College

If the random variable [tex]$X$[/tex] is normally distributed, what percentage of all observed values of [tex]$x$[/tex] will be within 1 standard deviation of the mean?

A. [tex]$68 \%$[/tex]
B. [tex]$95 \%$[/tex]
C. [tex]$99.7 \%$[/tex]
D. [tex]$100 \%$[/tex]

Answer :

When dealing with a normally distributed random variable [tex]\( X \)[/tex], we can use the empirical rule, which is a statistical rule describing the distribution of data.

Here's a step-by-step breakdown:

1. Normal Distribution Basics:
- A normal distribution is symmetric, with most of the data clustering around a central point — the mean.
- The spread or dispersion of the distribution is measured by the standard deviation.

2. Empirical Rule (68-95-99.7 Rule):
- This rule helps us understand how data is spread in a normal distribution.
- 68% of the data falls within 1 standard deviation ([tex]\( \sigma \)[/tex]) of the mean ([tex]\( \mu \)[/tex]).
- 95% of the data falls within 2 standard deviations of the mean.
- 99.7% of the data falls within 3 standard deviations of the mean.

3. Within 1 Standard Deviation:
- For this specific question, we are interested in the percentage of data within 1 standard deviation of the mean.
- According to the empirical rule, approximately 68% of the data values lie within 1 standard deviation above and below the mean.

Thus, the percentage of observed values of [tex]\( x \)[/tex] that fall within 1 standard deviation of the mean in a normal distribution is 68%.