Questions: Normal Distribution

Question 1

A normal distribution has μ = 100 and σ = 15. Approximately what percentage of values fall between 85 and 115?

• A. 50%
• B. 68%
• C. 95%
• D. 99.7%

Answer: (B)

85 and 115 are each exactly one standard deviation from the mean (μ ± σ = 100 ± 15). The empirical rule states that approximately 68% of values in a normal distribution fall within one standard deviation of the mean. 95% falls within two, and 99.7% within three.

Question 2

Any distribution that looks bell-shaped is a normal distribution.

• T. True
• F. False

Answer: False

Many distributions are bell-shaped but not technically normal. For example, t-distributions are symmetric and bell-shaped but have heavier tails than the normal. A distribution is normal only if it follows the specific exponential formula f(x) = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²)). Visual resemblance to a bell curve is not sufficient.

Question 3

How do the mean and standard deviation together completely determine the shape of a normal distribution?

Model answer: The mean sets the center (location) of the bell curve, while the standard deviation sets its width — larger σ produces a flatter, wider curve and smaller σ produces a taller, narrower one. No other parameters are needed.

Unlike many distributions that require additional shape parameters, the normal distribution is fully specified by just μ and σ. This is one of the reasons it is mathematically tractable and widely used — every normal distribution is just a shifted and scaled version of the standard normal (μ=0, σ=1).