Question

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?

Short answer

1. The average score of the class is 54.6.
2. The RMS score of the class is 61.07.
3. The RMS score is higher because there is more weight included in the RMS calculation.

Step-by-step solution

Identification of the given data.

Total number of students is 150.

(a) Determine the average score of the class.

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 .

(b) Determine the rms score of the class.

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 $61.07$.

(c) Which is higher: the average score or the rms score? Why?                     

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