The statistical quantities “average value” and “root-mean-square value” can be applied to any distribution. Figure P18.82 shows the scores of a class of 150 students on a 100 point quiz. (a) Find the average score for the class. (b) Find the rms score for the class. (c) Which is higher: the average score or the rms score? Why?

Total number of students is 150.

The multiplicity of each score is denoted by ${\mathrm{n}}_{\mathrm{i}}$.

The average score can be calculated as,

$$\begin{gathered} {\mathrm{S}}_{\mathrm{A}}=\left(\frac{1}{150}\right)\sum {\mathrm{n}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}} \\ \sum {\mathrm{n}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}=\left(11\right)\left(10\right)+\left(12\right)\left(20\right)+\left(24\right)\left(30\right)+\left(15\right)\left(40\right)+\left(19\right)\left(50\right)+\left(10\right)\left(60\right)+\left(12\right)\left(70\right)+\left(20\right)\left(80\right)+\left(17\right)\left(90\right)+\left(10\right)\left(100\right) \\ =110+240+720+600+950+600+840+1600+1530+1000 \\ =8190 \end{gathered}$$

Thus, the average score is,

$$\begin{gathered} {\mathrm{S}}_{\mathrm{A}}=\frac{8190}{150} \\ =54.6 \end{gathered}$$

Hence, the average score of the class is .

The multiplicity of each score is denoted by ${\mathrm{n}}_{\mathrm{i}}$.

The average score can be calculated as,

$$\begin{gathered} S_{rms}={\left[\left(\frac{1}{150}\right)\sum n_i{x}_i^2\right]}^{\frac{1}{2}} \\ \sum n_i{x}_i=\left(11\right)\left(100\right)+\left(12\right)\left(400\right)+\left(24\right)\left(900\right)+\left(15\right)\left(1600\right)+\left(19\right)\left(2500\right)+\left(10\right)\left(3600\right)+\left(12\right)\left(4900\right)+\left(20\right)\left(6400\right)+\left(17\right)\left(8100\right)+\left(10\right)\left(10000\right) \\ =1100+4800+21600+24000+47500+36000+58800+128000+137700+100000 \\ =559500 \end{gathered}$$

Thus, the rms score is,

role="math" localid="1664364669179" $$\begin{gathered} {\mathrm{S}}_{\mathrm{A}}={\left[\frac{559500}{150}\right]}^{\frac{1}{2}} \\ ={\left[3730\right]}^{\frac{1}{2}} \\ =61.07 \end{gathered}$$

Hence, the RMS score of the class is $61.07$.

The RMS score is clearly higher than the average score because there is more weight included in the RMS calculation.