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Question statement

Using a metre rule, you performed serveral repeat measurements of your friend's height :

Measurement	Height [m]
1	$\var{data[0]}$
2	$\var{data[1]}$
3	$\var{data[2]}$

Measurement	Height [m]
1	$\var{data[0]}$
2	$\var{data[1]}$
3	$\var{data[2]}$
4	$\var{data[3]}$
5	$\var{data[4]}$

Give any introductory information the student needs.

			Name	Type	Generated Value

data	list	List of 5 items
result	number	1.8454
error	number	0.0026472627
finalresult	number	1.845
finalerror	number	0.003
mu	number	1.845
n	integer	5
sigma	number	0.004
finalprecision	number	3
precision	integer	3
units	string	m
std	number	0.0059194594

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
repeat(precround(normalsample(mu,sigma), precision), n)

JME function reference

Description

Produce values of measurements by repeatively sampling a normal distribution with mean (mu) and standard deviation (sigma).

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`

[ 1.850, 1.847, 1.850, 1.846, 1.834 ]

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Mean±Error Gap-fill
1. Gap Mean Number entry
  Add
2. Gap Error Number entry
  Add
3. Gap Final mean Number entry
  Add
4. Gap Final Error Number entry
  Add
5. Gap Std Number entry
  Add

Gap-fill

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

Calculate:

Mean measurement = Mean

The standard deviation =Std

Error on the mean = Error

Now write the final result and error to the correct precision:

Final mean ± Final Error m

Advice

When studying a parameter of interest, scientists record many repeat measurements. This is to help account for the imperfect precision and accuracy of measurements.
The best estimate of a parameter, $x$, is the mean of 𝑁 repeat measurements:
\[ \bar{x} = \frac{1}{N}(x_1 + x_2 +\cdots+ x_N) = \frac{1}{N}\sum_{i=1}^N x_i \]

In this example of measuring the height of your friend, the values of $x_i$ are:

$x_1$ =	$\var{data[0]}$
$x_2$ =	$\var{data[1]}$
$x_3$ =	$\var{data[2]}$

Using Python, this can be calculated via:

import numpy as np

x = np.array([$\var{data[0]}$, $\var{data[1]}$, $\var{data[2]}$])

mean = np.mean(x)

Or if using a spreadsheet:

=average($\var{data[0]}$, $\var{data[1]}$, $\var{data[2]}$)

To determine how precise the mean is we need to calculate the average deviation of our measurements $x_i$ with the mean value $\bar{x}$.

In this case, we have repeated measurements of the same object (height of your friend) so we can use the population standard deviation:

\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2} \]

which can be calculated using Python via:

sd = np.std(x)

or via a spreadsheet approach:

=stdevp($\var{data[0]}$, $\var{data[1]}$, $\var{data[2]}$)

We can use this standard deviation to determine the error on the mean value:

\[ \alpha = \frac{\sigma}{\sqrt{N}} \]

which indicates that the more repeat measurements we perform, the smaller the error on the mean becomes.

Typically, the error on the mean is only precise to 1 significant figure. It requires 1000's of repeat measurements for an additional significant figure of precision. Hence we round $\alpha$ to 1 significant figure and then round the mean to the same number of decimal places to produce the final result:

\[ \bar{x} \pm \alpha \]

See this useful book for more details:

Hughes, I., Hase, T. , "Measurements and Their Uncertainties", Oxford University Press (2010)

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Mark Quinn 2 years, 8 months ago

Mark Quinn 2 years, 8 months ago

Mark Quinn 2 years, 8 months ago

Mark Quinn 2 years, 8 months ago

Mark Quinn 2 years, 8 months ago

Mark Quinn 2 years, 8 months ago

Error in variable testing condition

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Generated value: `list`

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Mean±error 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Mark Quinn 2 years, 8 months ago

Mark Quinn 2 years, 8 months ago

Mark Quinn 2 years, 8 months ago

Mark Quinn 2 years, 8 months ago

Mark Quinn 2 years, 8 months ago

Mark Quinn 2 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

Generated value: `list`