During an experiment some water was removed from each of the 6 tanks .If the Std deviation of the volumes of the water at the beginning of the experiment was 10 gallons what was the standard deviation of the volumes of the water after the experiment ?

1) For each tank 30% of the volume of the water that was in the tank before the beginning of the experiment were removed during the experiment

2) The average ( mean ) volume of water in the at the end of the experiment was 63 gallons

Please explain the approach ?

Make sure you know that, when ALL numbers in a set are multiplied or divided by some number,** the mean and standard deviation are multiplied/divided by the same number. If you don't see why this should be, consider a set of heights expressed in feet, and then expressed in inches (which is the same thing as saying 'all multiplied by 12'). Since the set contains the same heights, its mean and standard deviation will of course be the same - which means that the mean and standard deviation of the 'inches' set will both be exactly 12 times as big as those of the 'feet' set.

**This includes increasing or decreasing all the numbers in the set by some percentage (which can be accomplished by multiplication: e.g., 30% increase = multiplication by 1.3).

Using this principle, statement (1) tells us that both the mean and the standard deviation of the set will decrease by 30%. Therefore, the new standard deviation will decrease to 7 gallons. SUFFICIENT.

Statement (2) tells us nothing about standard deviation, which measures SPREAD of numbers. If we achieved the 63 gallons by taking most of the water out of the tanks that were already lowest, then the standard deviation will be huge (because you'll have some tanks almost full and some almost empty). If we got there by taking most of the water out of the fullest tanks, then the standard deviation will be a lot smaller. INSUFFICIENT.

So the answer is A.

Hi Ron,

Your explanations are by far the best available online. I have learnt a lot from your posts.


That said,

For the above post, is it also true that if we add or substract the numbers by a constant term the SD still remains the same.. I worked with 2,4,6 and 3,5,7 and it gives the same result. But was not sure..


Thanks much..

correct: if you add the same constant to (or subtract the same constant from) a set of numbers, then the resulting set will have the same standard deviation as did the original set.

this is because standard deviation represents spread, and the spread of a set of numbers doesn't change at all when the numbers are all transformed by the same addition/subtraction. (visualize the set on a number line: the transformation does nothing more than silde all the points the same distance to the left or right, a move that won't affect any of the spaces between them.)

Ron, please see the link below where this question has been discussed previously. One of the comments (from Emily) is


http://www.manhattangmat.com/forums/during-an-experiment-some-water-was-removed-from-each-of-t819.html



Harish, the source of your confusion is your statement "I know that the standard deviation of the sample doesn't change if we add or subtract the same constant value to the sample values." That is only true if all of the samples have the same quantity to begin with (std. dev. = 0)!

The more accurate statement would have been "The standard deviation of the sample changes by a known factor if we add or subtract the same percentage to each of the sample values." If the samples each decrease by 30%, the mean decreases by 30%, and the (X - mean) decreases by 30% for each term. You don't really have to complete the calculation to see that the resulting std. dev. will be smaller than the original 10 by some factor (I believe the result would be 7, but you can check my math).

Applying your explanation and emily's explanation to this, does this mean the option A is sufficient or actually not

Not too clear about the context.

ah, whoa, no, the first statement is incorrect.

here are the correct statements:

* if you ADD OR SUBTRACT A CONSTANT to/from all the values in a set, then the standard deviation will remain exactly the same.
visual equivalent: imagine sliding all the data points along a number line, by exactly the same amount, to the right or left. if you do this, then the average spread obviously won't change, because, in fact, none of the spreads anywhere in the set changes at all.
this is where the referenced post went wrong: even if you have a set in which the values are wildly different from one another, the standard deviation will not change if a constant is added to or subtracted from all the values.

* if you INCREASE OR DECREASE ALL THE VALUES BY A FIXED FACTOR / PERCENTAGE, then the standard deviation will increase or decrease by the same percentage.
visual equivalent: imagine drawing a number line with the set on an elastic band, and stretching or contracting the elastic band to mimic the % increase / decrease applied to the set. if you do this, then the average spread must increase / decrease by the same %, because all the spreads will increase / decrease by that %.