Why Normal Distributions with Different Standard Deviations Cannot Have Equal Peak Heights

Conditions

• The curves are normalized normal distributions.
• The total area under each curve is 1.
• Comparing curves on the same coordinate scale.

Reasoning, step by step

1. Recall the formula for the peak height of a normal distribution: $f(\mu)=1/(\sigma\sqrt{2\pi})$.
2. Observe that the peak height depends inversely on the standard deviation $\sigma$.
3. Understand that the total area under the curve is constrained to be 1.
4. Deduce that if $\sigma$ is larger, the horizontal spread is wider, so the height must be lower to maintain an area of 1.
5. Conclude that different standard deviations result in different peak heights.

Example

The video states: 'The editorial peak formula is $f(\mu)=1/(\sigma\sqrt{2\pi})$: the entire product of the standard deviation and the square root of 2π is in the denominator. Diagram B’s equal peaks fail this comparison...'

Common misconceptions

• Believing that area 1 alone determines the width and peak height for arbitrary densities.
• Thinking that the peak height is independent of the standard deviation.

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Connected concepts

Explore next

• What is the mathematical formula for the peak height of a normal distribution?
• How does the magnitude of the standard deviation affect the width and height of the normal distribution curve?
• What is the geometric meaning of the area under the normal distribution curve?