Effect of Larger Standard Deviation on Normal Curve Width and Height
When the standard deviation is larger, does the normal curve become wider or taller?
When the standard deviation is larger, the normal curve becomes wider and shorter (flatter). A larger standard deviation means the data is more dispersed, resulting in a wider spread along the horizontal axis and a lower peak to maintain a total area of 1 under the curve.
Conditions
- The curve is a normalized normal distribution.
- The total area under the curve is always 1.
- Comparing curves on the same coordinate scale.
Reasoning, step by step
- Recall the definition of standard deviation as a measure of dispersion.
- Understand that a larger standard deviation means data points are more spread out from the mean.
- Apply the constraint that the total area under the probability density curve must equal 1.
- Deduce that to cover a wider horizontal range while maintaining an area of 1, the peak height must decrease.
- Conclude that a larger standard deviation results in a wider and shorter curve.
Example
The video states: 'School A’s standard deviation 10 exceeds School B’s 5. Its normal curve is wider with a lower peak; School B’s is narrower with a higher peak.'
Common misconceptions
- Intuitively associating a 'larger' value with a 'taller' curve.
- Believing that the standard deviation only affects the width and not the height of the curve.
Watch the explanation
0:42 – 0:50Watch this moment ↗
Connected concepts
Explore next
- How does the magnitude of the standard deviation affect the width and height of the normal distribution curve?
- Why can't the peak heights be the same when comparing two normal distributions with different standard deviations?
- What is the mathematical formula for the peak height of a normal distribution?
Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.