Problem

The grades for 20 students on the most recent exam are given in the data table below. Round your answers to 2 decimal places as needed \begin{tabular}{|l|l|l|l|l|} \hline 42 & 57 & 73 & 72 & 52 \\ \hline 75 & 88 & 92 & 79 & 41 \\ \hline 46 & 41 & 49 & 54 & 92 \\ \hline 85 & 58 & 76 & 82 & 83 \\ \hline \end{tabular} Mean $=$ Median $=$ Standard deviation $=$ Minimum score $=$ Highest score $=$

Solution

Step 1 :First, list all the scores in ascending order: 41, 41, 42, 46, 49, 52, 54, 57, 58, 72, 73, 75, 76, 79, 82, 83, 85, 88, 92, 92

Step 2 :Calculate the mean by summing all the scores and dividing by the number of scores: \(\frac{41+41+42+46+49+52+54+57+58+72+73+75+76+79+82+83+85+88+92+92}{20} = 68.35\)

Step 3 :Find the median by taking the average of the 10th and 11th scores: \(\frac{58 + 72}{2} = 65\)

Step 4 :Calculate the squared differences from the mean: \((41-68.35)^2, (41-68.35)^2, (42-68.35)^2, ..., (92-68.35)^2\)

Step 5 :Calculate the average of these squared differences to get the variance: \(\frac{(41-68.35)^2 + (41-68.35)^2 + (42-68.35)^2 + ... + (92-68.35)^2}{20} = 291.1775\)

Step 6 :Take the square root of the variance to get the standard deviation: \(\sqrt{291.1775} = 17.06\)

Step 7 :The minimum score is the lowest score: \(\boxed{41}\)

Step 8 :The highest score is the highest score: \(\boxed{92}\)

Step 9 :So, the mean is \(\boxed{68.35}\), the median is \(\boxed{65}\), the standard deviation is \(\boxed{17.06}\), the minimum score is \(\boxed{41}\), and the highest score is \(\boxed{92}\)

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