Objectives: Distinguish between the distribution of population and distribution of its sample means Explain the importance of central limit theorem Compute and interpret the standard error of the mean. Sampling distribution of sample mean: A population is a collection or a set of measurements of interest to the researcher. For example a researcher may be interested in studying the income of households in Karachi. The measurement of interest is income of each household in Karachi and the population is a list of all households in Karachi and their incomes. Any subset of the population is called a sample from the population. A sample of ‘n’ measurements selected from a population is said to be a random sample if every different sample of size ‘n’ from the population is equally likelyto be selected. For the purpose of estimation of certain characteristics in the population we would like to select a random sample to be a good representative of the population. The set of measurements in the population may be summarized by a descriptive characteristic, called a parameter. In the above example the average income of households would be the parameter. The set of measurements in a sample may be summarized by a descriptive statistic, called a statistic . For example to estimate the average household income in Karachi, we take a random sample of the population in Karachi. The sample mean is a statistic and is an estimate of the population mean. Because no one sample is exactly like the next , the sample mean will vary from sample to sample ,and hence is itself a random variable. Random variables have distribution ,and since the sample mean is a random variable it must have a distribution. If the sample mean has a normal distribution ,we can compute probabilities for specific events using the properties of the normal distribution. Consider the population with population mean = μ and standard deviation = σ. Next, we take many samples of size n, calculate the mean for each one of them, and create a distribution of the sample means. This distribution is called the Sampling Distribution of Means. Technically, a sampling distribution of a statistic is the distribution of values of the statistic in all possible samples of the same size from the same population. Standard error of the mean: The quantity σ is referred to as the standard deviation .it is a measure of spread in the population . The quality σ/n is referred to as the standard error of the sample mean .It is a measure of spread in the distribution of mean A very important result of statistics referring to the sampling distribution of the sample mean is the Central Limit Theorem . Central Limit Theorem: Consider a population with finite mean  and standard deviation . If random samples of n measurements are repeatedly drawn from the population then, when n is large, the relative frequency histogram for the sample means ( calculated from repeated samples)

2. Sampling Distribution of
Sample Mean
Shakir Rahman
BScN, MScN, MSc Applied Psychology, PhD Nursing (Candidate)
Principal & Assistant Professor
Ayub International College of Nursing & AHS Peshawar
Visiting Faculty
Swabi College of Nursing & Health Sciences Swabi
Nowshera College of Nursing & Health Sciences Nowshera

3. Objectives
By the end of this session the students should be
able to:
• Distinguish between the distribution of
population and distribution of its sample
means
• Explain the importance of central limit
theorem
• Compute and interpret the standard error of the
mean
2

4. Sampling distribution of
sample mean
A population is a collection or a set of
measurements of interest to the researcher.
For example a researcher may be interestedin
studying the income of households inKarachi.
The measurement of interest is income ofeach
household in Karachi and the population is a
list of all households in Karachi and their
incomes.

5. Any subset of the population is called a sample
from the population. A sample of ‘n’
measurements selected from a population is said
to be a random sample if every different sample
of size ‘n’ from the population is equallylikelyto
be selected.
For the purpose of estimation of certain characteristics in
the population we would like to select a random sample tobe
a good representative of the population
Sampling distribution of
sample mean

6. Population and Sample
Measures
The set of measurements in the population may be
summarized by a descriptive characteristic, called a
parameter. In the above example the average income of
households would be theparameter.
The set of measurements in a sample may be summarized
by a descriptive statistic, called a statistic . For example to
estimate the average household income in Karachi, we take
a random sample of the population in Karachi. The sample
mean is a statistic and is an estimate of the population
mean.

7. Population and Sample
Measures
Parameters:
Mean of the Population =
Standard Deviation of the Population =
Variance of the Population =2
Statistics (sample estimates of theparameters):
Sample estimate of  =x
Sample Estimate of  =s
Sample Estimate of 2= s2

8. Normal Distributions
Byvaryingthe parameters  and , weobtain
different normal distributions
There are an infinite number ofnormal distributions

9. Sampling distribution of the mean
 Because no one sample is exactly like the next , the
sample mean will vary from sample to sample ,and
hence is itself a randomvariable.
 Random variables have distribution ,and sincethe
sample mean is a random variable it must have a
distribution.
 If the sample mean has a normal distribution ,we can
compute probabilities for specific events using the
properties of the normal distribution.

10. Population and Samples
Population
Sample 1 Sample 2 Sample 3 Sample 4
X1 X2
X3 X4
X8 X9
X10 X11
X19
X20
X21
X22 X28 X29
X30 X31

12. Sampling Distribution
Consider the population with population mean = μ
and standard deviation = σ.
Next, we take many samples of size n, calculate the
mean for each one of them, and create a distribution
of the sample means.
This distribution is called the Sampling Distribution of
Means.
Technically, a sampling distribution of a statistic is the
distribution of values of the statistic in all possible
samples of the same size from the same population.

