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The solution to the first part is here – the other parts can be done in the same way.

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For {exam1} we have the mean is:

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\$\\simplify[]{({r0[0]} + {r0[1]} + {r0[2]} + {r0[3]} + {r0[4]} + {r0[5]} + {r0[6]} + {r0[7]} + {r0[8]} + {r0[9]}) / {n} = {mean1}}\$

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The sample variance is given by the formula:

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\$\\textrm{Sample Variance} = \\frac{1}{n-1}\\left(\\sum_{j=1}^{n}x_j^2 -n\\mu^2\\right)\$

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where the $x_j$ are the exam scores for {exam1}, $n=\\var{n}$ the number of students and $\\mu=\\var{mean1}$ the sample mean.

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We find that
\$\\begin{eqnarray*}\\sum_{j=1}^{n}x_j^2 &=& \\simplify[]{({r0[0]}^2 + {r0[1]}^2 + {r0[2]}^2 + {r0[3]}^2 + {r0[4]}^2 + {r0[5]}^2 + {r0[6]}^2 + {r0[7]}^2 + {r0[8]}^2 + {r0[9]}^2)}\\\\ \n \n &=& \\var{ssq1}\\\\\n \n \\\\\n \n \\\\\n \n n\\mu^2 &=&\\var{n} \\times\\var{mean1}^2\\\\\n \n &=& \\var{n*mean1^2}\n \n \\end{eqnarray*}\n \n \$
Hence substituting these values into the formula we find that:

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\$\\begin{eqnarray*}\n \n \\textrm{Sample Variance} &=& \\frac{1}{\\var{n-1}}\\left(\\var{ssq1}-\\var{n*mean1^2}\\right)\\\\\n \n &=& \\var{var1}\n \n \\end{eqnarray*}\n \n \$ to 3 decimal places.

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The Sample Standard Deviation is then the square root of the Sample Variance i.e.

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Sample Standard Deviation = $\\sqrt{\\var{var1}} = \\var{stdev1}$ to one decimal place.

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#### {exam1}

\n \n \n \n

Sample Standard Deviation = [[0]] (to one decimal place)

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#### {exam2}

\n \n \n \n

Sample Standard Deviation = [[0]] (to one decimal place)

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#### {total}

\n \n \n \n

Sample Standard Deviation = [[0]] (to one decimal place)

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The following table gives the examination marks in {exam1} and in {exam2} and their total for a sample of $n=\\var{n}$ students.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 {exam1} Mean = $\\var{mean1}$ {exam2} Mean = $\\var{mean2}$ {total} Mean = $\\var{overallmean}$ $\\var{r0[0]}$ $\\var{r0[1]}$ $\\var{r0[2]}$ $\\var{r0[3]}$ $\\var{r0[4]}$ $\\var{r0[5]}$ $\\var{r0[6]}$ $\\var{r0[7]}$ $\\var{r0[8]}$ $\\var{r0[9]}$ $\\var{r1[0]}$ $\\var{r1[1]}$ $\\var{r1[2]}$ $\\var{r1[3]}$ $\\var{r1[4]}$ $\\var{r1[5]}$ $\\var{r1[6]}$ $\\var{r1[7]}$ $\\var{r1[8]}$ $\\var{r1[9]}$ $\\var{sscores[0]}$ $\\var{sscores[1]}$ $\\var{sscores[2]}$ $\\var{sscores[3]}$ $\\var{sscores[4]}$ $\\var{sscores[5]}$ $\\var{sscores[6]}$ $\\var{sscores[7]}$ $\\var{sscores[8]}$ $\\var{sscores[9]}$
\n

Find the sample standard deviation for each of {exam1}, {exam2} and Total Score.

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Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.

", "licence": "Creative Commons Attribution 4.0 International"}, "name": "Harry's copy of STAT7008 Sample standard deviation", "type": "question", "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}