1. From Distributions to Confidence
Practice & Communication of Science
From Distributions to Confidence

2. The Normal Distribution
Represents the idealised distribution of a large number of things we measure in biology
many parameters approximate to the ND
Is defined by just two things…
population mean
µ (mu)
the centre of the distribution (mean=median=mode)
population standard deviation (SD)
σ (sigma)
the distribution ‘width’ (mean  point of inflexion)
encompasses 68% of the area under the curve
95% of area found within 1.96 σ either side of mean

3. The Normal Distribution
Is symmetrical
mean=median=mode

4. The Normal Distribution
One SD either side of mean includes 68% of represented population
SD boundary is inflexion point
curvature changes direction
the ‘s’ bit
2* SD covers 95%
(*actually 1.96)
3 SD covers 99.7%

5. The Normal Distribution
All Normal Distributions are similar
differ in terms of…
mean
SD (governs how ‘spikey’ curve is)
Fig below…
4 different SDs, 2 different means

6. Standardising Normal Distributions
Regardless of what they measure, all Normal Distributions can be made identical by…
subtracting the mean from every reading
the mean then becomes zero
dividing each reading by the SD
a reading one SD bigger  +1
Called Standard Scores or z-scores
amazing! Different measurements  same ‘view’

7. Standard (z) Scores A ‘pure’ way to represent data distribution
the actual measurements (mg, m, sec) disappear!
replaced by number of SDs from the mean (zero)
For any reading, z = (x - µ) / σ
A survey of daily travel time had these results (in minutes):
26,33,65,28,34,55,25,44,50,36,26,37,43,62,35,38,45,32,28,34
The Mean is 38.8 min, and the SD is 11.4 min
To convert the values to z-scores…
eg to convert 26
first subtract the mean: = -12.8,
then divide by the Standard Deviation: -12.8/11.4 = -1.12
So 26 is Standard Deviations from the Mean

8. Familiarity with the Normal Distribution
95% of the class are between 1.1 and 1.7m tall
what is the mean and SD?
Assuming normal distribution…
the distribution is symmetrical, so mean height is ( ) / 2 = 1.4m
the range 1.1  1.7m covers 95% of the class, which equals ± 2 SDs
one SD = (1.7 – 1.1) / 4
= 0.6 / 4
= 0.15m

9. Familiarity with the Normal Distribution
One of that class is 1.85m tall
what is the z-score of that measurement?
Assuming normal distribution…
z-score = (x - µ) / σ
z = (1.85m - 1.4m) / 0.15m
= 0.45m / 0.15m
= 3
note there are no units
3 SDs cover 99.7% of the population
only 1.5 in 1000 of the class will be as tall/taller
a big class, with fractional students! 

10. Familiarity with the Normal Distribution
36 students took a test; you were 0.5 SD above the average; how many students did better?
from the curve, 50% sit above zero
from the curve, 19.1% sit between 0 and 0.5 SD
so 30.9% sit above you
30.9% of 36 is about 11

11. Familiarity with the Normal Distribution
Need to have a ‘feel’ for this…

12. Populations and Samples – a Diversion?
A couple of seemingly pedantic but important points about distributions…
population
the potentially infinite group on which measurements might be made
don’t often measure the whole population
sample
a sub-set of the population on which measurements are actually made
most studies will sample the population
n is the number studied
n-1 called the ‘degrees of freedom’
often extrapolate sample results to the population

13. Populations and Samples – so what?
They are described/calculated differently…
μ is the population mean, x is the sample mean
σ or σn is population SD, s or σn-1 is sample SD
Calculating the SD is different for each
most calculators do it for you…
as long as you choose the right type (pop vs samp)

14. Populations and Samples – choosing
Analysing the results of a class test…
population, since you don’t intend extrapolating the results to all students everywhere
Analysing the results of a drug trial…
sample, since you expect the conclusions to apply to the larger population
A national census collects information about age
population, since by definition the census is about the population taking part in the survey
If in doubt, use the sample SD
and as n increases, the difference decreases

15. Populations and Samples – implications
If the sample observed is the population, then the mean and SD of that sample are the population mean and the population SD
ie you calculate σ (or σn)
If the sample is part of the bigger population, then the sample mean and SD are estimates of the population mean and the population SD
ie you calculate σn-1 (or s)
(the presumption here is you chose your sample to reflect the population!)

