Data can be "distributed"
(spread out) in different ways.
SAMPLE SIZE
DETERMINATION
Or more on the right
It can be spread out more on the left
Or it can be jumbled
But there are many cases where the data
tends to be around a central value with
no bias left or right, and it gets close to a
"Normal Distribution" like this:
The Normal Distribution has:
• mean = median = mode
• symmetry about the center
• 50% of values less than the mean
and 50% greater than the mean
Standard Deviations
Example: 95% of students at school are between 1.1m and 1.7m tall.
Assuming this data is normally distributed can
you calculate the mean and standard deviation?
• The mean is halfway between 1.1m and 1.7m:
Mean = (1.1m + 1.7m) / 2 = 1.4m
95% is 2 standard deviations either side of the
mean (a total of 4 standard deviations) so:
1 standard deviation
= (1.7m-1.1m) / 4 = 0.6m / 4 = 0.15m
It is good to know the standard deviation,
because we can say that any value is:
• likely to be within 1 standard deviation (68
out of 100 should be)
• very likely to be within 2 standard deviations
(95 out of 100 should be)
• almost certainly within 3 standard deviations
(99.7 out of 100 should be)
Standard Deviations
So to convert a value to a Standard Score ("z’
score"):
first subtract the mean, then divide by the
Standard Deviation and doing that is called
"Standardizing":
SAMPLING DISTRIBUTION OF MEAN
According to the central limit theorem, the various
arithmetic means of a large number of random
samples of the same size will form normal
distribution curve. If an arithmetic mean of all
possible sample mean will be calculated, it will
coincide with the population mean.
• Calculation of mean
• Calculation of standard deviation
Sample Frequency Deviation
mean from d/0.5 fd d2 fd2
assumed
mean (d)
• Calculation of mean
• Calculation of standard deviation
Standard error
IMPORTANT TERMS
• Parameter: A parameter is a summary description of
fixed characteristics or measure of target population. A
parameter denotes the true value if a census rather
than a sample is undertaken.
• Statistic: It is a summary description of a
characteristics or measure of the sample. The sample
statistic is used as an estimate of the population
parameter.
• Finite Population correction (fpc): It is a correction for
overestimation of a population parameter, e.g., a mean
or proportion, when the sample size is 5% or more of the
population size.
Sample Size Determination
Major consideration for sample size
determination
• The extent of error or imprecision allowed
• The degree of confidence desired in the
estimate
• Estimate of the standard deviation of the
population
The formula to determine sample size is
Where, Z= Confidence Level
E= maximum error allowed
σ = standard deviation
DETERMINE THE SAMPLE SIZE WHEN
STANDARD DEVIATION IS KNOWN
You have been asked to select a simple random
sample from a population of 2,00,000 sales invoices
to estimate the average amount per invoices.
Suppose that the standard deviation of the
population is Rs. 500. Determine the sample size
required if allowable error is Rs. 100 and the
confidence coefficient is 95.4%.
DETERMINE THE SAMPLE SIZE WHEN
STANDARD DEVIATION IS KNOWN
You have been asked to select a simple random
sample from a population of 10,00,000 sales
invoices to estimate the average amount per
invoices. Suppose that the standard deviation of the
population is Rs. 1000. Determine the sample size
required if allowable error is Rs. 200 and the
confidence coefficient is 95.4%.
STRATIFIED SAMPLE SIZE
Determine the proportionate and disproportionate
stratified sample size to conduct the market research
project for luxury retailer and select the better choice.
Income stratum Standard Size of Stratum
Deviation
Affluent 40 6000
Rich 60 3000
HNI’s 70 1000
Assume a tolerable error of 10; Keep confidence level
95.4% (Z = 2.0)
Proportionate Stratified Random
Sampling
Proportionate stratified random
sampling
Income Standard Size of Stratum Weight Sample size
stratum Deviation from stratum
Affluent 40 6000
Rich 60 3000
HNI’s 70 1000
Disproportionate Stratified Random
Sampling
Table for DPRS
i Ni
Afflu 40
6000
ent
Rich 3000 60
HNI’s 1000 70
Comparison
Standard error for PSRS Standard Error for DSRS
STRATIFIED SAMPLE SIZE PROBLEM
2
Determine the proportionate and disproportionate
stratified sample size to conduct the market research
project for motorcycle manufacturing company for
perceptual mapping of the brand.
Age stratum Standard Size of Stratum
Deviation
Upto 25 years 4 years 50 million
25 to 40 years 6 years 40 million
40+ 8 years 30 million
Assume a tolerable error of 3; Keep confidence level
Prof. Punit Kumar Mishra
