The number of cartoons x, watched by mrs. kelly's first grade class on saturday morning is shown below.

x p(x)
0 0.15
1 0.20
2 0.30
3 0.10
4 0.20
5 0.05
give the standard deviation for the probability distribution above.

The standard deviation σ is the square root of the variance:σ=V(X)−−−−−√=E(X2)−(E(X)2−−−−−−−−−−−−−√)where E(X)is the expected value of X. By definition,E(X)=∑x∈Dxp(x)where D is the values X may take (in this case D={0,1,2,3,4,5})and p(x)=P(X=x)is the probability that X equals x.Then,E(X)==0⋅p(0)+1⋅p(1)+2⋅p(2)+3⋅p(3)+4⋅p(4)+5⋅p(5)2.15.
Similarly,E(X2)==02⋅p(0)+12⋅p(1)+22⋅p(2)+32⋅p(3)+42⋅p(4)+52⋅p(5)6.75,soV(X)=6.75−2.15=4.6andσ=4.6−−−√≈2.144.