1. Practice & Communication of Science
Measurement

2. What is Measurement?
Assigning comparative labels to things to help explain their relationships…
sounds a bit abstract…
…but things/relationships is all there is…
…and that’s all that science is about!
so measurement is rather central to science
Measurements are typically, but not exclusively, numerical
not all types of measurement are equivalent
four different levels of measurement
nominal, ordinal, interval and ratio

3. Nominal Scales The ‘lowest’ level of measurement
nominal implies ‘names’
Just labels to stick things into categories and separate them
No implicit order
eg yes/no
shirt numbers of footballers
1 – goalie, 10 – striker (but not ‘10x better’!)
can serve to separate and provide some info
if refer to #10, it’s likely to be about a striker not a goalie
blue, yellow, red, green etc
no implicit order (though underlying spectrum has)

4. Ordinal Scales The measurements can be ‘ordered’ (ranked)
the order of finishers in a race (first, second, etc)
but time between each can vary dramatically
equal gaps not implied
the Likert scale (1  5)
agree strongly, agree, neutral, disagree, disagree strongly
again, the ‘gaps’ between each are not equal
agree - neutral doesn’t ‘equal’ neutral - disagree

5. Interval Scales
The ‘gaps’ (intervals) between units of measurement are equal
the Centigrade scale
the temp difference between 20 and 30 C is the same as between 10 and 20 C
though they might not ‘feel’ that way!
there is no absolute reference point
0 C is arbitrary
water’s freezing point used to define the baseline
Much more common in science

6. Ratio Scales An interval scale that has an absolute reference point
the Kelvin temperature scale
0 K is C
the reference point is absolute
absolute zero (0 K) is, well, absolute!
for our everyday lives, time is a ratio scale
zero time is absolute
like interval scales, ratio scales common in science
These measurement scales are important as they determine the types of data-handling/statistics that can be performed

7. Summaries of Data Scientists seldom take single measurements
need repeated measurements to…
minimise error
permit extrapolation to the general case
eg my eyes are blue
32 out of 100 subjects studied had blue eyes
32% of the general population have blue eyes
Data is plural (datum is singular)
Could just report all measurements…
contains unadulterated ‘info’ about what you did
but doesn’t carry a ‘message’ about the findings
can’t see the wood for the trees

8. Summaries of Data Here is a set of ordered data… Mode Median Mean
17, 18, 18, 18, 19, 19, 20 21, 21
Mode
the most common value(s) of a list of data (18)
Median
the central value in the ordered data (19)
Mean
sum of values/sample size (171 / 9 = 19)
Range (or maybe Maximum and Minimum)
highest minus lowest (21 – 17 = 4)
starts to indicate variability, but biased by extremes

9. Indicating Variability
This is an important aspect of measurement
17, 18, 18, 18, 19, 19, 20 21, 21 and
18, 19, 19, 19, 19, 19, 19, 19, 20 and
19, 19, 19, 19, 19, 19, 19, 19, 19
all have the same mean
Need a way to summarise data both in terms of ‘central tendency’ and ‘spread’
mean and standard deviation
median and quartiles
Measures of variation covered in detail elsewhere

10. Summary
Measurements are labels assigned to things to explain relationships
Four levels…
Nominal – names; no inherent order
Ordinal – ordered; ‘gaps’ not equal
Interval – ordered; ‘gaps’ are equal
Ratio – ordered; equal gaps; absolute ref point
Summaries of data needed to ease interpretation – eg mode, mean, median, range
Need indicators of ‘spread’ as well as ‘centre’
eg range, max, min, standard deviation

11. Practice & Communication of Science
Probability

12. What is Probability? We cannot know everything about everything
we cannot measure everything
our measurements are prone to error
So uncertainty is a central feature of science
uncertainty in observations/measurements 
uncertainty in explanations 
uncertainty in predictions
Probability is a way of quantifying (un)certainty
scale of 0  1 (or 0%  100%)
Probability reflects random influences
‘randomness’ reflects our lack of knowledge

13. Randomness  Predictable Rules
Individual outcomes cannot be predicted, but repeated runs are very predictable
eg individual coin-toss  H or T
‘infinite’ repeats  50:50 H:T (if fair)
Modelling a system in terms of probabilities can be done through observation or from theory
Throwing dice
theory
Red vs Blue in sport
observation
(Hill & Barton)

14. Frequencies and Probabilities
Red and Blue football teams
in 140 matches, Red won 60 and drew 30
relative frequency = 60/140
probability of red winning (in the future) is also 60/140 = 0.429
For throwing a die
relative frequency of getting a ‘3’ is 1/6
probability is also 1/6 = 0.167

15. Combining Probabilities
For independent events, eg probability of throwing a five and then a two
multiply the individual probabilities
P(A and B) = P(A) x P(B)
eg 1/6 x 1/6 = 1/36 = 0.028
For incompatible events, eg probability of throwing a five or a two
add the individual probabilities
P(A or B) = P(A) + P(B)
eg 1/6 + 1/6 = 1/3 = 0.333

