# How to know the variability of Expected Value for a sample series of events with different outcome probabilities?

I have a quick question I would humbly like to ask for your help to solve: let's assume that I am analyzing a sample series of events with different probabilities of success. There are only 2 possible results: "success" or "failure".

See the table below for reference:

To find the EV of this set, I took the number of Successes (6) and from that, subtracted the sum of all percentages in the column a_prob (7,41). Therefore, 6 - 7,41 = -1,41, which should be the EV for this set.

What I would like to know now is by how much this EV can vary, considering that this data set is a sample of a bigger universe. My first idea was to calculate the Standard Deviation, but the EV is a constant, so I am not sure if I can apply StDev to that.

So first question would be: how can I know the variability of the number?

And second: how can I calculate that using Excel?

Thank you so much in advance!

EDIT 1: Grammar, correcting mistakes. And thanks for the answers so far - the question was worded poorly before. I hope it is understandable now.

• You say "The results can be either "success" or "failure", so the distribution is binomial." but also say the sucess probabilities are unequal. But then the distribution is not binoial, which assumes constant success probability. Commented Jul 30, 2015 at 19:27
• Thank you for your comment! As you can see, I lack knowledge on many basic statistics concepts - I thought that what made the distribution binomial was the idea that only 2 results were possible, and not necessarily the probability of each result happening. Thank you for helping me understand this =)! Commented Jul 30, 2015 at 20:00
• You seem to be using "expected value" to be something like "sample mean". It isn't. The expected value is a property of a probability distribution; if the situation is unaltered so that the underlying distribution is unchanged, expected value doesn't change.. Commented Aug 1, 2015 at 10:34
• Thanks Glen_b. This is very helpful knowledge. From what I understand, what I am trying to calculate is not needed then, correct? Expected Value should not have any variability as long as the probabilities taken into account to calculate it don't change. Please, let me know if this thought process is not correct. Commented Aug 1, 2015 at 18:28

## 1 Answer

First of all, "standard deviation of expected value" doesn't make sense. The expected value is a constant and its standard deviation is zero. Do you mean the s.d. of the sample proportion? If so this can be found in a straightforward way. Letting $X_i$ be an indicator equal to one if trial $i$ is a success and zero otherwise and assuming independence across trials \begin{align} \text{Var} \left ( \frac{\sum_{i=1}^{n} X_i}{n} \right ) &= \frac{1}{n^2} \sum_{i=1}^{n} \text{Var}(X_i) \\ &= \frac{1}{n^2} \sum_{i=1}^{n} p_i(1 - p_i) \end{align} and so the standard deviation is $n^{-1} \sqrt{\sum_{i=1}^{n}p_i (1 - p_i)}$.

• Thank you for the answer! From your reply: "Do you mean the s.d. of the sample proportion?" I am not sure, but I think that yes, this is what I need. What I wanted to know, put in simpler words, is: given that the EV in this particular case is -1,41, what is the variability of this EV? I assume it is not a number with zero variability, but as you mentioned, the variability actually comes from the sample proportion, and not from the EV itself, correct? And to measure this variability using the same unit, I also assumed that calculating its Standard Deviation would be the best way to go. Commented Jul 30, 2015 at 19:56
• Still learning how to use Stack Exchange - I had no idea I could only edit my comments 5 minutes after posting. Tried to skip one line by pressing Enter and it posted my comment. Very bad UX for such a good place to find knowledge. In any case: are my assumptions from the previous comment correct? Thank you! Commented Jul 30, 2015 at 19:58
• I'm confused as to what it is exactly you're trying to find, so it's hard to give helpful advice. Commented Jul 30, 2015 at 20:37
• Sorry for the confusion. To find the EV of this set, I took the number of Successes (6) and from that, subtracted the sum of all percentages in the column a_prob (7,41). Therefore, 6 - 7,41 = -1,41, which should be the EV for this set. However, assuming that these events are a sample of a bigger universe , shouldn’t this EV (-1,41) float (vary) a little bit? This is what I'm trying to understand: how much variability is there in the EV I found? And again, thank you for all the help! Commented Jul 31, 2015 at 5:38