Normal distribution and Empirical Rule

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What is a normal distribution?

Back

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

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What is the Empirical Rule?

Back

The Empirical Rule states that for a normal distribution, approximately 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

What is a z-score?

Back

A z-score indicates how many standard deviations an element is from the mean. It is calculated as z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

Front

What percentage of data lies within one standard deviation of the mean in a normal distribution?

Back

Approximately 68% of the data.

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CCSS.HSS.ID.A.4

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What percentage of data lies within two standard deviations of the mean in a normal distribution?

Back

Approximately 95% of the data.

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CCSS.HSS.ID.A.4

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What percentage of data lies beyond three standard deviations from the mean in a normal distribution?

Back

Approximately 0.3% of the data.

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CCSS.HSS.ID.A.4

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How do you calculate the z-score for a given value?

Back

Z-score is calculated using the formula: z = (X - μ) / σ.

Front

If the mean of a dataset is 50 and the standard deviation is 10, what is the z-score for a value of 70?

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The z-score is 2.0.

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What does a z-score of 0 indicate?

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A z-score of 0 indicates that the value is exactly at the mean.

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In a normal distribution, what does it mean if a z-score is positive?

Back

A positive z-score indicates that the value is above the mean.

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CCSS.HSS.ID.A.4