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Correct Answer:
2%.
a normal distribution, also known as a gaussian distribution, is a bell-shaped curve that is symmetric about the mean. it is a fundamental concept in statistics and is used to represent real-valued random variables with unknown distributions. the properties of a normal curve are determined by its mean and standard deviation. the mean indicates the center of the distribution, and the standard deviation shows the spread of the data around the mean.
according to the empirical rule, also known as the 68-95-99.7 rule, approximately 68% of the data within a normal distribution falls within one standard deviation (σ) of the mean (μ). about 95% of the data falls within two standard deviations (2σ), and about 99.7% falls within three standard deviations (3σ). this rule helps to understand the distribution of data in a normal curve without having detailed knowledge of the actual distribution.
to find the percentage of scores that are more than two standard deviations away from the mean, we can use the properties outlined by the empirical rule. since 95% of the data lies within two standard deviations from the mean, the remaining 5% of the data lies outside this range. this 5% is split equally between the two tails of the distribution because the normal curve is symmetric. therefore, 2.5% of the data lies beyond two standard deviations on the right side of the mean, and 2.5% lies beyond two standard deviations on the left side of the mean.
when asked what percentage of scores should be more than two standard deviations from the mean, we are considering both tails beyond the ±2σ range. thus, the total percentage is 2.5% + 2.5% = 5%. however, if the question specifically asks how much of the data is beyond two standard deviations in one direction (either above or below), the answer is 2.5%. this specificity needs to be clear in the context of the question to provide the correct response.
in the context provided, where it asks for scores more than two standard deviations from the mean without specifying a direction, the correct answer would typically be 5%, representing both tails. however, if the question or context implies considering only one tail (either above or below), then the correct answer would be 2.5%. it's crucial to understand the question's context to provide the correct percentage.
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