16. Probability and common sense
In playing the lottery, which choice of numbers is more likely to win?
3, 5, 15, 27, 29, 44
1, 2, 3, 4, 5, 6
I won the lottery last week; the chances of me winning the lottery this week are…
less, the same, greater?
I won the lottery last week; the chances of me winning the lottery twice in a row are…

17. Probability and common sense
What are the chances of two people on a football field sharing the same birthday?
1%, 11%, 21%, 31%, 41%, 51%
For 23 people, prob of not sharing b’day with previously considered people is…
person 1 : 365/365
person 2 : 364/365 …
person 23 (includes ref!) : 343/365
Multiply them all together  0.493
1 – = = 51%

18. Probability and common sense
Linda is thirty-one, single, outspoken and very bright
She studied political science at Uni; she was concerned with discrimination and social justice, and took part in CND demonstrations
Which of the following statements about Linda is more likely?
Linda works as an estate agent
Linda works as an estate agent and is active in the feminist movement

19. Probability is context-sensitive
In tossing a coin ten times, which sequence is most probable?
H T T H H T H T H T
H H H H H H H H H H
In coin-tossing, which sequence is more probable?
A mix of heads and tails
10 heads in a row
1 in 1024

20. Probability is context-sensitive
Derren Brown can toss a coin heads 10x in a row
Incredible motor control over ‘random’ variables?

21. Probability can be counter-intuitive
Flip a coin three times to get HH or HT
Are the two outcomes equally probable?
HHH  HH first
HHT  HH first
HTH  TH first
HTT
TTT
TTH  TH first
THT  TH first
THH  TH first

22. Probability can be counter-intuitive
Defendant’s DNA match was 1 in 1 billion
Lab’s false positive error rate (not disclosed) is 1%
What is the probability of the defendant being falsely convicted on that evidence?
1 in 1 billion
1 billion x 1% = 1 in 10 million
1 billion x 99% = 1 in 990,000,000
1 in

23. Probability can be counter-intuitive
The Monty Hall conundrum
You are on a game show
You have a choice of three doors
Behind one is a car, behind other two are goats
You choose a door
The host (who knows where the goats are) opens one to show you a goat
Should you now change the door you have chosen?

24. Probability can be counter-intuitive
A mother gives birth to twins (not identical)
What is the chance they will be of different sexes?
25%
33%
50%
One is a girl
What is the chance that they will both be girls?

25. Conditional Probability
Previous questions not conditional/conditional…
Not con - A mum gives birth to twins (not identical). What is the chance that they will both be girls?
Cond - A mum gives birth to twins (not identical). What is the chance that they will both be girls if one is a girl?
The if clause makes all the difference
It provides extra information that alters the odds
The influence of additional info on odds was developed by Thomas Bayes (b 1702)
Bayesian odds
Prior prob (1 in 4) and posterior prob (1 in 3)

26. Three variations on a theme…
A family has two children; what are the chances that both children are girls?
1 in 4, 1 in 3, 1 in 2?
A family has two children; what are the chances that both children are girls if one is a girl?
A family has two children; what are the chances that both children are girls, if one is a girl named Florida?

27. Three variations on a theme…
2 children; P that both children are girls?
GB, BG, BB, GG
1 in 4
2 children; P both girls if one is a girl?
1 in 3
2 children; P both girls if 1 girl named Florida?
BB,BGF,BGNF,GFB,GNFB,GFGNF,GNFGF,GNFGNF,GFGF
1 in 2

28. Conspiracy Theories and Probability
Are these equivalent?
1) The P of a series of events happening if due to a huge conspiracy
2) The P of a huge conspiracy existing if a series of events happened
P of 1) > P of 2)
a ‘single’ explanation vs ‘many’ other explanations
think 9/11, moon landings, paranoia, confabulation
Bayes’ theory supports this non-correspondence

29. Conditional Probability and Testing
A test for disease ‘X’ comes back positive
And the false-positive rate is low at only 1 in 1000
Only a 0.1% chance of not having the disease?! 
But. Is…
the chance of not having the disease if I tested positive
…the same as…
the chance of testing positive if I didn’t have the disease?
No. Think of the ‘sample space’
‘categories’ of people tested

30. The Test’s Sample Space
1) tested +ve and have ‘X’ (true positives)
2) tested +ve put don’t have ‘X’ (false positive)
3) tested –ve and don’t have ‘X’ (true negative)
4) tested –ve and have ‘X’ (false negatives)
Reported false positive rate is 1 in 1000
Incidence rate: say 1 in 10,000 tested have the disease (and false neg effectively 0)
incidence rate not usually mentioned/considered
For 10,000 tested, there will be 10 false positives and only 1 true positive
so 10/11 chance of not having the disease!

31. Summary
Probability estimates the odds of future events based on theory or observation
Probability cannot predict an individual event
Probability can predict pattern of events
Probability, P, 0  1 or 0%  100%
Probability is often not ‘intuitive’, it fools us
Combining probabilities
Independent events: p(A and B) = p(A) x p(B)
Incompatible events: p(A or B) = p(A) + p(B)
Conditional probabilities (prob of A if B)
Baysian probability
Prior probability + extra info  posterior probabilities
prob of A if B often different to prob of B if A