Which of the following is a key property of the normal distribution?

Question

BlackTom AI · Accepted Answer

First, restating the question and options to set the stage: the question asks which statement describes a key property of the normal distribution. The options are:
- a. It is always skewed to the right
- b. The mean, median, and mode are all equal
- c. The total area under the curve is always greater than 1
- d. It only applies to discrete data

Now, evaluating each option:
- Option a: It is always skewed to the right. This is incorrect. A defining feature of the normal distribution is its symmetry around the center; it is not skewed (neither right nor left). Skewness would violate the bell-shaped symmetry.
- Option b: The mean, median, and mode are all equal. This is a true property of the normal distribution. In a perfectly symmetric, unimodal distribution like the normal, these three measures of central tendency coincide at the center of the distribution, reflecting its symmetry.
- Option c: The total area under the curve is always greater than 1. This is false. For any probability density function, the total area under the curve is exactly 1, representing total probability. The normal distribution integrates to 1 over the entire real line.
- Option d: It only applies to discrete data. This is incorrect because the normal distribution is a continuous probability distribution, not restricted to discrete data. It models continuous outcomes such as heights, test scores, and measurement errors.

Putting these together, the reasoning shows that the statement describing a key property—symmetry with mean, median, and mode all equal at the center—is the one that aligns with the normal distribution’s defining characteristics, while the other options describe incorrect or misleading properties.

Statistics and Probability

Which of the following is a key property of the normal distribution?

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