Understanding the science of statistics begins with the ability to interpret sets or series of data. In the case of purely numerical data, this in turn begins with an understanding of the three Ms of statistics: mean, median and mode.
Mean is another word for average. To find it, add the value of all given numbers in the set and divide the result by the number of entries. Even if all entries are whole numbers, the result will not always be as such. Take, for instance, the following set: 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7 and 9. The sum of these numbers is 77, and there are 15 entries. Therefore, the average is 5.1333 or 5 2/15, even though all of them are whole numbers. When finding mean, the order of the entries is inconsequential. This set could be kept as is, or it could be rearranged in ascending order to appear as 1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 9, 9 and 9 and the result would still be the same.
For long lists with consistent intervals, there is a fast way to find the mean without using a calculator. Take, for instance, all integers from 1 through 100. Add the lowest and highest numbers, and we get 101. Mark those off and move inward, and we have 2 + 99 = 101. Do it once more and we have 3 + 98 = 101. With the pattern made clear, we simply count how many permutations occur before we reach the middle. There are 50, going from 1 + 100 to 50 + 51.Multiply the number of permutations, 50, by the consistent result of each, 101 and get 5050. Divide by 100, and we get 50.5.
Order is essential, however, in finding the median. The median is the middle entry on the list. This is easy to remember if you recall that the wall dividing the two sides of the interstate, putting it right in the center, is also called a median. Anyway, in the case of a list of whole numbers with consistent intervals (such as 1, 2, 3, 4 or 12, 18, 24, 30) the median will always be a whole number if the number of entries is odd. If the number of entries is even, the median will only be a whole number if the interval is an even number.
This is because there can only be one median. Thus, if our marking off entries two at a time, one from each end of the list, results in two remaining entries at the middle, we must take the average of those two. This occasion will arise only in the instance of an even number of entries. If we have a consistently even interval, the average of those two will always be a whole number. This is because an even interval means that either both numbers are even or both are odd. Two odds or two evens result in an even number which is divisible by two. If the interval is odd, one will be odd and the other even. The sum of an odd number and an even number is always an odd number which is not divisible by two.
Also, in the case of consistent intervals, we find that the mean and the median are the same. Look back at the example of 1 through 100. The quick way of calculating the mean involved marking our way through either end of the list until we reached the middle, at which sat 50 and 51. The average of these two, the halfway point between them, is 50.5, which was also the mean for the entire list. This is only the case with consistent intervals or a symmetrical pattern of intervals (such as +1, +2, +3, +2, +1).
An example of that parenthetical list of intervals would be 1, 2, 4, 5, 7, 8. The mean of these is 27/6 = 4.5. The median is halfway between 4 and 5, which is also 4.5. However, if we had a pattern such as 1, 2, 4, 7, 11, 16, the mean would be 41/6 = 6.8333, while the median would be halfway between 4 and 7, which is 5.5. See, intervals have no bearing on the median, but they do influence the mean. Conversely, order has no bearing on the mean, but it is essential to finding the median.
Intervals and order both have no impact whatsoever on the third M, mode. The mode of a list is the entry that appears the most, an easy definition to remember since the words mode and most sound quite similar. Unlike median and mean, there can be more than one mode, as there is no way to perform a tie-breaker. In the very first list given, we find when we put the entries in order (which makes the process easier, but is not essential) that 5 and 9 each appear three times. Thus the mode is both 5 and 9. If there were one more of either, than one number would become the mode. If there is the same number of all entries, there basically is no mode.
Keeping track of the mode is essentially the basis of counting cards in black jack. The count is based on what cards have turned up. Although it’s simplified, for the sake of quick tracking, to high cards, low cards and middle cards, a card counter still keeps track of whether the cards dealt have consisted more of lows or highs.
To conclude, we have mean as the average of all entries, dependent on intervals but not order, median as the middle entry, dependent on order but not interval, and mode as the most common entry, dependent on neither. Getting comfortable with these is the first of many steps towards understanding statistics. Granted, you’re still a long ways from understanding, for instance, BCS rankings, but you can’t get to that point without starting with the three Ms. For those curious, the next step is to understand standard deviation, which is influenced by both order and interval.