Basic Probability Distribution Sampling Distribution in Research

1. P R O B A B I L I T Y
D I S T R I B U T I O N
B A S I C S

2. Sreeraj S R
WEATHER FORECAST
Sunny: 80%
Rain: 10%
Snow: 10%
Probability?
Probability is the likelihood that any one event will occur, given
all the possible outcomes.
100%
Portney L (2015)

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INTRODUCTION TO PROBABILITY
• Probability is similar to the idea of Percentage.
• For example 50% ~ 0.5 probability
• Probabilities are numbers between 0 and 1
• Where 0 is Impossible event and 1 is Sure event. So
• ∑p = 1 (Sum of all probabilities is 1)
• Example: If Probability of a baby born as girl is 0.6 what will
be the probability of boy?
• Answer: 0.4

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• Arithmetically, we can calculate the probability (p) or chances
of occurrence of a positive event by the formula:
• p =
Number of events occurring
Total number of events
• Example: If a surgeon transplants kidney in 200 cases and
succeeds in 80 cases then probability of survival after
operation is calculated as:
• p =
80
200
= 0.4
Khanal AB. Chapter 7, Probability (Chance).

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• Example: What will be the probability of not surviving or
dying in the same case?
• q =
Number of events occurring
Total number of events
• q =
120
200
= 0.6
Khanal AB. (Chance).

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TERMINOLOGY
• Experiment: activity with observable result. Ex: flipping a
coin 3 times
• Trials: repetition of experiment. Ex: Every flip of a coin
• Outcome: result of each trial.
• Sample space: set of all possible outcome
• Sample points: elements of sample space
• Events: subset of the sample space.
• Example: Getting exactly 2 heads
Sample point
Sample point
Sample
Space
Events
HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.
EventsEvents

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UNIONS & INTERSECTIONS
1. Union
• Outcomes in either events A or B or both
• ‘OR’ statement denoted by ‘∪’ symbol (i.e., A ∪ B)
2. Intersection
• Outcomes in both events A and B
• ‘AND’ statement denoted by ‘∩’ symbol (i.e., A ∩ B)
3. Given
• P(B given that A has occurred).
• There is a special notation for this: P(B | A).

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DISCRETE & CONTINUOUS VARIABLES
• Discrete random variables have a countable number of outcomes.
• Example, when you roll a die, the possible outcomes are 1, 2, 3,
4, 5 or 6 and not 1.5 or 2.45.
• Binomial distribution and the Poisson distribution are discrete
probability distributions.
• Continuous random variables have an infinite continuum of
possible values.
• Example, The weight of a girl can be any value from 54 kg, or
54.5 kg, or 54.5436 kg.

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LAWS OF PROBABILITY
• Probability provides the basis for all the tests of significance.
• It is estimated usually on the basis of following five laws of
probability,
1. Addition law of probability
2. Multiplication law of probability
3. Binomial law of probability distribution
4. Probability (chances) from shape of normal distribution or
normal curve
Khanal AB. Chapter 7, Probability (Chance).

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ADDITION LAW OF PROBABILITY
• Mutually Exclusive
• If one event excludes the probability of occurrence of the other
specified event or events, the events are called mutually exclusive.
• If two events are mutually exclusive, then the probability of either
occurring is the sum of the probabilities of each occurring. i.e.
• P (A or B) = P (A) + P (B)
• Example: Chances of Odd or Even number
Khanal AB. Chapter 7, Probability (Chance).

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ADDITION LAW OF PROBABILITY
MUTUALLY EXCLUSIVE
• Example: King OR Queen
• In a Deck of 52 Cards:
• the probability of a King is 1/13, so P(King)=1/13
• the probability of a Queen is also 1/13, so P(Queen)=1/13
• When we combine these two Events:
• The probability of a King or a Queen is (1/13) + (1/13) = 2/13
• Which is written like this:
• P(King or Queen) = (1/13) + (1/13) = 2/13

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ADDITION LAW OF PROBABILITY
• Two non-mutually exclusive events
• Two or more events can occur simultaneously i.e. the occurrence
of one does not prevent the occurrence of the other
• In events which aren't mutually exclusive, there is some overlap.
• i.e. P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Example: Chances of an Odd number which is less than 4

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• Example: Hearts and Kings

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MULTIPLICATION LAW OF PROBABILITY
• This law is applied to two or more events occurring together
but they must not be associated, i.e. must be independent of
each other.
• According to the multiplication law of probability:
“The probability of occurring of two independent events is
equal to the product of the probabilities of those events
separately.”
• P(A and B)= P (A) X P (B | A)

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MULTIPLICATION LAW OF PROBABILITY
• Example: What is the probability of randomly drawing a card
from the deck that is an Ace AND a Heart?
• P(A and B) = P(A) X P(B|A)
• P(ace and heart) = P(ace) X P(heart | ace)
= (4/52) X (1/4)
= 1/52

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BINOMIAL LAW OF PROBABILITY DISTRIBUTION
• A binomial distribution can be thought of as the probability
of a SUCCESS or FAILURE outcome in an experiment or
survey that is repeated multiple times.
• Probability of success is p; probability of failure is 1 – p
• Example: The four possible outcomes that could occur if you
flip a coin twice are listed below in Table.
Outcome First Flip Second Flip
1 Heads Heads
2 Heads Tails
3 Tails Heads
4 Tails Tails
Number of Heads Probability
0 1/4
1 1/2
2 1/4
http://onlinestatbook.com/2/probability/binomial.html

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POISSON DISTRIBUTION
• It is a discrete probability distribution of the number of events
occurring in a given time period, given the average number of
times the event occurs over that time period.
• Example:
• A certain fast-food restaurant gets an average of 3 visitors to
the drive-through per minute. This is just an average,
however. The actual amount can vary.
http://onlinestatbook.com/2/probability/poisson.html
https://brilliant.org/wiki/poisson-distribution/

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POISSON DISTRIBUTION
• Conditions for Poisson Distribution:
1. An event can occur any number of times during a time
period.
2. Events occur independently.
3. The probability of an event occurring is proportional to the
length of the time period.

