Standard Deviation

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What is Standard Deviation?

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Standard Deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

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How is Standard Deviation calculated?

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Standard Deviation is calculated using the formula: \( \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \), where \( \sigma \) is the standard deviation, \( N \) is the number of observations, \( x_i \) is each individual observation, and \( \mu \) is the mean of the observations.

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What does a Standard Deviation of 0 indicate?

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A Standard Deviation of 0 indicates that all values in the dataset are identical and there is no variation.

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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: approximately 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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If a dataset has a mean of 100 and a standard deviation of 15, what percentage of data falls between 85 and 115?

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According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean (100 ± 15).

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What is the Z-score?

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The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated as: \( Z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

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How do you interpret a Z-score of -2?

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A Z-score of -2 indicates that the value is 2 standard deviations below the mean.

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What is the significance of a negative Standard Deviation?

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Standard Deviation cannot be negative. A negative value indicates an error in calculation or interpretation.

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How does increasing the standard deviation affect the shape of a normal distribution?

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Increasing the standard deviation makes the normal distribution wider and flatter, indicating more variability in the data.

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What percentage of data falls below a Z-score of 1 in a normal distribution?

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Approximately 84.13% of the data falls below a Z-score of 1 in a normal distribution.