Practice Quiz Calculate the standard deviation

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Charlie Earle 6 years, 3 months ago

Created this as a copy of Calculate the 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 number of participants in two gym classes; {exam1} and {exam2}, and the total attendance for both classes over each week ( $n=\var{n}$ ).

	Week 1	Week 2	Week 3	Week 4	Week 5	Week 6	Week 7	Week 8	Week 9	Week 10
Pilates	$\var{r0[0]}$	$\var{r0[1]}$	$\var{r0[2]}$	$\var{r0[3]}$	$\var{r0[4]}$	$\var{r0[5]}$	$\var{r0[6]}$	$\var{r0[7]}$	$\var{r0[8]}$	$\var{r0[9]}$	Mean = $\var{mean1}$
Circuits	$\var{r1[0]}$	$\var{r1[1]}$	$\var{r1[2]}$	$\var{r1[3]}$	$\var{r1[4]}$	$\var{r1[5]}$	$\var{r1[6]}$	$\var{r1[7]}$	$\var{r1[8]}$	$\var{r1[9]}$	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 Pilates, Circuits and Total Score.

Give any introductory information the student needs.

			Name	Type	Generated Value

overallmean	number	78.2
mean1	number	39.3
mean2	number	38.9
overallvar	number	303.5111111111
ssq1	number	15829
ssq2	number	16747
total	string	Total Score
exam2	string	circuits
tol	integer	0
exam1	string	pilates
stdev1	number	6.5
stdev2	number	13.4
var1	number	42.678
var2	number	179.433
sig1	integer	13
sig0	integer	9
stdevoverall	number	17.4
r0	list	List of 10 items
r1	list	List of 10 items
n	integer	10
mu	integer	40
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`

78.2

← Depends on:

sscores

This variable doesn't seem to be used anywhere.

Parts

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

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Pilates

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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Practice Quiz Calculate the standard deviation

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Taxonomy: mathcentre

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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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Pilates

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Practice Quiz Calculate the standard deviation

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Charlie Earle 6 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Pilates

Generated value: `number`