Measurements and Statistics Flashcards
What is Random Error?
measurement caused by factors which vary from one measurement to another
How can random error be reduced?
- by careful experimentation (e.g. controlling the temperature when measuring reaction rates)
- repeated measurements help reduce effects of random noise
What is Systematic Error?
they affect measurements by the same amount or by the same proportion
How to remove systematic error?
By identifying the flaw and eliminating it
What errors can be measured by % agreement?
systematic error
How to deal with mistakes in measurements?
- they are ignored!
- mistakes are often identified by repeating measurements
What does high precision imply?
a low spread of results (low random error)
What does high accuracy imply?
the average result is close to ‘true’ answer, therefore low systematic error
What does it mean for accuracy when taking differences?
accuracy may not be a priority (systematic error cancels) - looking for change
What does it mean for a distribution to be normalised?
The integral is 1
What is probability density?
Vertical axis when dealing with a continuous distribution
Discrete Distribution
a probability distribution that depicts the occurrence of discrete (individually countable) outcomes - e.g. when determining the probability distribution of a die
Continuous Distribution
describes the probabilities of the possible values of a continuous random variable - e.g. a person’s height
For poisson distribution, what happens when the fixed interval increases?
It tends towards a normal distribution
What is the sum of deviations of values from the mean?
ZERO
What happens to standard deviation as n increases?
the standard deviation tends towards the width parameter of the parent normal distribution.
Definition of mean
average - adding all numbers in the data set and then dividing by the number of values in the set
Definition of standard deviation
a measure of how dispersed the data is in relation to the mean
What does the standard error of the mean (SEM) tell us?
the uncertainty on the measured mean
What is special about uncertainties?
- quoted to one figure
- use 2 figures if leading digit is 1
- uncertainties are rounded up
What is the best estimate of a parameter?
the mean
What is also known as uncertainty?
Standard error in the mean (SEM)
How does averaging over a larger number of measurements affect measurement results?
the mean of the sample tends towards the true mean and the same goes for the standard deviation
What is the central limit theorem?
the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough
Explain qualitatively the significance of the central limit theorem for measurement uncertainties
It depicts precisely how much an increase in sample size diminishes sampling error which tells us the precision for estimates
What is R in the ideal gas law?
8.314 J K -1 mol -1
Define residuals
the difference between the observed and predicted values of data
Define regression
relates a dependent variable to one or more independent variables
Define the method of least squares
method that minimises the sum of the residuals of points from the plotted curve
What is a distribution?
a function that shows the possible values for a variable and how often they occur
Examples of discrete distributions
- the number of patients a doctor sees a day
- number of children in a family
- toss a coin 3 times and let X be the number of heads
Examples of continuous distributions
- modelling the rate of radioactive decay
- speed of sound waves
- measuring height and weight
How to find accuracy on a histogram?
the distance of the modal value of the data to the reference value
high accuracy is related to low systematic error
How to find precision on a histogram?
the width of the normal distribution
high precision is related to low random error
What does a graph of the poisson distribution look like with a large number of data?
like the normal distribution curve
What are the properties of the poisson distribution?
- events are independent of each other
- the average rate is constant
- two events cannot occur at the same time
In the context that “x and y are independent variables”, explain the term “independent variable”
x and y are independent variables if the random errors on their values are uncorrelated
Define confidence level
A probability that a parameter will fall between a set of values
