Statistics
In a normal distribution, which parameter indicates how wide the bell is?
The standard deviation.
Card 1 of 259. Answer: The standard deviation.
The standard deviation.
Card 1 of 259. Answer: The standard deviation.
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The standard deviation, σ. In a normal distribution N(μ, σ²), μ fixes the center and σ the width: the larger σ, the wider and flatter the bell. The variance σ² carries the same information on a squared scale.
σ has a geometric meaning: the curve's inflection points, where its curvature flips, sit at μ − σ and μ + σ. About 68% of values lie within one σ of the mean, 95% within two and 99.7% within three. Since the total area is 1, the peak height is 1 / (σ√(2π)), so doubling σ halves the peak. A machine filling boxes to 500 g with σ = 5 g puts about 95% of them between 490 and 510 g.
A classic notation bug: N(μ, σ²) is written with the variance, but NumPy's random.normal and SciPy's stats.norm take the standard deviation as scale, so passing 4 when you mean σ² = 4 doubles the width. A z-score, (x − μ) / σ, measures distance from the mean in units of σ.