Proportion of Total Area Under the Normal Curve within Two Standard Deviations

B

95% of the area under the curve lies within plus and minus one standard deviation of the mean.

The proportion of the total area under the normal curve within plus and minus two standard deviations is known as the empirical rule or the 68-95-99.7 rule. According to this rule, in a normal distribution:

• Approximately 68% of the total area under the curve falls within plus and minus one standard deviation from the mean.
• Approximately 95% of the total area under the curve falls within plus and minus two standard deviations from the mean.
• Approximately 99.7% of the total area under the curve falls within plus and minus three standard deviations from the mean.

In the given question, we are asked about the proportion within plus and minus two standard deviations. Therefore, the correct answer is option B, 95%.

This result can be visualized on a standard normal distribution curve, also known as a z-score curve. The curve is symmetric and bell-shaped, with the mean located at the center. The area under the curve represents the probability of observing a particular range of values.

By knowing that approximately 95% of the data falls within plus and minus two standard deviations, we can infer that the remaining 5% is distributed in the tails beyond that range.