Question

mrs. durand, the science teacher, informed her students of their scores on mondays test. score number of students 22 4 30 3 43 6 68 1 79 3 85 3 x is the score that a randomly chosen student scored. what is the standard deviation of x? round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the total number of students

$n=4 + 3+6 + 1+3+3=20$

Step2: Calculate the mean $\mu$

$\mu=\frac{22\times4 + 30\times3+43\times6+68\times1+79\times3+85\times3}{20}=\frac{88+90 + 258+68+237+255}{20}=\frac{996}{20}=49.8$

Step3: Calculate the variance $\sigma^{2}$

\[

$$\begin{align*} \sigma^{2}&=\frac{4\times(22 - 49.8)^{2}+3\times(30 - 49.8)^{2}+6\times(43 - 49.8)^{2}+1\times(68 - 49.8)^{2}+3\times(79 - 49.8)^{2}+3\times(85 - 49.8)^{2}}{20}\\ &=\frac{4\times(- 27.8)^{2}+3\times(-19.8)^{2}+6\times(-6.8)^{2}+1\times(18.2)^{2}+3\times(29.2)^{2}+3\times(35.2)^{2}}{20}\\ &=\frac{4\times772.84+3\times392.04+6\times46.24+1\times331.24+3\times852.64+3\times1239.04}{20}\\ &=\frac{3091.36+1176.12+277.44+331.24+2557.92+3717.12}{20}\\ &=\frac{11151.2}{20}=557.56 \end{align*}$$

\]

Step4: Calculate the standard - deviation $\sigma$

$\sigma=\sqrt{557.56}\approx23.61$

Answer:

$23.61$