13. EXAMPLE 1
The sample mean X, is to calculated from a random
sample of size 2 taken from a population consistingof
the five values ( $2, $3, $4, $5, $6). Find the sampling
distribution x, of based on a sample of size 2.
First note that the population mean,  is
 = 2 + 3 + 4 + 5 + 6 = 4
5

14. EXAMPLE 1
Mean ofX values =
2.5 + 3 + 3.5 + 4 + 3.5 + 4 + 4.5+ 4.5 + 5 +5.5
10
Possible
Samplesof
size 2
ValueofX
2,3 2.5
2,4 3 = 4
2,5 3.5
2,6 4
3,4 3.5
3,5 4
3,6 4.5
4,5 4.5
4,6 5
5,6 5.5

15. Sampling Distribution of x
X P (X)
2.5 1/10
3 1/10
3.5 2/10
4 2/10
4.5 2/10
5 1/10
5.5 1/10

16. Sampling distribution of sample mean
Now through the above example we have learnt that the
sample mean X has a nice mathematical property that is if
you average all possible sample means which are obtained
through repeating the experiment a number of times,you
will obtain the population mean,.
But the variance among the sample means obtained
through repeated sampling, is related to thepopulation
variance through the followingformula
Standard deviation ofX = X =  / n
which will be estimated using the samplestandard
deviation:
Standard error ofX = S / n

17. Standard error of the
mean
The quantity σ is referred to as the standard deviation .it is
a measure of spread in the population .
The quality σ/n is referred to as the standard error ofthe
sample mean .It is a measure of spread in the distribution
of mean
A very important result of statistics referring to the
sampling distribution of the sample mean is theCentral
Limit Theorem .

22. Central Limit Theorem
Consider a population with finite mean  and standard
deviation . If random samples of n measurements are
repeatedly drawn from the population then, when n is large, the
relative frequency histogram for the sample means ( calculated
from repeated samples) will be approximately normal (bell-
shaped) with mean  and standard deviation / n .

24. Transforming Normal to
Standard Normal
Distributions
Transforming Normal to StandardNormal
Distributions
x
z

 x   Note that now weuse
x and sigma for the
p.d.f. of x.

25. The Normal Distribution Family

i
z 
x i  
z  x  
 / n

26. Suppose hemoglobin level in adults isapproximately
normally distributed with mean 12.7 and standard
deviation 2.8
– A) What proportion of adults would you expect to haveHB
level between 10 & 13.

z  x -  10-12.7  -2..7
2.8

z  x -   13-12.7 = 0.3
2.8
Example

29. Answer
Z -2.7 = 0.0035
Z 0.3 = 0.1179
Area between -2.7 and 0.3 =
0.0035+0.1179= 0.1214 x 100 =12.14 %
12.14 % of adults would expect to have
hemoglobin level between 10 & 13.

30. Example
Suppose hemoglobin level in adults is
approximately normally distributed with mean
12.7 and standard deviation 2.8
– If random sample of 16 adults taken fromthe
above population, then obtain thefollowing:
The mean and standard error of sampling distribution of
sample mean

31. Mean and Standard error
n=16
_
S.E(X) = ?
_
X=?
The expected of sample
mean is equal to the
population mean
_
S.E(X) ==12.7

32. Example
 Suppose hemoglobin level in a random sample
of 16 adults is approximately normally
distributed with mean 12.7 and standard
deviation 2.8
oWhat proportion of adults would you expect to
have HB level between 10 &13.
– We will have same result as for population ora
different inference.

33. Example
z  x  
 / n
Hemoglobin level between 10 to 13 will be:
Z = 10-12.7/ 0.7 = -3.85
Z -3.85= .00006
Z = 13-12.7/ 0.7 = 0.42
Z 0.42= 0.1628
Adding two probabilities 0.00006+0.1628 = 0.16286
16.28% of individuals will have Hb level between 10 & 13

35. Difference between
Normal distribution
Why is the normal distributionso
important in the study of
statistics?
It’s not because things in nature
are always normally distributed
(although sometimes they are)
It’s because of the central limit
theorem—the sampling
distribution of statistics (like a
sample mean) often follows a
normal distribution if the sample
sizes are large
Samplingdistribution
Why is sampling distribution
important?
If a sampling distribution has a lot
of variability then if you took
another sample, it’s likely you
would get a very different result

36. Acknowledgements
Dr Tazeen Saeed Ali
RM, RM, BScN, MSc ( Epidemiology & Biostatistics),
Phd (Medical Sciences), Post Doctorate (Health Policy
& Planning)
Associate Dean School of Nursing & Midwifery
The Aga Khan University Karachi.
Kiran Ramzan Ali Lalani
BScN, MSc Epidemiology & Biostatistics (NICU)
Aga Khan University Hospital