Probability distributions, sampling distributions and central limit theorem

Probability distributions
Dr. S. A. Rizwan, M.D.
Public Health Specialist
SBCM, Joint Program – Riyadh
Ministry of Health, Kingdom of Saudi Arabia

Learning objectives
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Define probability distributions
• Describe the common types of probability distributions
• Describe sampling distribution
• Understand the central limit theorem

Probability distribution
• Probability distribution is a mathematical
function that can be thought of as providing
the probability of occurrence of different
possible outcomes in an experiment.
• The distribution of a statistical data set (or a
population) is a listing or function showing all
the possible values (or intervals) of the data
and how often they occur.

Probability distribution

Section 1:
Binomial distribution
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 5

Binomial distribution
• Following conditions need to be satisfied for a
binomial experiment/distribution:
• There is a fixed number of n trials carried out.
• The outcome of a given trial is either a “success” or
“failure”.
• The probability of success (p) remains constant from
trial to trial.
• The trials are independent, the outcome of a trial is
not affected by the outcome of any other trial.

Binomial distribution – example
• Suppose we have n = 40 patients who will be receiving an experimental
therapy which is believed to be better than current treatments which
historically have had a 5-year survival rate of 20%, i.e. the probability of
5-year survival is p = 0.20
• Thus the number of patients out of 40 in our study surviving at least 5
years has a binomial distribution, i.e. X ~ BIN(40, 0.20)

• Suppose that using the new treatment we find that 16 out of the 40
patients survive at least 5 years past diagnosis.
• Q: Does this result suggest that the new therapy has a better 5-year
survival rate than the current, i.e. is the probability that a patient survives
at least 5 years greater than .20 or a 20% chance when treated using the
new therapy?

• We essentially ask ourselves the following:
• If we assume that new therapy is no better than the current what is the
probability we would see these results by chance variation alone?
• More specifically what is the probability of seeing 16 or more successes
out of 40 if the success rate of the new therapy is .20 or 20% as well?

• This is a binomial experiment situation
• There are n = 40 patients and we are counting the number of patients
that survive 5 or more years. The individual patient outcomes are
independent and IF WE ASSUME the new method is NOT better, then the
probability of success is p = .20 or 20% for all patients.
• So X = # of “successes” in the clinical trial is binomial with n = 40 and p =
0.20, i.e. X ~ BIN(40, 0.20)

• X ~ BIN(40,.20), find the probability that 16 or more patients survive at
least 5 years. probabilities are computed
automatically for greater than
or equal to and less than or
equal to x.
Enter
n = sample size
x = observed # of “successes”
p = probabilityof “success”

• The chance that we would see 16 or more patients out of 40 surviving at
least 5 years if the new method has the same chance of success as the
current methods (20%) is VERY SMALL, 0.0029.

Section 2:
Normal distribution

Normal distribution
• The normal distribution is a descriptive model that describes real world
situations.
• It is defined as a continuous frequency distribution of infinite range (can
take any value).
• This is the most important probability distribution in statistics and
important tool in analysis of epidemiological data

Normal distribution - properties
• The normal distribution is defined by two parameters, μ and σ.
• You can draw a normal distribution for any μ and σ combination.
• There is one normal distribution, Z, that is special.
• It has μ = 0 and σ = 1.
• Also called standard normal distribution.

• Mean = Median = Mode
• Spread determined by SD
• Bell-shaped
• Symmetry about the center
• 50% of values less than the mean and
50% greater than the mean
• It approaches horizontal axis
asymptotically: - ∞ < X < + ∞
• Area under the curve is 1

Normal distribution – example
• Assuming the normal heart rate (H.R) in normal healthy individuals is
normally distributed with Mean = 70 and Standard Deviation = 10
• Q1. What area under the curve is above 80 beats/min?
• Q2. What area of the curve is above 90 beats/min?
• Q3. What area of the curve is between 50-90 beats/min?
• Q4. What area of the curve is above 100 beats/min?
• Q5. What area of the curve is below 40 beats per min or above 100 beats
per min?

Normal distribution – example

Section 3:
Sampling distribution

• Sampling distribution of the mean – A theoretical probability distribution
of sample means that would be obtained by drawing from the population
all possible samples of the same size.

Central Limit Theorem
• No matter what we are measuring, the distribution of any measure
across all possible samples we could take approximates a normal
distribution, as long as the number of cases in each sample is about 30 or
larger.
• If we repeatedly drew samples from a population and calculated the
mean of a variable or a percentage or, those sample means or
percentages would be normally distributed.
• It enables us to calculate Standard error from a single sample

Section 4:
Percentiles

Percentiles
• Value below which a percentage of data falls.
• For example: 80% of people are shorter than you, That means you are at
the 80th percentile. If your height is 1.85m then "1.85m" is the 80th
percentile height in that group.

Percentiles

Percentiles
• Quantiles are cutpointsdividing the range of a probability distribution
into contiguous intervals with equal probabilities
• Median, tertiles, quartiles, quintiles, sextiles, septiles, octiles, deciles,
percentiles or centiles
• Inter-quartile range

Take home messages
• Understanding the distributions
lets us understand the inferential
statistics better

Thank you!
Email your queries to sarizwan1986@outlook.com

Probability distributions, sampling distributions and central limit theorem

More Related Content

What's hot

Similar to Probability distributions, sampling distributions and central limit theorem

More from Rizwan S A

Recently uploaded

Probability distributions, sampling distributions and central limit theorem