Measures of central tendency from a list 2

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Vicky Hall

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Needs to be tested

6 years, 6 months ago

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Tom Lowe commented 4 years, 3 months ago

Advice contains an answer to a part b), which doesn't exist in the question.

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George Stagg 4 years, 4 months ago

Saved a checkpoint:

Fixed types issue

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Chris Graham 6 years, 5 months ago

Saved a checkpoint:

Checkpointing before removing part (b)

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Vicky Hall 6 years, 6 months ago

Gave some feedback: Needs to be tested

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Lauren Frances Desoysa 6 years, 8 months ago

Created this as a copy of Measures of central tendency-Q3.

Name	Status	Author	Last Modified
Weight of a scoop in two ice cream parlours	Ready to use	Stanislav Duris	20/11/2019 14:45
Measures of central tendency-Q3	draft	Lauren Frances Desoysa	08/08/2018 13:08
Measures of central tendency from a list 2	Needs to be tested	Lauren Frances Desoysa	11/11/2020 10:03
Michael's copy of Weight of a scoop in two ice cream parlours	draft	Michael Proudman	07/12/2021 09:49

There are 2 other versions that do you not have access to.

Question statement

There is subtle variation from one step to the next when you walk. When the neuromuscular control of walking deteriorates due to ageing or neurological disease, steps become less consistent. Two people were each asked to walk $10$ steps and the lengths of each of their steps have been recorded in the following table.

Sally (cm)	$\var{a[0]}$	$\var{a[1]}$	$\var{a[2]}$	$\var{a[3]}$	$\var{a[4]}$	$\var{a[5]}$	$\var{a[6]}$	$\var{a[7]}$	$\var{a[8]}$	$\var{a[9]}$
Kate (cm)	$\var{b[0]}$	$\var{b[1]}$	$\var{b[2]}$	$\var{b[3]}$	$\var{b[4]}$	$\var{b[5]}$	$\var{b[6]}$	$\var{b[7]}$	$\var{b[8]}$	$\var{b[9]}$

Give any introductory information the student needs.

			Name	Type	Generated Value

random2	integer	1
a1	integer	58
a2	integer	59
a3	integer	59
a4	integer	61
a5	integer	61
a6	integer	61
a7	integer	62
a8	integer	62
a9	integer	63
a10	integer	64
alist	list	List of 10 items
a	list	List of 10 items
asort	list	List of 10 items

			Name	Type	Generated Value

random	integer	6
b1	integer	52
b2	integer	61
b3	integer	59
b4	integer	51
b5	integer	53
b6	integer	55
b7	integer	58
b8	integer	67
b9	integer	64
b10	integer	64
blist	list	List of 10 items
b	list	List of 10 items
bsort	list	List of 10 items

			Name	Type	Generated Value

meanab	integer	61
meana	rational	61
modea	integer	61
mediana	number	61
meanb	rational	292/5
modeb	integer	64
medianb	number	58.5
rangea	number	6
rangeb	number	16
suma	number	610
sumb	number	584

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1..3)

JME function reference

Description

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

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Generated value: `integer`

→ Used by:

Parts

Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
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4. Gap Gap 3. Number entry
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6. Gap Gap 5. Number entry
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7. Gap Gap 6. Number entry
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8. Gap Gap 7. Number entry
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Gap-fill

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

Using the data above, fill in the following table, entering numbers with decimal places if required.

	Sally (cm)	Kate (cm)
Mean length	Gap 0.	Gap 1.
Median length	Gap 2.	Gap 3.
Modal length	Gap 4.	Gap 5.
Range	Gap 6.	Gap 7.

Advice

We denote Sally as $s$ and Kate as $k$.

a)

We are going to start with completing the column for Sally.

First, we need to find the sum of the step lengths:

\[\begin{align} \sum s &= \var{a[0]} + \var{a[1]} + \var{a[2]} + \var{a[3]} + \var{a[4]} + \var{a[5]} + \var{a[6]} + \var{a[7]} + \var{a[8]} + \var{a[9]} \\&= \var{suma} \text{.}
\end{align}\]

The total number of measurements $n$ is $10$.

