Calculate the measures of central tendency for a sample

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Xiaodan Leng 5 years, 8 months ago

Created this as a copy of Calculate the measures of central tendency for a sample.

Name	Status	Author	Last Modified
Calculate the measures of central tendency for a sample	Ready to use	Stanislav Duris	20/11/2019 14:34
Ioannis's copy of Calculate the measures of central tendency for a sample	draft	Ioannis Lignos	17/09/2019 12:45
Calculate the measures of central tendency for a sample	draft	Hugh O'Donnell	16/01/2018 13:54
Measures of central tendency from a list	Needs to be tested	Lauren Frances Desoysa	04/10/2021 12:37
Michael's copy of Measures of central tendency from a list	draft	Michael Proudman	07/12/2021 09:49
kopie mediaan gemiddelde aantal 20	draft	Johan Maertens	09/12/2018 13:05
Calculate descriptive statistics for a sample	Has some problems	Mario Orsi	08/03/2019 14:07
Calculate mean, median and standard deviation for a sample	draft	Mario Orsi	13/03/2019 14:02
Tom's copy of Calculate the measures of central tendency for a sample	draft	Tom Bold	18/06/2019 10:50
Ruth's copy of Calculate descriptive statistics for a sample	draft	Ruth Allison	02/07/2019 12:01
Calculate the measures of central tendency for a sample	draft	Xiaodan Leng	10/07/2019 21:35
Calculate the measures of central tendency for a sample	draft	Adelle Colbourn	13/08/2020 00:48
Zentralmaße für einen Datensatz bestimmen	Should not be used	Andreas Vohns	02/12/2020 15:42
1 Calculate the measures of central tendency for a sample	draft	Adelle Colbourn	08/06/2023 10:40
SA9 Calculate Median from a list	Ready to use	Upuli Wickramaarachchi	04/04/2024 11:10
SA6 Calculate Range	Ready to use	Lauren Desoysa	04/04/2024 11:08
SA8 Calculate Mode From a list	Ready to use	Lauren Desoysa	04/04/2024 11:05

There are 49 other versions that do you not have access to.

Question statement

A random sample of 20 residents from Newcastle were asked about the number of times they went to see a play at the theatre last year.

Here is the list of their answers:

$\var{a[0]}$	$\var{a[1]}$	$\var{a[2]}$	$\var{a[3]}$	$\var{a[4]}$	$\var{a[5]}$	$\var{a[6]}$	$\var{a[7]}$	$\var{a[8]}$	$\var{a[9]}$
$\var{a[10]}$	$\var{a[11]}$	$\var{a[12]}$	$\var{a[13]}$	$\var{a[14]}$	$\var{a[15]}$	$\var{a[16]}$	$\var{a[17]}$	$\var{a[18]}$	$\var{a[19]}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	list	List of 20 items
a_s	list	List of 20 items
mean	number	4.65
median	number	4.5
mode	list	[ 4 ]
mode1	integer	4
range	number	7
modetimes	list	[ 0, 1, 1, 3, 5, 4, 3, 1, 2 ]

			Name	Type	Generated Value

modea1	list	[ 4 ]
modea2	list	[ 0, 3, 6 ]
a1	list	List of 20 items
a2	list	List of 20 items
a3	list	List of 20 items

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
if(len(modea1) = 1, a1, if(len(modea2) = 1, a2, a3))

JME function reference

Description

The final list.

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

Can an exam override the value of this variable?

Generated value: `list`

[ 4, 5, 1, 8, 5, 4, 4, 5, 2, 5, 3, 6, 3, 8, 6, 4, 3, 7, 6, 4 ]

← Depends on:

This variable doesn't seem to be used anywhere.

Advice

a)

The mean is the sum of all the responses ($\sum x$) divided by the number of responses ($n$).

Here, $n = 20$.

\begin{align}
\sum x &= \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{a[10]} + \var{a[11]} +\var{a[12]} +\var{a[13]} +\var{a[14]} +\var{a[15]} +\var{a[16]} +\var{a[17]} +\var{a[18]} +\var{a[19]} \\
&= \var{sum(a)} \text{.}
\end{align}

Therefore we calculate the mean

\begin{align}
\overline{x} &= \frac{\sum x}{n} \\[0.5em]
&= \frac{\var{sum(a)}}{20} \\[0.5em]
&= \var{mean} \text{.}
\end{align}

b)

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

\[ \var{a_s[0]}, \quad \var{a_s[1]}, \quad \var{a_s[2]}, \quad \var{a_s[3]}, \quad \var{a_s[4]}, \quad \var{a_s[5]}, \quad \var{a_s[6]}, \quad \var{a_s[7]}, \quad \var{a_s[8]}, \quad \var{a_s[9]}, \quad \var{a_s[10]}, \quad \var{a_s[11]}, \quad \var{a_s[12]}, \quad \var{a_s[13]}, \quad \var{a_s[14]}, \quad \var{a_s[15]}, \quad \var{a_s[16]}, \quad \var{a_s[17]}, \quad \var{a_s[18]}, \quad \var{a_s[19]} \]

There is an even number of responses, so there are two numbers in the middle (10^th and 11^th place). To find the median, we need to find the mean of these two numbers $\var{a_s[9]}$ and $\var{a_s[10]}$:

\begin{align}
\frac{\var{a_s[9]} + \var{a_s[10]}}{2} &= \frac{\var{a_s[9] + a_s[10]}}{2} \\
&= \var{median} \text{.}
\end{align}

c)

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

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

$\var{a_s[0]}, \var{a_s[1]}, \var{a_s[2]}, \var{a_s[3]}, \var{a_s[4]}, \var{a_s[5]}, \var{a_s[6]}, \var{a_s[7]}, \var{a_s[8]}, \var{a_s[9]}, \var{a_s[10]}, \var{a_s[11]}, \var{a_s[12]}, \var{a_s[13]}, \var{a_s[14]}, \var{a_s[15]}, \var{a_s[16]}, \var{a_s[17]}, \var{a_s[18]}, \var{a_s[19]}$.

We notice that $\var{mode1}$ occurs the most ($\var{modetimes[mode1]}$ times) so $\var{mode1}$ is the mode.

d)

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:

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

Give a worked solution to the whole question.

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Calculate the measures of central tendency for a sample

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Taxonomy: mathcentre

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Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

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Warning

Generated value: `list`

Parts

Number entry

Advanced settings

a)

b)

c)

d)

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Calculate the measures of central tendency for a sample

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Number entry

Advanced settings

a)

b)

c)

d)

Generated value: `list`