1. The Detail of the Normal Distribution
Scientific Practice
The Detail of the Normal Distribution

2. The Binomial Distribution
This distribution can be seen when the outcomes have discrete values…
eg rolling dice
Assumptions…
Fixed number of trials
eg we will roll the dice 10 times
Independent trials
one roll cannot influence another
Two different classifications
rolled/didn’t roll a 12 = ‘success/failure’
Probability of success stays the same for all trials
didn’t add extra dice half way through

3. Rolling Dice One die… outcome values are 1, 2, 3, 4, 5 or 6
each equally probable (1 in 6)
distribution is…
boring!

4. Rolling Dice Two dice… outcome values are 2,3,4,5,6,7,8,9,10,11,12
36 ways of making these
each not equally probable
only 1 way to get 2 (1+1), 3 ways to get 4, etc
distribution is…
slightly less boring!

5. Rolling Dice Three dice…
outcome values are 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18
216 ways of making these
each not equally probable
27 ways to throw a 10 or 11, only 1 to get a 3 or 18
distribution is…
starting to curve

6. Rolling Dice Four dice…
outcome values are 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,18,20,21, 22,23,24
1926 ways of making these
each not equally probable
distribution is…
looking familiar!

7. Rolling Dice 24 dice… outcome values are 24 144
× 1018 ways of making these!
each not equally probable
distribution is…
looking very familiar!
still discrete outcomes

8. Rolling Dice Infinite number of dice…
outcome values are no longer discrete but continuous
the Binomial Distribution becomes known as…
…the Normal Distribution/Bell Curve
Substitute dice for something like height and…
height, being determined by the sum effect of a large number of factors (genes, nutrition, etc)…
looks like a continuous variable
approximates the Normal Distribution
Ie its variation becomes definable/predictable
we can expect our data to behave in a certain way

9. 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

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

11. 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%
3 SD covers 99.7%

12. 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

13. 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’

14. 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

15. 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

16. 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! 

17. 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

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

19. 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

20. Populations and Samples – so what?
The two 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)

21. 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

22. Populations and Samples – implications
The sample mean and SD are estimates of the population mean and the population SD
ie you calculate σn-1 (or s)
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)

23. Implications of Estimating Pop Mean
For a sample, the ‘quality’ of the estimate of the population mean and SD depends on the number of observations made
if you sampled, say, 1 member of the population, it’s unlikely to be close to the population mean
if you sampled the whole population, your estimate is the population mean
in between, adding extra samples will improve estimate
sampling different amounts  a variety of means
that set of means will have its own SD (!)
called the Standard Error of the Mean (SEM)

24. The Standard Error of the Mean
Recap…
each sampling of a distribution will produce a different estimate of the population mean
the variation in those estimates called the SEM
Surprisingly easy to calculate
SEM = sample standard dev / square root of number of samples
SEM = s / √ N
eg if N=16, then SEM is 4x smaller than SD

25. Summary The Binomial Distribution is a basic distribution
eg rolling dice
With lots of dice Binomial Dist  Normal Dist
Normal Dist fully defined just by mean and SD
Transformation to z-scores makes all NDs identical
SD calculation differs for sample vs population
sample is a subset of the whole population
population is, erm, the whole population
Estimation of population mean from a sample is always prone to uncertainty
Standard Error of Mean (s/√N) reflects uncertainty