Theory:
When values of the random variable #xi#, normally distributed with mathematical expectation #mu# and standard deviation #sigma#, are used to find an approximation for #mu#, there is a dependency between #sigma# and confidence interval of a radius #Delta# around #mu#, where the values of #xi# must be with certain probability, so we can say that
"With probability #P# the values of random variable #xi# will fall in an interval from #mu - Delta# to #mu + Delta#."
Given any required probability #P#, we can find confidence interval #Delta# if we know the standard deviation #sigma#.
Obviously, the higher required probability with a given standard deviation #sigma# - the wider confidence interval must be.
On the other hand, if probability #P# is fixed, the smaller standard deviation #sigma# is - the narrower confidence interval should be to satisfy the probability.
Radius of the confidence interval #Delta# is called a margin of error.
Example:
100 people measure the length of a car.
Their results #X_1,X_2...X_100# can be interpreted as values of normally distributed random variable with mathematical expectation equaled to the real length of a car #mu# and some standard deviation #sigma# that depends on precision of the measuring instrument and accuracy of the people.
Let's approximate unknown #mu# as
#mu ~= (X_1+X_2+...+X_100)/100#
and approximate unknown #sigma# as
#sigma = sqrt(((X_1-mu)^2+(X_2-mu)^2+...+(X_100-mu)^2)/100)#
Knowing these approximate values and fundamental properties of the normal distribution, for a given probability #P=0.95# we can derive that confidence interval should be within an interval from #mu - 2sigma# and #mu + 2sigma#, Radius of the confidence interval (that is a margin of error #Delta#) is #2sigma#.
So, we can say that with probability #95%# the car length is between #mu - 2sigma# and #mu + 2sigma#.
As you see, standard deviation and margin of error are functionally dependent.
