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Sampling distribution by Dr. Ruchi Jain
1. Frequency distribution
Dr. Ruchi jain
Associate professor
Department of Commerce-Financial Studies
Iis (deemed to be university) Jaipur
2. • The probability distribution of a random variable may be:
Theoretical listing of outcomes and probabilities which can be obtained
from a mathematical model representing some phenomenon of interest.
An empirical listing of outcomes and their observed relative frequencies.
3. • The observed frequency distributions are based on observation and
experimentation.
• It can be obtained by grouping data. They help in understanding properly the
nature of data.
• For exp a shoe maker must know something about the distribution of sizes
of customers' feet, similarly an owner of an restaurant must know people’s
like and dislike .
4. Frequency distribution
• When it is possible to deduce mathematically what the frequency
distributions of certain populations should be.
• Such distributions as are expected on the basis of previous experience or
theoretical considerations are known as theoretical distributions or
probability distributions.
• It is for a discrete random variable is a mutually exclusive listing of all
possible numerical outcomes for that random variable such that a particular
probability of occurrence is associated with each outcome.
5. Random Variable-Discrete or Continuous
• A random variable can be either discrete or continuous.
• A random variable is said to be discrete if the set of values defined by it over
the sample space is finite.
• A random variable is continuous if it can assume any real value in an interval.
• If random variable is Discrete---its distribution as discrete Probability
distribution.
• If random variable is Continuous-its distribution as continuous probability
distribution.
6. • Knowledge of the expected behavior of a phenomenon or, in other words,
the expected frequency distribution is of great help in a large number of
problems in practical life.
• They serve as benchmarks against which to compare observed distributions
and act as substitutes for actual distributions when the latter are closely to
obtain or cannot be obtained at all.
• They provide decision-makers with a logical basis for making decision and
are useful in making predictions on the basis of limited information or
theoretical considerations.
7. • For example: the proprietor of a shoe store must know something about the
distribution of the size of his potential customers’ feet;
• The manufacturer of ready-made garments must know the size of collars for which
he expects maximum demand so that the has no stock of unwanted sizes.
• Teachers in the school, college or university should know what they expect of the
students.
• It is only then that they would be in a position to comment on good or bad
performance.
8. Types of Frequency distributions
• Binomial Distribution
• Poisson distribution
• Normal distribution
9. Binomial Distribution
• Also known as Bernoulli distribution.
• Associated with the name of a Swiss mathematician James Bernoulli.
• Binomial distribution is a probability distribution expressing the probability
of one set of dichotomous alternatives, i.e. success or failure.
• It is discrete probability distribution.
• Used in business and social sciences
10. Binomial distribution can be used when:
The outcome or result of each trial in the process are characterized as one of
two types of possible outcomes. In other words , they are attributes.
The possibility of outcome of any trial does not change and is independent
of the results of previous trials.
11. Assumptions
1. An experiment is performed under the same conditions for a fixed number of trials; n
2. In each trial , there are only two possible outcomes of the experiment. Success or Failure.
3. The probability of a success denoted by p remains constant from trial to trial.
4. The probability of failure denoted by q is equal to 1-p.
5. If probability of success is not the same in each trial we will not have binomial
distribution.(exp. Balls should be replaced before drawing again in second trial)
6. The trials are statistically independent. The outcomes of any trial do not affect the
outcomes of subsequent trials
12. • The binomial distribution:
• P(r )= 𝑟=0
𝑛 𝑛
𝐶
𝑟
𝑝 𝑟 𝑞 𝑛−𝑟
• P= probability of success in a single trial
• q= 1-p
• n = number of trials
• r=number of success in n trials
13. • If we want to obtain the probable frequencies of the various outcomes in N
sets on n trials, the following expression shall be used:
• N(q+p)n= N[qn +nC1 qn-1.p+ nc2qn-2 p2+…+ nCr qn-rpr+…+pn]
• Mean of the binomial distribution= np
• Standard deviation= 𝑛𝑝𝑞
14. Poisson Distribution
• Poisson distribution may be expected in cases where the chance of any individual event
being a success is very small. The distribution is used to describe the behavior of rare
events . Such as the number of accidents on road, number of printing mistakes in a book
etc.
• It has been called the law of improbable events.
• Exp serious floods, accidental release of radiation in nuclear sector etc.
• It is discrete distribution.
• Poisson distribution is applicable where the successful event in the total events are few.
• When n is very large n----∞ and the value of p is too small--0 and np is a finite number
, here Poisson distribution is suitable not binomial.
15. λ or m =mean of poisson distribution i.e np or the average number of occurrence of any
event
16. Poisson Distribution
All Poisson
distributions are
skewed to right
• That’s why they are
called the
distribution of rare
events
17. Constants of poisson distribution
• Since p is very small in case of pisson distribution so the value of q is almost
equal to 1. we can put q=1 in case of binomial distribution.
The mean of Poisson distribution is m
The standard deviation is σ = 𝑚 or variance = m
The rule most often is that the poisson is a good approximation of the
binomial when n is equal to or greater than 20 and p is equal to or less than
0.05, in case that meet these conditions, we can substitute (np) in place of
mean of the poisson distribution(m).
18. Role of Poisson distribution
• It is applicable where there are infrequently occurring events with respect to
time,area,volume or similar units.
• Quality control-no. of defective units
• Biology---no. of bacteria count
• Physics---no. of particles emitted from radioactive substance
• Insurance----no. of casualities
• Waiting line/queuing----no. of incoming telephone calls or incoming customers
• No. of traffic arrivals such as trucks at terminal, aeroplanes at airports, ship or docks.
• Deaths in a district/ year
19. Normal Distribution
• Also called Normal Probability distribution, most useful for theoretical
distribution for CONTINUOUS variables.
• Rediscovered by Gauss , hence also called Gaussian distribituion.
20. • The Normal distribution is an approximation to binomial distribution.
• When n - ∞ (n becomes very large)
• p is equal to q
21. Importance of Normal Distribution
• Central Limit Theorem: Acc. To this theorem as the sample size n increases the
distribution of mean , 𝑋 of a random sample taken from practically any population
approaches a normal distribution(with mean μ and standard deviation .
• Thus if a sample of large size n are drawn from a population that is not normally
distributed ,nevertheless the successive sample mean; will form themselves a
distribution that is approximately normal.
• As the size of sample is increased the sample means will tend to be normally
distributed.
22. Basic form of Normal distribution
• Basic form relating to a curve with mean μ and standard deviation σ
24. Properties of Normal Distribution
• Normal distribution isa limiting case of binomial distribution when
1. n----∞
2. Neither p nor q is very small.
• The mean of a normally distributed population lies at the center of its normal
curve.
• The two tails of the normal probability distribution extend indefinitely and never
touch the horizontal axis(which implies a positive probability for finding values of
the random variable within any range from minus infinity to plus infinity)
25. • Bell shaped and symmetrical in its appearance.
• Height of the normal curve is at its maximum at the mean, so mode is equal
to mean. if the number of cases below the mean in a normal distribution is
equal to the number of cases above the mean ,which means the mean
median coincide.
• One maximum point is mean, the normal curve is unimodal i.e. it has only
one mode.
