Question

Approximately what percent of normally distributed data values lie within 2 standard deviation to either side of the mean?

Answer 1

Let the random variable follow the normal distribution with mean  , and standard deviation, . Mathematically,

The problem here is to determine the area within two standard deviations, on the either side of the mean, . In other words, this denotes the probability that the absolute value of is less than . that is, needs to be determined.

Simplifying the expression, , using the folowing steps:

Step 1: Eliminate the modulus sign after making the necessary adjustments. This is shown below:

Step 2: Divide the expression inside the parenthesis by , as shown below:

Step 3: The middle term, that is   is equivalent to the standard normal variable, .

This implies that probability of the standard normal random variable lying between the point -2 and 2 needs to be determined.

Step 4: Simplifying the required probability further as shown below:

The probability of the left tail of the normal distribution can be calculated using the command "=NORMSDIST()" in MS-Excel. The screenshot is shown below:

This implies .

Substitutng these values in the above formula:

Therefore, of the values lie within 2 standard deviations towards the either side of the mean.