Therefore the mean is

\[ \begin{align} \overline{s} &= \frac{\sum s}{n} \\[3pt]&= \frac{\var{suma}}{10} \\&= \var{meana} \text{.} \end{align}\]

The median is the middle value. We need to sort the list in order:

Sally (cm)	$\var{asort[0]}$	$\var{asort[1]}$	$\var{asort[2]}$	$\var{asort[3]}$	$\var{asort[4]}$	$\var{asort[5]}$	$\var{asort[6]}$	$\var{asort[7]}$	$\var{asort[8]}$	$\var{asort[9]}$

To find the middle position within a data set, we take the sample size, add $1$, then divide by $2$. For our data set, the middle position is

$\displaystyle\frac{n+1}{2}=\frac{10+1}{2}=5.5.$

As there is not actually a $5.5$^th position, we need to find the mean of the $5$^th and $6$^th values:

\[ \displaystyle \begin{align} \frac{\var{asort[4]} + \var{asort[5]}}{2} &= \frac{\var{asort[4] + asort[5]}}{2} \\&= \var{mediana} \text{.} \end{align}\]

The mode is the value that occurs the most often in the data.

To find a mode, we can look at our sorted list:

Sally (cm)	$\var{asort[0]}$	$\var{asort[1]}$	$\var{asort[2]}$	$\var{asort[3]}$	$\var{asort[4]}$	$\var{asort[5]}$	$\var{asort[6]}$	$\var{asort[7]}$	$\var{asort[8]}$	$\var{asort[9]}$

We notice that $\var{modea}$ occurs the most times and so $\var{modea}$ is the mode.

The range is the difference between the highest and the lowest value in the data.

To find this, we subtract the lowest value from the highest value:

\[ \var{max(a)} - \var{min(a)} = \var{rangea} \text{.}\]

So the first column is

	Sally (cm)
Mean length	$\var{meana}$
Median length	$\var{mediana}$
Modal length	$\var{modea}$
Range	$\var{rangea}$

Similarly for Kate,

\[\begin{align} \sum k &= \var{b[0]} + \var{b[1]} + \var{b[2]} + \var{b[3]} + \var{b[4]} + \var{b[5]} + \var{b[6]} + \var{b[7]} + \var{b[8]} + \var{b[9]} \\&= \var{sumb} \text{.}
\end{align}\]

The total number of measurements $n$ is $10$ again.

Therefore the mean is

\[ \begin{align} \overline{k} &= \frac{\sum k}{n} \\[3pt]&= \frac{\var{sumb}}{10} \\&= \var{meanb} \text{.} \end{align}\]

For median, we sort the list in order:

Kate (cm)	$\var{bsort[0]}$	$\var{bsort[1]}$	$\var{bsort[2]}$	$\var{bsort[3]}$	$\var{bsort[4]}$	$\var{bsort[5]}$	$\var{bsort[6]}$	$\var{bsort[7]}$	$\var{bsort[8]}$	$\var{bsort[9]}$

Agin, we need to find the mean of the $5$^th and $6$^th values:

\[ \displaystyle \begin{align} \frac{\var{bsort[4]} + \var{bsort[5]}}{2} &= \frac{\var{bsort[4] + bsort[5]}}{2} \\&= \var{medianb} \text{.} \end{align}\]

For mode, we look at our sorted list:

Kate (cm)	$\var{bsort[0]}$	$\var{bsort[1]}$	$\var{bsort[2]}$	$\var{bsort[3]}$	$\var{bsort[4]}$	$\var{bsort[5]}$	$\var{bsort[6]}$	$\var{bsort[7]}$	$\var{bsort[8]}$	$\var{bsort[9]}$

We notice that $\var{modeb}$ occurs the most times and so $\var{modeb}$ is the mode.

To find the range, we subtract the lowest value from the highest value:

\[ \var{max(b)} - \var{min(b)} = \var{rangeb} \text{.}\]

So the complete table is

	Sally (cm)	Kate (cm)
Mean length	$\var{meana}$	$\var{meanb}$
Median length	$\var{mediana}$	$\var{medianb}$
Modal length	$\var{modea}$	$\var{modeb}$
Range	$\var{rangea}$	$\var{rangeb}$

b)

Kate is the correct answer, because she has a larger range of values, therefore her step length is less consistent.

Give a worked solution to the whole question.

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This question is used in the following exams:

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Tom Lowe commented 4 years, 3 months ago

George Stagg 4 years, 4 months ago

Chris Graham 6 years, 5 months ago

Vicky Hall 6 years, 6 months ago

Lauren Frances Desoysa 6 years, 8 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

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Measures of central tendency from a list 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Tom Lowe commented 4 years, 3 months ago

George Stagg 4 years, 4 months ago

Chris Graham 6 years, 5 months ago

Vicky Hall 6 years, 6 months ago

Lauren Frances Desoysa 6 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

Generated value: `integer`