User:Kiplynbrenton

Mathematical Statisticsby B. BrooksMean, Median, Mode, and Range Mean, median, and mode are three kinds of "averages". There are many "averages" in statistics, but these are, I think, the three most common, and are certainly the three you are most likely to encounter in your pre-statistics courses, if the topic comes up at all. The "mean" is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers. The "median" is the "middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order, so you may have to rewrite your list first. The "mode" is the value that occurs most often. If no number is repeated, then there is no mode for the list.The "range" is just the difference between the largest and smallest values.Find the mean, median, mode, and range for the following list of values:13, 18, 13, 14, 13, 16, 14, 21, 13The mean is the usual average, so:(13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15Note that the mean isn't a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers.The median is the middle value, so I'll have to rewrite the list in order:13, 13, 13, 13, 14, 14, 16, 18, 21There are nine numbers in the list, so the middle one will be the(9 + 1) ÷ 2 = 10 ÷ 2 = 5th number:13, 13, 13, 13,14,14, 16, 18, 21So the median is14. The mode is the number that is repeated more often than any other, so13is the mode.The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8.mean:15 median:14 mode:13 range: 8Note: The formula for the place to find the median is "( [the number of data points] + 1) ÷ 2", but you don't have to use this formula. You can just count in from both ends of the list until you meet in the middle, if you prefer. Either way will work.Find the mean, median, mode, and range for the following list of values:8, 9, 10, 10, 10, 11, 11, 11, 12, 13The mean is the usual average:(8 + 9 + 10 + 10 + 10 + 11 + 11 + 11 + 12 + 13) ÷ 10 = 105 ÷ 10 = 10.5The median is the middle value. In a list of ten values, that will be the(10 + 1) ÷ 2 = 5.5th value; that is, I'll need to average the fifth and sixth numbers to find the median:(10 + 11) ÷ 2 = 21 ÷ 2 = 10.5The mode is the number repeated most often. This list has two values that are repeated three times.The largest value is 13 and the smallest is 8, so the range is 13 – 8 = 5.mean:10.5 median:10.5 modes:10 and11 range: 5While unusual, it can happen that two of the averages (the mean and the median, in this case) will have the same value.Note: Depending on your text or your instructor, the above data set may be viewed as having no mode (rather than two modes), since no single solitary number was repeated more often than any other. I've seen books that go either way; there doesn't seem to be a consensus on the "right" definition of "mode" in the above case. So if you're not certain how you should answer the "mode" part of the above example, ask your instructor before the next test.About the only hard part of finding the mean, median, and mode is keeping straight which "average" is which. Just remember the following:mean: regular meaning of "average" median: middle value mode: most often(In the above, I've used the term "average" rather casually. The technical definition of "average" is the arithmetic mean: adding up the values and then dividing by the number of values. Since you're probably more familiar with the concept of "average" than with "measure of central tendency", I used the more comfortable term.)A student has gotten the following grades on his tests: 87, 95, 76, and 88. He wants an85 or better overall. What is the minimum grade he must get on the last test in order to achieve that average?The unknown score is "x". Then the desired average is:(87 + 95 + 76 + 88 + x) ÷ 5 = 85Multiplying through by 5 and simplifying, I get:87 + 95 + 76 + 88 + x = 425  346 + x = 425  x = 79