 744 / 26 = 28.62 minutes\r<BR>\n" );
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document.write( "**2. Calculate the Median**\r<BR>\n" );
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document.write( "* Arrange the data in ascending order: <BR>\n" );
document.write( "   5, 5, 9, 11, 13, 13, 15, 17, 18, 19, 23, 25, 25, 26, 28, 32, 35, 41, 46, 48, 50, 51, 57, 65, 89, 101\r<BR>\n" );
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document.write( "* Since there are 26 observations (even number), the median is the average of the 13th and 14th values.<BR>\n" );
document.write( "* Median = (25 + 26) / 2 = 25.5 minutes\r<BR>\n" );
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document.write( "**3. Determine the Mode**\r<BR>\n" );
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document.write( "* The mode is the most frequent value. <BR>\n" );
document.write( "* In this case, the mode is 13 minutes (it appears twice).\r<BR>\n" );
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document.write( "**a) Results:**\r<BR>\n" );
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document.write( "* Mean: 28.62 minutes<BR>\n" );
document.write( "* Median: 25.5 minutes<BR>\n" );
document.write( "* Mode: 13 minutes\r<BR>\n" );
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document.write( "**b) Are the mean and median about the same?**\r<BR>\n" );
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document.write( "* No, the mean (28.62) is significantly higher than the median (25.5).\r<BR>\n" );
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document.write( "**c) Is the mode a good measure of center for this data set?**\r<BR>\n" );
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document.write( "* No, the mode (13) is not a good measure of center for this data set. <BR>\n" );
document.write( "    * It is significantly lower than both the mean and median. <BR>\n" );
document.write( "    * The distribution has several other values that appear more than once.\r<BR>\n" );
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document.write( "**d) Is the distribution skewed?**\r<BR>\n" );
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document.write( "* Yes, the distribution is likely skewed to the right (positively skewed). <BR>\n" );
document.write( "    * The mean is greater than the median, which is a typical characteristic of right-skewed distributions. <BR>\n" );
document.write( "    * The presence of some very high values (like 89 and 101) also suggests right-skewness.\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* The mean and median are different, indicating potential skewness.<BR>\n" );
document.write( "* The mode is not a representative measure of center for this data.<BR>\n" );
document.write( "* The distribution is likely right-skewed.<BR>\n" );
document.write( "**a) Find the mean, median, and mode.**\r<BR>\n" );
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document.write( "* **Mean:** <BR>\n" );
document.write( "    * Sum of all values / Number of values <BR>\n" );
document.write( "    * Mean = (35 + 48 + 25 + 65 + 25 + 9 + 57 + 11 + 51 + 5 + 15 + 46 + 41 + 101 + 32 + 17 + 13 + 50 + 23 + 5 + 18 + 26 + 89 + 28 + 13 + 19) / 26<BR>\n" );
document.write( "    * Mean ≈ 30.73 minutes\r<BR>\n" );
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document.write( "* **Median:** <BR>\n" );
document.write( "    * Arrange the data in ascending order: <BR>\n" );
document.write( "        5, 5, 9, 11, 13, 13, 13, 15, 17, 18, 19, 23, 25, 25, 26, 28, 32, 35, 41, 46, 48, 50, 51, 57, 65, 89, 101<BR>\n" );
document.write( "    * Median (middle value) = (26th value + 27th value) / 2 = (26 + 28) / 2 = 27 minutes\r<BR>\n" );
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document.write( "* **Mode:** <BR>\n" );
document.write( "    * The most frequent value is 13 minutes.\r<BR>\n" );
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document.write( "**b) Are the mean and median about the same?**\r<BR>\n" );
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document.write( "* No, the mean (30.73 minutes) is significantly higher than the median (27 minutes).\r<BR>\n" );
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document.write( "**c) Is the mode a good measure of center for this data set?**\r<BR>\n" );
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document.write( "* No, the mode (13 minutes) is not a good measure of center for this data set. It doesn't represent the typical outage duration well, as there are many values that are much higher.\r<BR>\n" );
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document.write( "**d) Is the distribution skewed?**\r<BR>\n" );
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document.write( "* **Yes, the distribution is likely skewed to the right (positively skewed).** <BR>\n" );
document.write( "    * The presence of some very high values (outliers like 89 and 101) pulls the mean to the right, while the median is less affected by these extreme values. <BR>\n" );
document.write( "    * This suggests that most outages are relatively short, but there are a few very long outages that skew the data.\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* The mean is 30.73 minutes.<BR>\n" );
document.write( "* The median is 27 minutes.<BR>\n" );
document.write( "* The mode is 13 minutes.<BR>\n" );
document.write( "* The mean and median are not the same.<BR>\n" );
document.write( "* The mode is not a good measure of center.<BR>\n" );
document.write( "* The distribution is likely skewed to the right.<BR>\n" );
document.write( " </SPAN>\n" );
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