Standard Deviation word problem in Probability Theory

In probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. standard deviation formula is a widely used measure of the variability or dispersion, being algebraically more tractable though practically less robust than the expected deviation or average absolute deviation.We can use Standard deviation calculator to make it easy.

In simple terms, it shows how much variation there is from the "average" (mean). It may be thought of as the average difference of the scores from the mean of distribution, how far they are away from the mean. A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.Let's see an example from probability problems
Question

A volunteer ambulance service handles 0 to 5 service calls on any given day .The probability distribution for the number of service calls is as follows :

Number of calls : 0 probability 0.08,1-0.16,2-0.27,3-0.20,4-0.15and 5-0.14

a)What is the expected number of service calls(2 decimals)?

b)What is the variance in the number of service calls(to 2 decimals)?

c)What is the standard deviation (to 2 decimals)?

Answer

a)

Expected number= 0(0.08)+1(0.16)+2(0.27)+3(0.20)+4(0.15)+5(0.14)

=0+0.16+0.54+0.60+0.60+0.70

=2.6


b)

Var(x) = E(x²)-[E(x)]²

=8.94-(2.60)²

=8.94+6.76

=2.18

c)

Standard deviation = v2.18 = 1.48