Calculate the standard deviation

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Charlie Earle 6 years, 3 months ago

Created this as a copy of Standard deviation.

Name	Status	Author	Last Modified
Sample standard deviation	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Sample standard deviation	draft	Nigel Atkins	01/03/2017 13:41
Vicky's copy of Sample standard deviation	draft	Vicky Hall	12/06/2017 15:08
Shortened Sample standard deviation	draft	Catherine Palmer	20/09/2023 10:46
STAT7008 Sample standard deviation	draft	Catherine Palmer	16/10/2018 21:44
STAT7008 Sample standard deviation	draft	Catherine Palmer	06/11/2018 21:33
Harry's copy of STAT7008 Sample standard deviation	draft	Harry Flynn	15/11/2018 10:01
Standard deviation	draft	Paul Finley	16/01/2019 13:57
Calculate the standard deviation	draft	Charlie Earle	28/01/2019 15:25
Practice Quiz Calculate the standard deviation	draft	Charlie Earle	28/01/2019 15:26
Charlie's copy of Calculate the standard deviation	draft	Charlie Earle	23/01/2019 16:47
Vicky's copy of Standard deviation	Ready to use	Vicky Hall	10/12/2021 19:02
STAT8013 Sample standard deviation	draft	Catherine Palmer	20/02/2023 11:17

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Question statement

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

	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10
{exam1}	$\var{r0[0]}$	$\var{r0[1]}$	$\var{r0[2]}$	$\var{r0[3]}$	$\var{r0[4]}$	$\var{r0[5]}$	$\var{r0[6]}$	$\var{r0[7]}$	$\var{r0[8]}$	$\var{r0[9]}$	Mean = $\var{mean1}$
{exam2}	$\var{r1[0]}$	$\var{r1[1]}$	$\var{r1[2]}$	$\var{r1[3]}$	$\var{r1[4]}$	$\var{r1[5]}$	$\var{r1[6]}$	$\var{r1[7]}$	$\var{r1[8]}$	$\var{r1[9]}$	Mean = $\var{mean2}$
{total}	$\var{sscores[0]}$	$\var{sscores[1]}$	$\var{sscores[2]}$	$\var{sscores[3]}$	$\var{sscores[4]}$	$\var{sscores[5]}$	$\var{sscores[6]}$	$\var{sscores[7]}$	$\var{sscores[8]}$	$\var{sscores[9]}$	Mean = $\var{overallmean}$

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

Give any introductory information the student needs.

			Name	Type	Generated Value

overallmean	number	117.5
mean1	number	57.7
mean2	number	59.8
overallvar	number	146.2777777778
ssq1	number	33777
ssq2	number	36616
total	string	Total Score
exam2	string	Cell Biology
tol	integer	0
exam1	string	Anatomy
stdev1	number	7.3
stdev2	number	9.8
var1	number	53.789
var2	number	95.067
sig1	integer	11
sig0	integer	9
stdevoverall	number	12.1
r0	list	List of 10 items
r1	list	List of 10 items
n	integer	10
mu	integer	59
s	integer	2
sscores	list	List of 10 items

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
mean(sscores)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `number`

117.5

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Number entry
  Add
Part b) Gap-fill
1. Gap Number entry
  Add
Part c) Gap-fill
1. Gap Number entry
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

{exam1}

Sample Standard Deviation = Unnamed gap (to one decimal place)

Advice

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

For {exam1} we have the mean is:

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

The sample variance is given by the formula:

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

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

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

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

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

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

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Charlie Earle 6 years, 3 months ago

Error in variable testing condition

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Generated value: `number`

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{exam1}

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Calculate the standard deviation

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

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Comment

Checkpoint description

Charlie Earle 6 years, 3 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

{exam1}

Generated value: `number`