Standard Deviation (Probability Distribution)

Factual Questions
1

Yes, this is homework. No, I’m not looking for you to give me the answer.
I’m having a problem. I’m trying to do the chapter practice to prepare for a test. My textbook has a tear-out folding thingie that has formulas and tables. I am using that. I do not have a scientific calculator and I don’t want to use one. I’d prefer to be able to do the work myself.

Anyway, this is an online class and the stupid program is telling me that I’m getting the wrong answer for the standard deviation of a probability distribution. I am using the formula from the card which is:

σ=√{Σ[x²(Px)]-µ}
The program is telling me to use this:

σ²=Σ[(x-µ)²(Px)]

I’ve tried them both and they’re definitely giving different answers. Can someone please tell me which one is the right formula?

2

The latter is the correct formula. The former should have a square on the µ.

That is, both σ=√{Σ[x²(Px)]-µ²} and σ²=Σ[(x-µ)²(Px)] are correct.

3

I knew I was going to forget something.

Amazingly enough, my professor already got back to me.

Apparently they’re both right (except for the missing square). One is for variance, the other is for standard deviation. There are 2 ways to find the variance and I forgot that. One is supposed to be easier to compute and one easier to understand. The one in the OP is supposed to be easier to understand.

So, I should be getting the same answer with both and I’m not and I don’t know why. I’ve done it multiple times and I’m getting the same answers. I must be making the same mistake every time.

Thanks for answering.
On preview: In other words - what you said. I was habitually using the easier to compute variance formula since I’m doing it with a regular calculator so I completely forgot that there was another way. Still need to find where I’m making my mistake though.

4

The second formula gives the variance. The (corrected) first formula gives standard deviation. Standard deviation is square root of variance.

Maybe you’re thinking about this formula:

Var(X) = E(X^2) - (E(X))^2 (where E denotes expectation).

This formula is derived from the definition of variance:

Var(X) = E((x-mu)^2).

5

Well, show us an example of what you’re doing and perhaps we could point out the mistake. I think that would count more as tutoring you then doing your homework for you.

ETA: This is in reply to the OP, not to statsman1982. Incidentally, the formula you give, statsman, is exactly what the first formula in the OP says (when the missing square on the mu is reinserted), only with the square root added to take standard deviation. Both formulas can just as well be read as giving standard deviation, if you solve for sigma, or as giving variance, if you solve for sigma^2; it’s just that one happens to be written “sigma = sqrt(…)” and one happens to be written “sigma^2 = …”, which is a minor presentational choice.

6

I finally figured out what stupid mistake I was making over an over.

I’m using the standard deviation formula (σ=√{Σ[x^2*P(x)]-µ^2}

Even though I keep reading x^2*P(x) as x squared times the probability of x, I was writing it down as x squared times the probability times x. Big difference. I knew what it meant but I was still writing it down wrong.

Once I figured out what I was doing, I reworked the problem and got the correct answer.

Thank you for answering. I don’t mind online classes but the one problem I do have is that I never know when my professors will answer. In this case I was lucky and he answered quickly but if he hadn’t, you guys were here and I appreciate it.

7

That’s part of the reason why I always denote the probability mass function or probability density function as f(x). For some reason, there’s not as much confusion for my students when I use f(.). This is interesting to me because many of my students were making the exact same mistake as you were last week :). I think it is because many of them are fresh out of college algebra, and are used to parentheses meaning multiply, as in (3)(4) = 3 x 4. Except, though, when it comes to f(x). No one ever seems to want to multiply f and x! :slight_smile:

A lot of textbooks, including the one I’m forced to use, use P(x), because for discrete distributions, P(x) gives the probability that X = x, so the mnemonic works. But for continuous distributions, the value at x is not probability, so the mnemonic fails in that case, and things get even more confusing.

8

And this is exactly what I was thinking was the cause of my problem. I’m not fresh out of college algebra. I’m taking it right now. Actually, I’m supposed to be studying for my Algebra midterm but the Stats practice is taking way longer than I expected. This is the first time that one has interfered with the other.