The normal distribution. Binomial distribution is discrete events, (infected, not infected) The normal distribution is a probability density function for a continuous variable, and is represented by a continuous curve. Density = relative frequency of varites on the Y (horizontal) axis.
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Binomial distribution is discrete events, (infected, not infected)
The normal distribution is a probability density function for a continuous variable, and is represented by a continuous curve.
Density = relative frequency of varites on the Y (horizontal) axis.
Area under curve is equal to the sum of expected frequencies
Cannot evaluate the probability of the variable being exactly equal to some value (that area of the curve is soooo small)
Must estimate the frequency of observations falling between two limits
Y is the height of the ordinate
μ is the mean
σ is the standard deviation
π is the constant 3.14159
e is the base of natural logarithms and is equal to 2.718282x can take on any value from -infinity to +infinity.
They both have a MEAN MEW.
The shape of a given normal curve results from different values of and
The mean, , determines the midpoint
The standard deviation, , changes the shape, it affects the spread or the dispersion of scores
The larger the value of the more dispersed the scores; the smaller the value, the less dispersed.
The mean, , determines the midpoint
A smaller , means less dispersion
Bonus question. What’s wrong with this graph?
How to determine what proportion of a normal population lies above/below a certain level
If distribution of Hobbit heights is normal with mean = 120 cm, SD = 20
Half < 120 & half >120
What is probability of finding a Hobbit taller than 130 cm??
The average Hobbit
- Any point on normal curve
- Here, 130 cm
- Normal deviate
- Test statistic
Z = (130-120)/20 = 0.5
Table A S & R
P (probability) (Xi >130 cm) = P (Z>0.50) = 0.3085 or 30.85%
68.27% of measurements lie w/in ( 1)
99.73% of measurements lie w/in ( 3)
50% lie w/in ( 0.67 )
95% lie w/in ( 1.96 )
- hence the 95% confidence interval of a sample = X 1.96 * s
- the range within one is 95% confident that the true population mean, , is to be found
Binomial distribution (p + q)k
Imagine a trait is controlled by many factors, ex skin pigmentation.
When a factor is present, it contributes 1 unit of pigmentation
If 3 factors were present, the animal would have skin that was 3 units dark
Assume 0.5 probability of each factor being present: p (hence 0.5 probability of each factor being absent): q
Half the animals would have it, half would not
expected proportion w 0 pigmentation unit=0.5
expected proportion w 1 pigmentation unit=0.5
If two factors existed, (p + q)2; k=2
There will now be 3 classes: pp, pq, qq
Frequency of pp = 0.25 or (0.5)2
Frequency of pq = 0.5 or 2[0.5*0.5]
Frequency of qq = 0.25 or (0.5)2
If k (number of independent factors) becomes large, the distribution produced by binomial expansion would come very close to the normal distribution
Many biological variables behave like this
When samples are large, this occurs even when the factors are not strictly independent, or not all equal in magnitude of effect.
I'm not an outlier; I just haven't found my distribution yet
Mean not equal to median
kurtosis: the proportion of a curve located in the center, shoulders and tails
How fat or thin the tails are
Revisit variance and SD relative to normal curve
Mean SS =variance
Total sum of squares
I side =SD
If you take repeated samples of size N from a normally distributed population, the distribution of the the means of those samples will be normal
If you take repeated samples of size N from a non-normally distributed population, the distribution of the the means of those samples will tend towards normality
Central Limit Theorem
Variance of mean =
Square root of the variance of the mean is the SD of the mean, also called the standard error
But rarely know pop parameters, so…….
We’ll come back to this with more on testing differences between a mean and a value or between 2 means
Become more accurate approximations of “true”
SE becomes smaller with increased sampling
SE, SD/ n
Sample size (n)
SD is a parameter of a natural population (even though real populations are constantly changing). Its size reflects real, natural variance. Big is not good, small is not good. SD just is. Natural dispersion of population
SE becomes smaller with increasing sample size, therefore reflects sampling effort. Accuracy of mean.
Both frequently reported (graphed) in ecological / biological literature. SE smaller, so often favored– but this is wrong reasoning!
Practically either OK, if you state which is shown and report n!