# What is the minimum score score for each percentage

by dfarris   Last Updated September 11, 2019 17:20 PM

Suppose the people living in a city have a mean score of $$53$$ and a standard deviation of $$4$$ on a measure of concern about the environment. Assume that these concern scores are normally distributed. Using the ​$$50\%$$-$$​34\%$$-$$​14\%$$ ​figures, what is the minimum score a person has to have to be in the top:

​ (a) $$2\%$$,​ (b) $$16\%$$,​ (c) $$50\%$$,​ (d) $$84\%$$, and​ (e) $$98\%$$?

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$$50$$-$$34$$-$$14$$ is the empirical rule for calculating percentiles in a normal distribution. This means that $$50\%$$ of the data can be found on either side of the mean, $$34\%$$ of the data is found within one standard deviation on either side of the mean, and $$14\%$$ of the data can be found between one and two standard deviations on either side of the mean.

Algebraically, for a distribution $$\mathcal{N}(\mu,\sigma^2)$$:

1. $$50\%$$ of the data can be found within the interval $$(\mu-\sigma$$, $$\mu+\sigma)$$;
2. $$34\%$$ of the data can be found within the intervals $$(\mu$$, $$\mu+\sigma)$$ or $$(\mu-\sigma$$, $$\mu)$$; and
3. $$14\%$$ of the data can be found within the intervals $$(\mu+\sigma$$, $$\mu+2\sigma)$$ or $$(\mu-2\sigma$$, $$\mu-\sigma)$$.

For your question, you have the distribution $$\mathcal{N}(53,4^2)$$. Use the above to figure out your answers, and share your solution.

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