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A teacher marks five short tests out of 10…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A teacher marks five short tests out of . She remembers that the mean of the five marks was and the variance was . She also remembers that three of the marks were , and , but the other two marks were written on a sheet that she lost. Every mark is a whole number. The teacher wants to rebuild the full list before she reports the results to the head of department. She decides to use the total of the marks and the total of the squares of the marks. These two totals follow from the mean and the variance, and they are enough to find the two missing marks. Find the sum of the two missing marks, the sum of their squares, and the two marks.
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The worked solution
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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The total. . The known marks add to , so the two missing marks add to .
Step 2. The total of the squares. , so . The known marks give , so .
Step 3. The product. .
Step 4. The two marks. They are the roots of , which is . So the marks are and .
Answer: the missing marks add to , their squares add to , and the marks are and .
Check. The full list has mean and squared deviations , which add to . Then the variance is .
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