IB Math Applications 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 the mean, median, and mode are all equal. It is defined by its mean (μ) and standard deviation (σ).

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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 (σ) of the mean (μ) 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 (2σ) of the mean (μ) 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 (3σ) of the mean (μ) in a normal distribution.

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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 is the significance of the z-score in a normal distribution?

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The z-score indicates how many standard deviations a data point is from the mean. It helps in determining the probability of a value occurring within a normal distribution.

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If the mean waist size is 29 inches and the standard deviation is 1.4 inches, what is the z-score for a waist size of 31 inches?

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The z-score for a waist size of 31 inches is calculated as: z = (31 - 29) / 1.4 = 1.43.

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What is the empirical rule in statistics?

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The empirical rule states that for a normal distribution: 68% of data falls within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.

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How do you find the area under the normal curve for a given z-score?

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The area under the normal curve for a given z-score can be found using z-tables or statistical software, which provide the cumulative probability associated with that z-score.

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What is the mean and standard deviation of the heights of male students given as 1.7 m and 0.2 m?

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The mean height is 1.7 m and the standard deviation is 0.2 m.