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Chapter 6: Problem 7

How do the width and height of a normal distribution change when its mean remains the same but its standard deviation decreases?

Short Answer

Expert verified
When the standard deviation of a normal distribution decreases while the mean remains unchanged, the width of the distribution decreases, or in other words, the distribution becomes narrower. Simultaneously, the height of the distribution, or the peak, increases.

Step by step solution

01

Understand the Normal Distribution

A normal distribution is described by a bell shaped curve. The curve is symmetric around the mean, and is decided by two parameters: the mean (\(\mu\)) and the standard deviation (\(\sigma\)). The mean determines the center of the distribution whereas the standard deviation determines the spread around this mean. Larger the standard deviation, wider the spread and fatter the tails of the distribution. Conversely, smaller the standard deviation, narrower the spread and thinner the tails.
02

Analyze the Effect of Decreasing Standard Deviation

When the standard deviation (\(\sigma\)) decreases, the values in the distribution are less spread out. This causes reduction in the width of the distribution. The bell curve becomes narrower because values are closer to the mean.
03

Determine the Height Change

As the standard deviation decreases, the same amount of data is presented, but it is less spread out around the mean. This decrease in spread (width) needs to be compensated in order for the area under the curve to remain 1 (property of probability distribution). So, the height of the distribution increases when standard deviation decreases while keeping the mean constant.

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Most popular questions from this chapter

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