Normal Distributions

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What is a normal distribution?

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A normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve, where most of the observations cluster around the central peak (mean), and probabilities for values further away from the mean taper off equally in both directions.

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What are the key parameters of a normal distribution?

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The key parameters of a normal distribution are the mean (μ), which indicates the center of the distribution, and the standard deviation (σ), which measures the spread or dispersion of the distribution.

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CCSS.6.SP.A.3

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What is the empirical rule (68-95-99.7 rule)?

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The empirical rule states that for a normal distribution: about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.

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CCSS.HSS.ID.A.4

Front

How do you calculate a score that lies a certain number of standard deviations from the mean?

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To calculate a score that lies 'z' standard deviations from the mean, use the formula: Score = Mean + (z * Standard Deviation).

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What percent of a normal distribution lies within ±1 standard deviation from the mean?

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Approximately 68% of a normal distribution lies within ±1 standard deviation from the mean.

Tags

CCSS.HSS.ID.A.4

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What percent of a normal distribution lies outside of ±2 standard deviations?

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Approximately 5% of a normal distribution lies outside of ±2 standard deviations.

Tags

CCSS.HSS.ID.A.4

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What is the significance of the standard deviation in a normal distribution?

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The standard deviation indicates how spread out the values are around the mean; a smaller standard deviation means the values are closer to the mean, while a larger standard deviation indicates more spread.

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If a dataset has a mean of 75 and a standard deviation of 15, what score is at -2 standard deviations?

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The score at -2 standard deviations is 45 (75 - 2*15).

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In a normal distribution, what is the probability of a value falling between the mean and one standard deviation above the mean?

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The probability of a value falling between the mean and one standard deviation above the mean is approximately 34%.

Tags

CCSS.HSS.ID.A.4

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How do you find the area under the normal curve for a specific range?

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To find the area under the normal curve for a specific range, you can use the z-score formula to convert the values to standard scores and then refer to the standard normal distribution table.

Tags

CCSS.HSS.ID.A.4