Advanced concept in Statistics

We know these three concepts of statistics: – Arithmetic mean is the number that can represent/replace all the numbers of the sequence. It lies somewhere in between the smallest and the largest values. – Median is the middle number (in case the total number of numbers is odd) or the average of two middle numbers (in case the total number of numbers is even). – Standard deviation is a measure of the dispersion of the values around the mean. A conceptual question is how these three measures change when all the numbers of the set are varied is a similar fashion. For example, how does the mean of a set change when all the numbers are increased by say, 10? How does the median change? And what about the standard deviation? What happens when you multiply each element of a set by the same number? Let’s discuss all these cases in detail but before we start, we would like to point out that the discussion will be conceptual. We will not get into formulas though you can arrive at the answer by manipulating the respective formulas. When you talk about mean or median or standard deviation of a list of numbers, imagine the numbers lying on the number line. They would be spread on the number line in a certain way. For example, ——0—a———b—c———————d———e————————f—g——————— Case I: When you add the same positive number (say x) to all the elements, the entire bunch of numbers moves ahead together on the number line. The new numbers a’, b’, c’, d’, e’, f’ and g’ would look like this ——0——————a’———b’—c’———————d’———e’————————f’—g’—————— The relative placement of the numbers does not change. They are still at the same distance from each other. Note that the numbers have moved further to the right of 0 now to show that they have moved ahead on the number line. The mean lies somewhere in the middle of the bunch and will move forward by the added number. Say, if the mean was d, the new mean will be d’ = d + x. So when you add the same number to each element of a list,  New mean = Old mean + Added number. On similar lines, the median is the middle number (d in this case) and will move ahead by the added number. The new median will be d’ = d + x So when you add the same number to each element of a list,  New median = Old median + Added number Standard deviation is a measure of dispersion of the numbers around the mean and this dispersion does not change when the whole bunch moves ahead as it is. Standard deviation does not depend on where the numbers lie on the number line. It depends on how far the numbers are from the mean. So standard deviation of 3, 5, 7 and 9 is the same as the standard deviation of 13, 15, 17 and 19. The relative placement of the numbers in both the cases will be the same. Hence, if you add the same number to each element of a list, the standard deviation will stay the same. Case II: Let’s now move on to the discussion of multiplying each element by the same positive number. The original placing of the numbers on the number line looked like this: ——0—a———b—c———————d———e————————f—g——————— The new placing of the numbers on the  number line will look something like this: ——0———a’——————b’———c’————————————d’—————————e—- etc The numbers spread out. To understand this, take an example. Say, the initial numbers were 10, 20 and 30. If you multiply each number by 2, the new numbers are 20, 40 and 60. The difference between them has increased from 10 to 20. If you multiply each number by x, the mean also gets multiplied by x. So, if d was the mean initially, d’ will be the new mean which is x*d. New mean = Old mean * Multiplied number Similarly, the median will also get multiplied by x. New median = Old median * Multiplied number What happens to standard deviation in this case? It changes! Since the numbers are now further apart from the mean, their dispersion increases and hence the standard deviation also increases. The new standard deviation will be x times the old standard deviation. You can also establish this using the standard deviation formula. New standard deviation = Old standard deviation * Multiplied number The same concept is applicable when you increase each number by the same percentage. It is akin to multiplying each element by the same number. Say, if you increase each number by 20%, you are, in effect, multiplying each number by 1.2. So our case II applies here. Now, think about what happens when you subtract/divide each element by the same number.