How to calculate Standard Deviation of a sample

Question

I have 100 measurements of a device being either 0 or 1. Measurements are independent.

Assuming P(0) = P(1), that is probability of getting 0 is the same as getting a 1, how would I calculate standard deviation of total number of "0" measurements (or total number of "1" measurements) ?

Answer 1

First of all you have to tell us if the 100 measurements are independent of not. If they are, the sum of your 100 measurements follows a binomial distribution known with mean and variance

In other words, if $X$ is the rv describing the measurement of a device,

$$Y=X_1+X_2+\dots+X_{100}\sim B(100;0.5)$$

and thus

$$\mathbb{V}[Y]=\frac{100}{4}$$

that is

$$\sigma_Y=5$$

here $Y$ represent the total "1" you have in your 100 measurements.

The random variable describing the total "0" can be written as

$$Z=100-Y$$

with evidently the same st dev.

---

To calculate $V(Y)$ you have two ways:

Given that your single measurement can be 0 or 1 with the same probability 0.5, the random variable is a bernulli with parameter 0.5.

Thus $V(X_1)=0.5\times0.5=0.25$ and obviously $V(Y)=V(X_1)+\dots+V(X_{100})=\frac{100}{4}=25$

Another way is to observe that the sum of independent bernulli is a binomial with variance $V(Y)=np(1-p)=100\times\frac{1}{2}\times\frac{1}{2}=25$