Normal Distribution & Empirical Rule

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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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What are the parameters of a Normal Distribution?

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The two 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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What is the Empirical Rule?

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

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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 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 of the mean in a Normal Distribution.

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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 of the mean in a Normal Distribution.

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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 of the mean in a Normal Distribution.

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If the mean height of men is 68.6 inches and the standard deviation is 2.8 inches, what is the height range for the middle 68% of men?

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The height range for the middle 68% of men is between 65.8 inches and 71.4 inches.

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What is the significance of the top 5% in a Normal Distribution?

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The top 5% represents the highest scores or values in a distribution, often used as a cutoff for acceptance criteria, such as in university admissions.

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In a Normal Distribution, what does it mean if a score is at the 95th percentile?

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A score at the 95th percentile means that the score is higher than 95% of the scores in the distribution.