Question

For a normal (Gaussian) distribution, decreasing the spread and increasing the height would lead to a:

• A. smaller value of standard deviation
• B. higher value of standard deviation
• C. smaller value of mean
• D. higher value of mean

Hint:

For a normal distribution, the standard deviation determines the spread of the data. A smaller standard deviation means less variability, a taller curve, and a narrower spread around the mean.

Answer 1

The Correct Option is A

Step 1: Understanding Gaussian Distribution.
In a normal (Gaussian) distribution, the shape of the curve is determined by the mean (central location) and the standard deviation (spread or width) of the distribution. The larger the standard deviation, the wider and flatter the distribution curve will be. Conversely, the smaller the standard deviation, the narrower and taller the curve becomes.
Step 2: Impact of changing spread and height.
Decreasing the spread refers to reducing the standard deviation, which leads to a more concentrated distribution around the mean.
Increasing the height of the curve corresponds to a taller distribution, which directly happens when the standard deviation is reduced, as the probability density increases near the mean.
Step 3: Conclusion.
When you decrease the spread (standard deviation) of a normal distribution, the curve becomes narrower, and its height increases. This is because the area under the curve remains constant (total probability = 1), so a narrower distribution leads to a higher peak. Therefore, decreasing the spread results in a smaller value of standard deviation.