Normal distribution

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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 and probabilities for values further away from the mean taper off equally in both directions.

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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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What are the mean and standard deviation in a Normal Distribution?

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The mean is the average of all data points, and the standard deviation measures the dispersion or spread of the data points around the mean.

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What percentage of data falls within one standard deviation of the mean in a Normal Distribution?

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Approximately 68% of the data falls within one standard deviation (±1σ) of the mean.

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CCSS.HSS.ID.A.4

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What percentage of data falls within two standard deviations of the mean in a Normal Distribution?

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Approximately 95% of the data falls within two standard deviations (±2σ) of the mean.

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CCSS.HSS.ID.A.4

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What percentage of data falls within three standard deviations of the mean in a Normal Distribution?

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Approximately 99.7% of the data falls within three standard deviations (±3σ) of the mean.

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CCSS.HSS.ID.A.4

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How do you calculate the z-score?

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The z-score is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

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What does a z-score represent?

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A z-score represents the number of standard deviations a data point is from the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean.

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What is the empirical rule in statistics?

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

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CCSS.HSS.ID.A.4

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How do you find the area under the curve in a Normal Distribution?

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The area under the curve can be found using z-scores and standard Normal distribution tables or software to determine probabilities.

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CCSS.HSS.ID.A.4

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What is the significance of percentiles in a Normal Distribution?

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Percentiles indicate the relative standing of a value within a distribution, showing the percentage of data points that fall below that value.

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CCSS.HSS.ID.A.4