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POISSON DISTRIBUTION
• Example: Number of phone calls an office would receive during
the noon hour are average 4 calls per hour.
1. Although the average is 4 calls, they could theoretically get any
number of calls during that time period.
2. The events are effectively independent since there is no reason
to expect a caller to affect the chances of another person calling.
3. The probability of getting a call in the first half hour is the same
as the probability of getting a call in the final half hour.
https://brilliant.org/wiki/poisson-distribution/

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NORMAL DISTRIBUTION
OR NORMAL CURVE
• The normal probability
model applies when the
distribution of the continuous
outcome conforms
reasonably well to a normal
or Gaussian distribution,
which resembles a bell
shaped curve.

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CHARACTERISTICS OF A NORMAL DISTRIBUTION
• Approximately 68% of the values
fall between the mean and one
standard deviation (in either
direction)
• Approximately 95% of the values
fall between the mean and two
standard deviations (in either
direction)
• Approximately 99.9% of the
values fall between the mean and
three standard deviations (in
either direction)

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EXAMPLE
• In a data for height of all adult men
alive today, the mean height was 69
in., with a SD of 3 in. What is the
probability that the man will be
between 66 and 72 in. tall, or within
± 1 standard deviation of the mean?
• There is a 68% probability (p = .68)
that any one man we select will fall
within this range.
• Similarly, the probability of selecting
a man of 78 in. or taller (scores
beyond +3 standard deviations) is
.0013, as this area represents 0.13%
of the total distribution.

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FEATURES OF NORMAL DISTRIBUTIONS
• Seven are listed below:
1. Normal distributions are symmetric around their mean.
2. The mean, median, and mode of a normal distribution are equal.
3. The area under the normal curve is equal to 1.0.
4. Normal distributions are denser in the centre and less dense in the
tails.
5. Normal distributions are defined by two parameters, the mean (μ) and
the standard deviation (σ).
6. 68% of the area of a normal distribution is within one standard
deviation of the mean.
7. Approximately 95% of the area of a normal distribution is within two
standard deviations of the mean.

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NON NORMAL DISTRIBUTION
“Normality is a myth; there never was, and never will be, a
normal distribution”.
—Robert C. Geary, quoted from Testing from Normality

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TYPES OF NON NORMAL DISTRIBUTION
• Beta Distribution.
• Exponential Distribution.
• Gamma Distribution.
• Inverse Gamma Distribution.
• Log Normal Distribution.
• Logistic Distribution.
• Maxwell-Boltzmann Distribution.
• Poisson Distribution.
• Skewed Distribution.
• Symmetric Distribution.
• Uniform Distribution.
• Unimodal Distribution.
• Weibull Distribution

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REASONS FOR THE NON NORMAL DISTRIBUTION
1. Outliers: Try removing any extreme high or low values and testing your data again.
2. Multiple distributions may be combined in your data, giving the appearance of
a bimodal or multimodal distribution. For example, two sets of normally distributed
test results are combined in a graph.
3. Insufficient Data can cause a normal distribution to look completely scattered.
• An extreme example: if you choose three random students and plot the results on a
graph, you won’t get a normal distribution. You might get a uniform distribution (i.e.
62 62 63) or you might get a skewed distribution (80 92 99). If you are in doubt
about whether you have a sufficient sample size, collect more data.
4. Data may be inappropriately graphed.
• For example, if you were to graph people’s weights on a scale of 0 to 1000 lbs, you
would have a skewed cluster to the left of the graph. Make sure you’re graphing
your data on appropriately labeled axes.

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DEALING WITH NON NORMAL DISTRIBUTIONS
• Several tests, including the one sample Z test, T test
and ANOVA assume normality.
• You may still be able to run these tests if your sample size is
large enough.
• You can choose to transform the data with a function, forcing it to
fit a normal model.
• However, if you have a very small sample, a sample that is
skewed or one that naturally fits another distribution type, you may
want to run a non parametric test.
• Non parametric tests include the Wilcoxon signed rank test,
the Mann-Whitney U Test and the Kruskal-Wallis test.

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THE CENTRAL LIMIT THEOREM.
• The Central Limit Theorem states that the sampling
distribution of the sample means approaches a normal
distribution as the sample size gets larger.

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SKEWED DISTRIBUTIONS

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SKEWNESS AND KURTOSIS
• Skewness is a
measure of symmetry,
or more precisely, the
lack of symmetry.
• Kurtosis is a measure
of whether the data are
heavy-tailed or light-
tailed relative to a
normal distribution.

32. Sreeraj S R
REFERENCES
1. Portney L, Watkins M. Chapter 18. Statistical Inference . In:
Foundations of Clinical Research: Applications to Practice.
3rd ed. Philadelphia: F A Davis, 2015. : 405.
2. Khanal AB. Chapter 7, Probability (Chance). In: Khanal A,
ed. Mahajan's Methods In Biostatistics For Medical
Students And Research Workers. 9th ed. New Delhi:
Jaypee Brothers Medical Publishers; 2016:141-163.
3. http://www.dummies.com/education/science/biology/probabi
lity-distributions-in-biostatistics/