Section 3 (Pg 25 - end of section 1) Flashcards
Name the 3 types of distribution?
Normal
Binomial
Poisson
When is normal distribution used?
For continuous variables
When is binomial distribution used?
For binary data
When is poisson distribution used?
For events occurring at random intervals of time or space and rare events such as drug side effects
What is another name for normal distribution?
Gaussian distribution
Give the 6 characteristics of normal distribution?
Bell-shaped Single central peak Symmetrical Equal mean, median and mode Continous Takes values between - infinity and + infinity
What 2 descriptive statistics are used to describe the normal distribution?
Mean
Variance
How would you write that X is a normally distributed variable with mean mu and standard deviation sigma?
X~N(μ, σ^2)
What is the mean and standard deviation of the standard normal distribution?
Mean = 0
Standard deviation = 1
How to standardise a normally distributed variable?
Subtract the mean and divide by the standard deviation
What is the standard normal variable referred to as?
z
What does the total area under the normal distribution density function curve equal?
1
If an observation selected at random from the population lies outside of the 95% range, what does this suggest about the population mean?
Casts doubt on the fact that the population mean is mu
Why standardise a normal distribution?
To calculate probabilities for normal (probability tables on exist for the standard normal)
What is the z-score?
The standardised normal distribution
What probability is associated with the mean?
0.5 (there is a 50% chance you will get a score that is less than the mean)
In the standard normal distribution, approximately what % of values lie within 1 standard deviation of the mean?
68%
In the standard normal distribution, approximately what % of values lie within 2 standard deviation of the mean?
95%
In the standard normal distribution, approximately what % of values lie within 3 standard deviation of the mean?
99.9%
Do standardised normal distribution tables give the probability that z is less than or more than the specified value?
Less than
How can we assess normality of distribution of a variable? (3)
Informal review of the properties of the normal distribution
Inspection of a normal plot
Formally through a statistical test e.g. Shapiro-Wilk
What is a normal plot?
A diagram constructed to show the extent of the departure of a data distribution from the normal
What shape is the cumulative frequency distribution of a normally distributed variable?
s-shaped
What does any departures form the straight line in normal plot suggest?
Deviation from normality
For shapiro-wilk, what does a result less than 0.05 indicate?
The distribution is significantly different to the normal
For shairo-Wilk, what does the closer the P-value is to 1 indicate?
The closer it is to being normally distributed
Give the 5 possible ways to transform data to be normally distributed?
Logarithmic transformation Square root transformation Reciprocal transformation Cube transformation Logit transformation
What type of transformation is used for data that is fairly skewed or groups of data in which the variances are proportional to the mean?
Logarithmic transformation
What type of transformation is used for data that is slightly skewed or counts?
Square root transformation
What type of transformation is used for data that is highly skewed?
Reciprocal transformation
What type of transformation is used for data that relates to volume?
Cube transformation
What type of transformation is used for proportions?
Logit transformation
What is the logit transformation equation?
logit (p) = ln (p/ 1-p)
What would be the most likely transformation appropriate for the number of units of alcohol consumed per week?
Square root
What would be the most likely transformation appropriate for the proportion of women in favour of breast screening?
Logit transformation
What would be the most likely transformation appropriate for the stimulated saliva flow (cc per minture)?
Cube root transformation
If 2 normally distributed variables are added or subtracted, what does the variance of the outcome equal?
The sums of the variances
If 2 normally distributed variables are added, what does the mean of the sum equal?
The sum of the means
If 2 normally distributed variables are subtracted, what does the mean of the differences equal?
The difference of the means
What is a prospective study?
A study that watches for outcomes during the study
How to calculate the expected number of events for binomial data if you know n and p?
n X p
How to calculate variance of the expected number of events for binary data?
n X p X (1-p)
How to calculate the probability that there will be x events for binary data?
P (x) = n!/ x! (n-x)! X p^x (1-p) ^n-x
What is factorial of 0?
1
Does the binomial distribution get closer or further from normality as the size of the group increases?
Closer
When can the binomial distribution with parameters n and p be approximated as normal?
When both:
np >5
n(1-p) >5
How can the binomial distribution with parameters n and p be approximated as normal?
N(np, square root (np(1-p))
For poisson data, what symbol is used for the average number of occurences in a fixed interval?
Llamda
For poisson data, what is the equation for the probability of r events?
e^-llambda X llamda^r / r(r-1)(r-2)… 1
What is e?
The exponential constant
What are the mean and standard deviation of the poisson distribution?
llamda
What happens for poisson distribution as llamda increase?
The poisson distribution approaches normality
