### 標準差的應用

 (a) Standard score For a set of data with mean $\overline{{x}}$ and standard deviation $\sigma$, the standard score ${z}$ of a given datum ${x}$ is given by ${z}=\dfrac{{{x}-\overline{{x}}}}{\sigma}$ . (b) Normal distribution Characteristics of a normal curve: ${1}.$ It is bell-shaped. ${2}.$ It has reflectional symmetry about ${x}=\overline{{x}}.$ ${3}.$ The mean, median and mode are all equal to $\overline{{x}}.$

 Example ${1}$ Based on the information given in the table below, in which test does Peter perform better? Briefly explain your answer.

Test ${1}$Test ${2}$
Peter's marks${66}$${75}$
Mean of the class${60}$${70}$
Standard deviation of the class${6}$${8}$

 Solution For test ${1}$ , ${z}=\dfrac{{{66}-{60}}}{{6}}={1}$ For test ${2}$ , ${z}=\dfrac{{{75}-{70}}}{{8}}={0.625}$ ∵   standard score of test ${1}$ $>$ standard score of test${2}$ ∴   Peter performs better in test ${1}$ .

${x}$${99.7}\%$${95}\%$${68}\%$$\overline{{x}}-{3}\sigma$$\overline{{x}}-{2}\sigma$$\overline{{x}}-\sigma$$\overline{{x}}$$\overline{{x}}+\sigma$$\overline{{x}}+{2}\sigma$$\overline{{x}}+{3}\sigma$

    

 Example ${2}$ The heights of ${120}$ children in a kindergarten are normally distributed with a mean of ${85}$ $\text{cm}$ and a standard deviation of ${5.5}$ $\text{cm}$.

 (a) Find the percentage of children with heights between ${79.5}$ $\text{cm}$ and ${90.5}$ $\text{cm}$. (b) Find the number of children who are taller than ${74}$ $\text{cm}$.

Solution

 (a) Let the mean and standard deviation be $\mu$ and $\sigma$ respectively. $\mu={85}$ $\text{cm}$ and $\sigma={5.5}$ $\text{cm}$ ${79.5}$ $\text{cm}={85}$ $\text{cm}-{5.5}$ $\text{cm}=\mu-\sigma$ ${90.5}$ $\text{cm}={85}$ $\text{cm}+{5.5}$ $\text{cm}=\mu+\sigma$ ∴   The required percentage $={68}\%$ (b) ${74}$ $\text{cm}={85}$ $\text{cm}-{2}{\left({5.5}\right)}$ $\text{cm}=\mu-{2}\sigma$ The percentage of children taller than${74}$ $\text{cm}$ $={\left(\dfrac{{95}}{{2}}+{50}\right)}\%={97.5}\%$ ∴   The required number of children  $={120}\times{97.5}\%$ $={117}$

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