Standard Deviation

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Standard Deviation
Standard Deviation, (or SD or Sigma, represented by the symbol σ) shows how much variation or dispersion exists from the average (mean, or expected value). In other words, it is a measure of how spread out the numbers of a set are and the GMAT tests how to read these numbers and their relationship to the entirety of the ‘spread’. An analogy:  Imagine a paintball gun in your hand and a clean, blank, taintless, squeaky white wall in front of you. Yes, I see the temptation too. And, I would do the same. Point and Shoot! Just for fun, release the inner child who wants to wreak havoc! Where is the math in here? Pay attention to each word in the following sentence. SD is a ‘measure’ of how ‘spread out’ the splash is from the center. A smaller SD means the splatter is small. A larger one means the splatter is big. Since this is GMAT quant, the splatter is comprised of a set of numbers. SIMPLE!
The actual numbers in SD problems don’t matter as much. What is important is how far each term is from the ‘mean.’ The ‘mean’ is: Woah, a math concept where actual numbers hardly matter. DELIGHTFUL! Then what matters – again – ”˜how far is each term from the mean’ matters. Basically, SD on the GMAT is Simple and Delightful. Now I’ll show you why:
Common areas of testing:

(a) If you multiply or divide every term in the set by the same number, the SD will change. SD will change by that same number. The mean will also change by the same number. Imagine the splatter to animatedly increase in size; but proportionately. The ”˜measure of spread’ will change. If every term is doubled, the distance between each term and the mean doubles, BUT also the distance between each term doubles and thus standard deviation increases. If each term is divided by two, the SD decreases.

(b) Adding a number to the set such that the number is very close to the mean generally reduces the SD. Imagine the splatter of paint on the wall. If I were to paint a droplet very close to the center, would it increase or decrease the spread? – Decrease, because now more droplets are closer to the center than before.

(c) Adding a number to the set such that the number is further away from the mean, will increase the SD. This is because the average distance of the numbers from the mean increases.

(d) SD of a set is zero if the range of the set is zero. If all the numbers in the set are the same then the range is zero and the SD is also zero. Consider the set {3, 3, 3, 3, 3}

(e) You need all these three to calculate SD.
– The population of the data set
– The mean of the data set and
– The data points themselves.

Definitions: Now, let’s see if you remember these definitions: arithmetic mean, median, mode, and range of a set of numbers. GMAT uses these building blocks and a few predictable ways to put them together to put together into seemingly complicated problems. ”¨”¨Symbols: It is important to know these so as to translate the math in the question into English.

1) σ read as Sigma is a symbol for Standard Deviation

2) μ pronounced as ”˜myu’ is a symbol for Arithmetic Mean

3) ”˜n’ denotes the number of elements in the set

Common Formula: SD = Square Root of the difference of the mean of the numbers and the square of the mean of the numbers.

Since SD problems in GMAT appear equally as Problem Solving and Data Sufficiency, here is one example of each kind.

Problem Solving:

If 25 students took a test, the average student scored 80% (arithmetic mean), the standard deviation was 5%, and the teacher awards ”˜A’s for those who scored 3 or more standard deviations above the mean, what is the lowest score for which a student could earn an A?
μ = 80% since this is arithmetic mean.”¨Lowest Score to Earn an A: Mean + 3(Standard Deviation)”¨Lowest Score to Earn an A: 80% + 3(5%) = 95%”¨”¨”¨Do you see what I see? The question did not ask for the value of SD. We didn’t calculate the value of SD. But, we needed to know the concept that a student needed to be further away – or more standard deviations away – from the average to get an A. ”¨”¨Data Sufficiency:

Set T consists of odd integers divisible by 5. Is standard deviation of T positive?

(1) All members of T are positive

(2) T consists of only one member

Statement 1 says that all the numbers of the set are positive but they may all be identical. If all the numbers of the set are identical, the range is zero and so is SD. Does that answer the question,’ Is SD positive?’. No, you cannot be sure – SD is either positive or zero. So (1) is not sufficient. ”¨Statement 2 says that the set has only one number. This makes the range zero and hence the SD also zero. Does that answer the question,’ Is SD positive?’. Yes, SD is definitely zero. (2) is sufficient data.  ”¨B is the answer then! ”¨”¨”¨Do you see what I see? No calculation at all. Applying the basic concept – that the DISTANCE of the variables from the mean is more important that the actual value of each variable and the mean – is enough.”¨”¨In B-school and later in your career, you will often use this concept in corporate finance and marketing. Again, always think about the ”˜distance between the terms’.

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