Measurements and Statistics

Question 1

What is Random Error?

Answer 1

measurement caused by factors which vary from one measurement to another

Question 2

How can random error be reduced?

Answer 2

• by careful experimentation (e.g. controlling the temperature when measuring reaction rates)
• repeated measurements help reduce effects of random noise

Question 3

What is Systematic Error?

Answer 3

they affect measurements by the same amount or by the same proportion

Question 4

How to remove systematic error?

Answer 4

By identifying the flaw and eliminating it

Question 5

What errors can be measured by % agreement?

Answer 5

systematic error

Question 6

How to deal with mistakes in measurements?

Answer 6

• they are ignored!
• mistakes are often identified by repeating measurements

Question 7

What does high precision imply?

Answer 7

a low spread of results (low random error)

Question 8

What does high accuracy imply?

Answer 8

the average result is close to ‘true’ answer, therefore low systematic error

Question 9

What does it mean for accuracy when taking differences?

Answer 9

accuracy may not be a priority (systematic error cancels) - looking for change

Question 10

What does it mean for a distribution to be normalised?

Answer 10

The integral is 1

Question 11

What is probability density?

Answer 11

Vertical axis when dealing with a continuous distribution

Question 12

Discrete Distribution

Answer 12

a probability distribution that depicts the occurrence of discrete (individually countable) outcomes - e.g. when determining the probability distribution of a die

Question 13

Continuous Distribution

Answer 13

describes the probabilities of the possible values of a continuous random variable - e.g. a person’s height

Question 14

For poisson distribution, what happens when the fixed interval increases?

Answer 14

It tends towards a normal distribution

Question 15

What is the sum of deviations of values from the mean?

Answer 15

ZERO

Question 16

What happens to standard deviation as n increases?

Answer 16

the standard deviation tends towards the width parameter of the parent normal distribution.

Question 17

Definition of mean

Answer 17

average - adding all numbers in the data set and then dividing by the number of values in the set

Question 18

Definition of standard deviation

Answer 18

a measure of how dispersed the data is in relation to the mean

Question 19

What does the standard error of the mean (SEM) tell us?

Answer 19

the uncertainty on the measured mean

Question 20

What is special about uncertainties?

Answer 20

• quoted to one figure
• use 2 figures if leading digit is 1
• uncertainties are rounded up

Question 21

What is the best estimate of a parameter?

Answer 21

the mean

Question 22

What is also known as uncertainty?

Answer 22

Standard error in the mean (SEM)

Question 23

How does averaging over a larger number of measurements affect measurement results?

Answer 23

the mean of the sample tends towards the true mean and the same goes for the standard deviation

Question 24

What is the central limit theorem?

Answer 24

the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough

Question 25

Explain qualitatively the significance of the central limit theorem for measurement uncertainties

Answer 25

It depicts precisely how much an increase in sample size diminishes sampling error which tells us the precision for estimates

Question 26

What is R in the ideal gas law?

Answer 26

8.314 J K ^-1 mol ^-1

Question 27

Define residuals

Answer 27

the difference between the observed and predicted values of data

Question 28

Define regression

Answer 28

relates a dependent variable to one or more independent variables

Question 29

Define the method of least squares

Answer 29

method that minimises the sum of the residuals of points from the plotted curve

Question 30

What is a distribution?

Answer 30

a function that shows the possible values for a variable and how often they occur

Question 31

Examples of discrete distributions

Answer 31

• 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

Question 32

Examples of continuous distributions

Answer 32

• modelling the rate of radioactive decay
• speed of sound waves
• measuring height and weight

Question 33

How to find accuracy on a histogram?

Answer 33

the distance of the modal value of the data to the reference value

high accuracy is related to low systematic error

Question 34

How to find precision on a histogram?

Answer 34

the width of the normal distribution

high precision is related to low random error

Question 35

What does a graph of the poisson distribution look like with a large number of data?

Answer 35

like the normal distribution curve

Question 36

What are the properties of the poisson distribution?

Answer 36

• events are independent of each other
• the average rate is constant
• two events cannot occur at the same time

Question 37

In the context that “x and y are independent variables”, explain the term “independent variable”

Answer 37

x and y are independent variables if the random errors on their values are uncorrelated

Question 38

Define confidence level

Answer 38

A probability that a parameter will fall between a set of values