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Questions and Answers
What is the purpose of a Z-score?
What does a Z-score of 0 indicate?
What is the formula to calculate a Z-score?
What is a characteristic of Z-scores?
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What is an example of a use of Z-scores?
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What does a positive Z-score indicate?
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Study Notes
What is a Z-score?
A Z-score, also known as a standard score, is a measure of how many standard deviations an observation is away from the mean of a normal distribution.
Formula:
Z = (X - μ) / σ
Where:
- Z is the Z-score
- X is the observation or data point
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Interpretation:
- A Z-score of 0 indicates that the observation is equal to the mean.
- A positive Z-score indicates that the observation is above the mean, and the number of standard deviations it is above the mean.
- A negative Z-score indicates that the observation is below the mean, and the number of standard deviations it is below the mean.
Example:
If a student scored 120 on a test with a mean of 100 and a standard deviation of 10, their Z-score would be:
Z = (120 - 100) / 10 = 2
This means the student's score is 2 standard deviations above the mean.
Uses of Z-scores:
- Comparing observations from different normal distributions
- Identifying outliers or unusual data points
- Standardizing data to a common scale
- Calculating percentiles and probabilities
Key Properties:
- Z-scores are dimensionless, meaning they have no units
- Z-scores are sensitive to changes in the mean and standard deviation
- Z-scores can be used to compare data from different distributions
What is a Z-score?
- A Z-score, also known as a standard score, measures how many standard deviations an observation is away from the mean of a normal distribution.
Formula:
- Z = (X - μ) / σ
- Where:
- Z is the Z-score
- X is the observation or data point
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Interpretation:
- A Z-score of 0 indicates that the observation is equal to the mean.
- A positive Z-score indicates that the observation is above the mean, and the number of standard deviations it is above the mean.
- A negative Z-score indicates that the observation is below the mean, and the number of standard deviations it is below the mean.
Example:
- If a student scored 120 on a test with a mean of 100 and a standard deviation of 10, their Z-score would be 2, indicating the score is 2 standard deviations above the mean.
Uses of Z-scores:
- Comparing observations from different normal distributions
- Identifying outliers or unusual data points
- Standardizing data to a common scale
- Calculating percentiles and probabilities
Key Properties:
- Z-scores are dimensionless, meaning they have no units
- Z-scores are sensitive to changes in the mean and standard deviation
- Z-scores can be used to compare data from different distributions
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Description
Learn about Z-scores, also known as standard scores, and how to calculate them using the formula. Understand the interpretation of Z-scores and their significance in normal distributions.
