Which statement is true about the two-parameter normal distribution?

Which statement is true about the two-parameter normal distribution?

• A. It is determined by variance and skewness
• B. It is determined by mean and variance
• C. It is determined by mean and range
• D. It is determined by mean and standard deviation

Answer: (D)

A normal distribution is completely specified by where it sits and how spread out it is. In the two-parameter form, those two pieces are the mean (which centers the distribution) and the standard deviation (which controls its spread). The standard form shows this clearly: f(x) = (1/(σ√(2π))) exp(-(x−μ)²/(2σ²)). Here μ is the center of the bell curve, and σ is the scale that sets how wide the curve is. Because the standard deviation has the same units as the data, it’s the natural way to describe spread. The variance is just σ², so you could describe the spread with mean and variance, but the conventional two-parameter specification uses mean and standard deviation. Skewness is not a parameter you set for a normal distribution; it is zero for all normal distributions. The range is not a defining parameter because a normal distribution extends to infinity in both directions, making range an uninformative descriptor for its shape. So, the statement that the two-parameter normal distribution is determined by the mean and the standard deviation is the correct way to specify it.

A normal distribution is completely specified by where it sits and how spread out it is. In the two-parameter form, those two pieces are the mean (which centers the distribution) and the standard deviation (which controls its spread).

The standard form shows this clearly: f(x) = (1/(σ√(2π))) exp(-(x−μ)²/(2σ²)). Here μ is the center of the bell curve, and σ is the scale that sets how wide the curve is. Because the standard deviation has the same units as the data, it’s the natural way to describe spread. The variance is just σ², so you could describe the spread with mean and variance, but the conventional two-parameter specification uses mean and standard deviation.

Skewness is not a parameter you set for a normal distribution; it is zero for all normal distributions. The range is not a defining parameter because a normal distribution extends to infinity in both directions, making range an uninformative descriptor for its shape.

So, the statement that the two-parameter normal distribution is determined by the mean and the standard deviation is the correct way to specify it.