A normal score is the expected z-score of a data value, assuming the distribution of the random variable is normal. Is this statement true or false?
The correct answer and explanation is:
Correct Answer: True
Explanation (Approximately 300 words):
The statement “A normal score is the expected z-score of a data value, assuming the distribution of the random variable is normal” is true.
To understand why, let’s break it down:
1. Z-Score Basics
A z-score measures how many standard deviations a data point is from the mean of a distribution. It is calculated using the formula: z=x−μσz = \frac{x – \mu}{\sigma}
Where:
- xx is the data value,
- μ\mu is the mean,
- σ\sigma is the standard deviation.
Z-scores help standardize values from a normal distribution so that they can be compared or analyzed more easily.
2. Normal Distribution
In a normal distribution, data values are symmetrically distributed around the mean, forming the classic “bell curve.” About 68% of data lies within 1 standard deviation, 95% within 2, and 99.7% within 3.
3. Normal Score
A normal score refers to the expected value of the z-score associated with a data value when the underlying data is assumed to follow a normal distribution. In practical statistics, normal scores are especially useful in normal probability plots and quantile-quantile (Q-Q) plots to assess normality.
In these contexts, each data value is ranked, and a corresponding expected z-score (based on its position or percentile in a standard normal distribution) is calculated. These expected z-scores are what we refer to as normal scores.
4. Conclusion
Since a normal score represents the expected z-score of a data value under the assumption that the data follows a normal distribution, the original statement is accurate.
Therefore, the statement is true. Normal scores are indeed the expected z-scores when data is assumed to be normally distributed.