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Harry Flynn 7 years, 11 months ago

Created this as a copy of Find sample mean, standard deviation, median and interquartile range, , .

Name	Status	Author	Last Modified
Find sample mean, standard deviation, median and interquartile range, ,	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Find sample mean, standard deviation, median and interquartile range, ,	draft	Nigel Atkins	01/03/2017 13:19
Harry's copy of Find sample mean, standard deviation, median and interquartile range, ,	draft	Harry Flynn	16/05/2017 17:13
Harry's copy of Find sample mean, standard deviation, median and interquartile range, ,	draft	Harry Flynn	15/11/2018 10:11
Find sample mean, standard deviation, median and interquartile range - small sample	Ready to use	Chris Graham	13/12/2019 15:52

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Question statement

The following data are taken from a survey of primary school class sizes

$\var{r0[0]}$	$\var{r0[1]}$	$\var{r0[2]}$	$\var{r0[3]}$	$\var{r0[4]}$	$\var{r0[5]}$	$\var{r0[6]}$	$\var{r0[7]}$
$\var{r0[8]}$	$\var{r0[9]}$	$\var{r0[10]}$	$\var{r0[11]}$	$\var{r0[12]}$	$\var{r0[13]}$	$\var{r0[14]}$	$\var{r0[15]}$

Give any introductory information the student needs.

			Name	Type	Generated Value

r0	list	List of 16 items
median	number	28
mean	number	28.38
standard_deviation	number	3.93
r1	list	List of 16 items
IQR	number	6
lower_quartile	number	25
upper_quartile	number	31

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
repeat(random(23..36),16)

JME function reference

Description

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

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Generated value: `list`

[ 26, 30, 35, 29, 23, 25, 31, 27, 24, 31, 25, 24, 32, 29, 27, 36 ]

This variable doesn't seem to be used anywhere.

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4. Gap Gap 3. Number entry
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Sample Mean (2 dp)	Sample Standard Deviation (2 dp)	Median (exact value)	Interquartile Range (exact value)
Gap 0.	Gap 1.	Gap 2.	Gap 3.

Advice

Sample mean

The sample mean is $\frac{\var{sum(r0)}}{\var{len(r0)}} = \var{mean} $, to 2 decimal places.

Sample standard deviation

The formula required may be written as

\[s = \sqrt{\frac{ \Sigma(x-\bar{x})^2}{n-1}}\]

\text{where }\bar{x}\text{ is the sample mean and n is the number of items in the data set (16 in this example). }

This is very time consuming for all but the simplest of questions so you should use your calculator, in statistics mode, to find this.

If you rearrange the data in ascending order you get the following table:

$\var{r1[0]}$	$\var{r1[1]}$	$\var{r1[2]}$	$\var{r1[3]}$	$\var{r1[4]}$	$\var{r1[5]}$	$\var{r1[6]}$	$\var{r1[7]}$
$\var{r1[8]}$	$\var{r1[9]}$	$\var{r1[10]}$	$\var{r1[11]}$	$\var{r1[12]}$	$\var{r1[13]}$	$\var{r1[14]}$	$\var{r1[15]}$

Denote the ordered data by $x_j$, thus $x_{10}=\var{r1[9]}$ for example.

Median: The median is the middle of the data once it has been ordered, it therefore lies between the 8th and 9th entries in the ordered table and is given by:

\[ \frac{x_{8}+ x_{9}}{2} = \frac{\var{r1[7]}+ \var{r1[8]}}{2}=\var{median}\]

Interquartile range: There is not one definitive method for finding the the interquartile range (IQR). We will use the following: as there is an even number of values, the Lower Quartile will be the 'median' of the first 8 values. Its position is calculated by finding

\[\frac{n+1}{2}=\frac{\var{8+1}}{2}=4\frac{1}{2}\]

Hence the Lower Quartile lies between the 4th and 5th entries in the ordered table.

It is \[x_4 + \frac{1}{2}(x_5 - x_4) = \var{r1[3]}+ \frac{1}{2}(\var{r1[4]} - \var{r1[3]})=\var{lower_quartile}\]

Once again as there is an even number of values, the Upper Quartile will lie between two values and its position is calculated by finding

\[x_{12} + \frac{1}{2}(x_{13} - x_{12}) = \var{r1[11]}+ \frac{1}{2}(\var{r1[12]} - \var{r1[11]})=\var{upper_quartile}\]

The interquartile range (IQR) is defined to be

\[ \text{Upper Quartile} – \text{Lower Quartile} \]

and so in this case we have:

\[ \text{Interquartile range} = \var{upper_quartile}-\var{lower_quartile}=\var{iqr} \]

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Harry's copy of Find sample mean, standard deviation, median and interquartile range, ,

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Taxonomy: mathcentre

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Checkpoint description

Harry Flynn 7 years, 11 months ago

The following data are taken from a survey of primary school class sizes

$\var{r0[0]}$

$\var{r0[1]}$

$\var{r0[2]}$

$\var{r0[3]}$

$\var{r0[4]}$

$\var{r0[5]}$

$\var{r0[6]}$

$\var{r0[7]}$

$\var{r0[8]}$

$\var{r0[9]}$

$\var{r0[10]}$

$\var{r0[11]}$

$\var{r0[12]}$

$\var{r0[13]}$

$\var{r0[14]}$

$\var{r0[15]}$

Error in variable testing condition

Error

Warning

Generated value: `list`

Parts

Gap-fill

Loading...

Error

Harry's copy of Find sample mean, standard deviation, median and interquartile range, ,

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Harry Flynn 7 years, 11 months ago

The following data are taken from a survey of primary school class sizes

$\var{r0[0]}$

$\var{r0[1]}$

$\var{r0[2]}$

$\var{r0[3]}$

$\var{r0[4]}$

$\var{r0[5]}$

$\var{r0[6]}$

$\var{r0[7]}$

$\var{r0[8]}$

$\var{r0[9]}$

$\var{r0[10]}$

$\var{r0[11]}$

$\var{r0[12]}$

$\var{r0[13]}$

$\var{r0[14]}$

$\var{r0[15]}$

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

Generated value: `list`