# What is the difference between the standard deviation and margin of error?

Jan 28, 2016

Both characterize the quality of approximation of some unknown constant with results of its statistical evaluation.
Knowing one of them, we can determine another.

#### Explanation:

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
$\mu \cong \frac{{X}_{1} + {X}_{2} + \ldots + {X}_{100}}{100}$
and approximate unknown $\sigma$ as
$\sigma = \sqrt{\frac{{\left({X}_{1} - \mu\right)}^{2} + {\left({X}_{2} - \mu\right)}^{2} + \ldots + {\left({X}_{100} - \mu\right)}^{2}}{100}}$

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 $\mu - 2 \sigma$ and $\mu + 2 \sigma$, Radius of the confidence interval (that is a margin of error $\Delta$) is $2 \sigma$.

So, we can say that with probability 95% the car length is between $\mu - 2 \sigma$ and $\mu + 2 \sigma$.

As you see, standard deviation and margin of error are functionally dependent.