How to calculate sample size?
How to calculate sample Size?
Sample design addresses two basic issues; how elements
of the population will be selected & how many elements will
be selected.
• Sampling techniques - Probability and non – probability
sampling
• Estimating sample size - How large sample is required?
How to calculate sample Size?
• Inadequate, or excessive sample sizes influence the quality
and accuracy of research.
• Sample Size refers to the number of items to be selected
from the universe to constitute a sample.
• If the sample size is too small, it may not serve to achieve
the objectives
How to calculate sample Size?
Too large sample size might cause huge cost
• How large or small should be the sample size?
As a general rule, the sample must be of an optimum size
Optimum means the ideal sample provides the most precision
for the least cost
Criteria for sample size
• Population size: Population size is the total number of people in the
group you are interested in, or in this case, trying to reach with your
survey.
• For example, if you were taking a random sample of students in a high
school, your population would be the total number of students attending
that school.
• Similarly, if you wanted a survey on the United States, the size of the
population would be the total number of people in the USA and if you
are surveying your company, the size of the population is the total
number of employees.
Criteria for sample size
• Margin of error: Margin of error, also called the confidence interval, is
a statistical measurement of difference between survey results and the
population value, expressed as a percentage.
• Within the survey ecosystem, the margin of error measures the
difference between your survey results and how accurately they
reflect the views of the overall population.
• The smaller the margin of error, the closer you are to having the
exact answer at a given confidence level.
Criteria for sample size
• Confidence level (not to be confused with confidence interval) is
a measure of how certain you are that your sample accurately
reflects the population, within its margin of error.
• Using the high school example, let’s say you conducted a survey
among a sample of students to see if breakfast is popular
amongst high scholars.
Criteria for sample size
• The survey results conclude that 60% of your respondents like to
eat breakfast every morning. With a 4% margin of error rate and a
95% confidence level, it is safe to say that if the survey was repeated
numerous times, the answers would reflect the same results 95% of
the time(i.e. a 95% confidence level means that you can be 95%
certain the results lie between x and y numbers).
• Common standards used by researchers are 90%, 95%, and 99%.
Criteria for sample size
• Degree of variability refers to the distribution of attributes in
the population.
• The more heterogeneous a population, the larger the sample
size required
• The less variable (more homogeneous) a population, the
smaller the sample size.
Estimating Sample Size
Sample size for infinite population
• In a population with standard deviation σ, with an error e, &
a confidence
corresponding to z, sample size, n, equals:
where;
n = size of sample
e = acceptable error (the precision)
σ = standard deviation of population
z = number relating to the degree of confidence you wish to have in the
result.
Sample Size for finite population
• For finite population, sample size is given by:
where
N = size of population
n = size of sample
e = acceptable error (the precision)
σp = standard deviation of population
z = number relating to the degree of confidence you wish to
have in the result.
Example:
• Determine the sample size for estimating the true weight of the
cereal containers for the universe with N = 5000 on the basis of
the following information:
• The variance of weight is 4 ounces on the basis of past records.
• Estimate should be within 0.8 ounces of the true average weight
with 99% probability.
will there be a change in the size of the sample if we assume
infinite population in the given case?
Solution
N = 5000;
σp = 2 ounces (since the variance of weight = 4 ounces);
e = 0.8 ounces (since the estimate should be within 0.8 ounces of the true
average weight);
z = 2.57 (as per the table of area under normal curve for at 99% confidence
level ).
Solution
N = 5000;
σp = 2 ounces (since the variance of weight = 4 ounces);
e = 0.8 ounces (since the estimate should be within 0.8 ounces of the true
average weight);
z = 2.57 (as per the table of area under normal curve for at 99% confidence
level ).
(ii) infinite population
(2.57)2 (2) 2 26.4196
n=
=
= 41.28 41
2
(0.8)
0.64
Thus in the given case the sample size remains the same
even if we assume infinite population
Example:
The government would like to estimate the mean family size for all families in a
particular state at a 99% confidence level. It is known that the standard
deviation for all sizes of all families in that state is 0.7.How large a sample
should the government select if it wants its estimate to be within 1% of the
population mean?
Solution:
Dr. Monika Saini, Department of Mathematics & Statistics,
Manipal University Jaipur
Example:
The government would like to estimate the mean family size for all families in a
particular state at a 99% confidence level. It is known that the standard
deviation for all sizes of all families in that state is 0.7.How large a sample
should the government select if it wants its estimate to be within 1% of the
population mean?
Solution:
e = 0.01, Z = 2.57,  = 0.7
2
2
(2.57) (0.7)
n=
= 32, 616
2
(0.01)
Dr. Monika Saini, Department of Mathematics & Statistics,
Manipal University Jaipur
Solvin’s Formula
Solvin’s formula is used when nothing about the behavior of the
population is known at all.
Where n is the sample size, N is the population size, and e is the
level of precision:
Example:
Suppose our evaluation of farmers' adoption of the new practice only
affected 2,000 farmers allowing for a 5% margin or error.
.
The sample size:
Dr. Monika Saini, Department of Mathematics & Statistics,
Manipal University Jaipur
Example:
Suppose our evaluation of farmers' adoption of the new practice only
affected 2,000 farmers allowing for a 5% margin or error.
.
The sample size:
Dr. Monika Saini, Department of Mathematics & Statistics,
Manipal University Jaipur
Example:
Calculate the survey size for a population of 240, allowing for a
4% margin or error.
Hint: N=240, e=0.04
Answer: n=173