In a standard normal distribution, the mean is 0 and the standard deviation is 1. One of the important properties of this distribution is the proportion of data within certain ranges.
To determine the range within which 95% of the data lies, we look at the standard deviation intervals. For a standard normal distribution:
1. We want to find the number of standard deviations, [tex]\( x \)[/tex], such that the area to the left of [tex]\( x \)[/tex] and the area to the right of [tex]\( -x \)[/tex] sums up to 95%.
In simpler terms, we want the area between [tex]\( -x \)[/tex] and [tex]\( x \)[/tex] to be 95%. Given the symmetry of the standard normal distribution:
- The area outside this interval (i.e., less than [tex]\( -x \)[/tex] and greater than [tex]\( x \)[/tex]) will sum up to 5%.
- Each tail (left and right) will have 2.5%.
From the given table, we can identify that:
- At [tex]\( x = 2.0 \)[/tex], the area to the left of 2.0 is 0.4772.
- Since the distribution is symmetrical, the area to the left of [tex]\( -2.0 \)[/tex] is also 0.4772.
So, the area between [tex]\( -2.0 \)[/tex] and [tex]\( 2.0 \)[/tex] is:
[tex]\[ 0.4772 + 0.4772 = 0.9544 \quad \text{(or 95.44%)} \][/tex]
This closely matches our requirement for 95%.
Therefore, in a standard normal distribution, 95% of the data is within approximately 2 standard deviations of the mean.
