Questions and Answers
What is the purpose of a z-score in a normal distribution?
To compare observations from different normal distributions
What does a z-score of 2 represent?
The observation is 2 units above the mean
What is the formula for calculating the standard deviation of a normal distribution?
σ = √(Σ(xi - μ)^2 / (n - 1))
What percentage of data points fall within 2 standard deviations of the mean in a normal distribution?
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What is the interpretation of a z-score of 0?
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What is the primary role of the standard deviation in a normal distribution?
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What happens to the standard deviation if all data points are equal to the mean?
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Study Notes
Normal Distributions
Z-scores
- A z-score is a measure of how many standard deviations an observation is away from the mean
- Formula: z = (x - μ) / σ
- x: observation value
- μ: mean of the distribution
- σ: standard deviation of the distribution
- Interpretation:
- z > 0: observation is above the mean
- z < 0: observation is below the mean
- z = 0: observation is equal to the mean
- Z-scores allow for comparison of observations from different normal distributions
Standard Deviation (σ)
- Measures the spread or dispersion of a normal distribution
- Represents how much individual data points deviate from the mean
- Formula: σ = √(Σ(xi - μ)^2 / (n - 1))
- xi: individual data points
- μ: mean of the distribution
- n: sample size
- Properties:
- σ = 0: all data points are equal to the mean (no variation)
- σ > 0: data points are spread out from the mean
- 68-95-99.7 Rule:
- Approximately 68% of data points fall within 1 standard deviation of the mean
- Approximately 95% of data points fall within 2 standard deviations of the mean
- Approximately 99.7% of data points fall within 3 standard deviations of the mean
Normal Distributions
Z-scores
- Z-scores measure the number of standard deviations an observation is away from the mean
- Formula: z = (x - μ) / σ
- Z-score interpretation:
- Positive z-score: observation is above the mean
- Negative z-score: observation is below the mean
- Zero z-score: observation is equal to the mean
- Z-scores enable comparison of observations from different normal distributions
Standard Deviation (σ)
- Standard deviation measures the spread or dispersion of a normal distribution
- Represents how much individual data points deviate from the mean
- Formula: σ = √(Σ(xi - μ)^2 / (n - 1))
- Standard deviation properties:
- Zero standard deviation: all data points are equal to the mean (no variation)
- Positive standard deviation: data points are spread out from the mean
- 68-95-99.7 Rule:
- Approximately 68% of data points fall within 1 standard deviation of the mean
- Approximately 95% of data points fall within 2 standard deviations of the mean
- Approximately 99.7% of data points fall within 3 standard deviations of the mean