16. Implications of Estimating Pop Mean
For a sample, the ‘quality’ of its estimate of the population mean and SD depends on the number of observations made
the mean from a sample of 1 member of the population is unlikely to be close to the pop mean
if you sampled the whole population, your sample estimate is the population mean
in between, adding extra readings to a sample will improve your estimate of the population mean
also, repeated sampling  a collection of means
eg you sampled 200 FVCs and did this 100 times
the 100 means are ND and will have its own SD (!)
called the Standard Error of the Mean (SEM)

17. The Standard Error of the Mean
Immediate recap…
repeated samples of a distribution will produce different estimates of the population mean
the SD of those estimates of pop mean called the SEM
Surprisingly easy to calculate
we don’t need to keep repeating our sampling
can derive it from SD of data in one sample
SEM = sample standard dev / square root of number of samples
SEM = s / √ N
eg if N=16, then SEM is 4x smaller than SD

18. Remember, Samples  Estimates
When we sample a population, we end up with a sample mean, x
our ‘best guess’ estimate of the real pop mean, µ
the ‘real’ mean of the population is ‘hidden’
Our sample also has a measure of the variability of the data that comprises it
the sample Standard Deviation, s
which is also an estimate of the population SD, σ
s can be also be used to indicate the variability of the population mean itself
SEM = s / √ N
can then use SEM to determine confidence limits

19. Confidence Limits and the SEM
The SEM reflects the ‘fit’ of a sample mean, x , to the underlying population mean, µ
if we calculate two sample means and they are similar, but for one the SEM is high,
we are less ‘confident’ about how well that sample mean estimates the population mean
Just like the ‘raw’ data used to calculate a sample mean follows a distribution, so do repeat estimates of the population mean itself
this is the t Distribution
similar to the Normal Distribution

20. The t Distribution Yet another distribution! 
but distributions are important because they define how we expect our data to behave
if we know that, then we gain insight into our expts!
Generally ‘flatter’ than the Normal Distribution
any particular area is more ‘spread out’ (less clear)
the more ‘pointed’ a curve, the clearer the peak

21. t Distribution Pointedness Varies!
Logic…
the number of samples influences the ‘accuracy’ of our estimate of the population mean from the sample mean
as N increases, the ‘peak’ becomes sharper
a given area of the curve is less ‘spread out’
At large N, t Distribution = Normal Distribution
and for both, 95% of curve contained in ± 1.96 SD

22. Using the t Distribution
When we calculate a sample mean and call it our estimate of the population mean…
it’s nice to know how ‘confident’ we are in that estimate
One measure of confidence is the 95% Confidence Interval (95%CI)
the range over which we are 95% confident the ‘true’ population mean lies
derived from our sample mean (we calculate)
and our SEM (we calculate)
and the N (though it’s the ‘degrees of freedom’, N-1)
we use this to look up a ‘critical t value’

23. From t Distribution to 95%CI
The t Distribution is centred around our mean and its shape is influenced by N-1
95%CI involves chopping off the two 2.5% tails
Need a t table to look up how many SEMs (SDs) along the x-axis this point will be
Value varies with N-1
And level of confidence sought

24. Step 1: The t Table t value varies with… For N = 10, α = 0.05 Row…
DoF is N-1
Column…
level of ‘confidence’
95%CI involves chopping off the two 2.5% tails
α = 0.05 (5%)
For N = 10, α = 0.05
t(N-1),0.05 = 2.262
when N large, t=1.96

25. Step 2: Using the t value
The t value is the number of SEMs along the x-axis (in each direction) that encompasses that % of the t distribution centred on our mean
2.262 in the case of t(N-1),0.05
Eg we measure the FVC (litres) of 10 people…
mean = 3.83, SD = 1.05, N = 10
SEM = 1.05/√10 = litres
t(N-1),0.05 = standard errors to cover 95% curve
So, litres either side of the mean = * 0.332
= litres either side of mean covers 95% of dist
So, 95%CI is 3.83 ± =  litres
(3.079, 4.581)

26. Effect of Bigger N
A larger sample size gives us greater confidence in any population mean we estimate
so 95%CI should be smaller
In previous example…
mean = 3.83, SD = 1.05, N =10, SEM = 0.332
95%CI is (3.079, 4.581)
But say we measured 90 more people…
mean = 3.55, SD = 0.915, N = 100
mean and SD similar to before, but
SEM now a lot smaller, at 0.915/√100 =
so too is t(N-1),0.05 = 1.96 (rather than 2.262)
95%CI = 3.55 ± (1.96 * ) =  3.729

27. Effect of Bigger N
A bigger N ‘sharpens’ the t distribution so that the 95% boundaries are less far apart
ie our confidence interval will become smaller
95%CI also shrinks because SEM = SD/√N

28. Summary Normal Dist fully defined just by mean and SD
Transformation to z-scores makes all NDs identical
SD calculation differs for sample vs population
Estimation of population mean from a sample is always prone to uncertainty
Standard Error of Mean (s/√N) reflects uncertainty
Estimates of means follow the t Distribution
t Distribution becomes ‘sharper’ with higher N
‘Width’ of t dist covering 95% is called 95%CI
range in which 95/100 mean estimates would fall
95%CI = mean ± (t(N-1),0.05 * SEM)
t is the number of SEMs along dist covering that %