Normal Distribution Math Example 1

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Example 1

medium
Scores on a test are normally distributed with mean μ=75\mu = 75 and standard deviation σ=10\sigma = 10. What percentage of students scored between 6565 and 8585?

Solution

  1. 1
    Identify the mean μ=75\mu = 75 and standard deviation σ=10\sigma = 10. Check whether 6565 and 8585 are within one standard deviation.
  2. 2
    Verify: 65=75−10=μ−σ65 = 75 - 10 = \mu - \sigma and 85=75+10=μ+σ85 = 75 + 10 = \mu + \sigma, so the interval [65,85][65, 85] is exactly μ±σ\mu \pm \sigma.
  3. 3
    By the empirical rule (68-95-99.7 rule), approximately 68%68\% of data in a normal distribution falls within one standard deviation of the mean.

Answer

≈68%\approx 68\%
The empirical rule provides quick approximations for normal distributions: about 68%68\% within 1σ1\sigma, 95%95\% within 2σ2\sigma, and 99.7%99.7\% within 3σ3\sigma of the mean.

About Normal Distribution

The normal distribution (also called the Gaussian distribution or bell curve) is a continuous probability distribution that is symmetric about its mean, with data tapering off equally on both sides following a precise mathematical rule.

Learn more about Normal Distribution →

More Normal Distribution Examples

Example 2 medium

Heights of adult women are normally distributed with [formula] cm and [formula] cm. What percentage

Example 3 hard

A machine fills bottles with a mean of [formula] mL and standard deviation [formula] mL (normally di

Example 4 medium

A standardized exam has scores that are approximately normal with mean [formula] and standard deviat

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