Sample Size Formula
Last Updated :
02 Jan, 2024
In the field of statistics, the sample size is defined as the number of observations used to calculate population estimates for a specific population. In other words, it refers to the number of individual samples utilized in a data study. It uses the difference between the population and the sample to calculate the correct sample size. It is the process of selecting a group of people from a population to estimate the characteristics of the entire population, known as sampling. It is denoted by the symbol n.
Sample Size Formula
For a smaller sample size, the concept of the T distribution is used in place of normal distribution. Specifically, this distribution is used when the value of the sample size is less than 30. In this test, we utilize the t statistic to test the null hypothesis using both one-tailed and two-tailed tests if the population variance is unknown and the sample size is small. It is also known as adjusted sample size.
A = n / (1 + (n – 1)/P)
Where,
- A is the adjusted sample size,
- n is the sample size,
- P is the population size.
For infinite population size, the formula is expressed in terms of z-value and error margin.
n = Z2p(1 – p)/m2
Where,
- n is the sample size,
- Z is the z-value,
- p is the proportion of population (generally taken as 0.5),
- m is the margin of error.
Sample Problems
Problem 1: Calculate the adjusted sample size for a sample size of 300 and a population of 50000.
Solution:
We have,
n = 300
P = 50000
Using the formula we have,
A = n / (1 + (n – 1)/P)
= 300 / (1 + 299/50000)
= 300/1.00598
= 298.216
Problem 2: Calculate the adjusted sample size for a sample size of 100 and a population of 25000.
Solution:
We have,
n = 100
P = 25000
Using the formula we have,
A = n / (1 + (n – 1)/P)
= 100 / (1 + 299/25000)
= 100/1.001196
= 99.88
Problem 3: Calculate the adjusted sample size for a sample size of 76 and a population of 2000.
Solution:
We have,
n = 76
P = 2000
Using the formula we have,
A = n / (1 + (n – 1)/P)
= 76 / (1 +75/2000)
= 76/1.0375
= 73.25
Problem 4: Calculate the population size if the adjusted sample size is 102.2 for a sample size of 104.
Solution:
We have,
A = 102.2
n = 104
Using the formula we have,
A = n / (1 + (n – 1)/P)
=> 102.2 = 104 / (1 + 103/P)
=> 1 + 103/P = 1.01
=> 103/P = 0.01
=> P = 10300
Problem 5: Calculate the sample size for z-value as 1.5 and the margin of error as 4.2%.
Solution:
We have,
z = 1.5
m = 4.2% = 0.042
p = 0.5
Using the formula we have,
n = Z2p(1 – p)/m2
= (1.5)2 × 0.5 × (1 – 0.5)/(0.042)2
= 0.5625/0.001764
= 318.87
Problem 6: Calculate the sample size for z-value as 1.2 and the margin of error as 3.5%.
Solution:
We have,
z = 1.2
m = 3.5% = 0.035
p = 0.5
Using the formula we have,
n = Z2p(1 – p)/m2
= (1.2)2 × 0.5 × (1 – 0.5)/(0.035)2
= 0.36/0.001225
= 293.877
Problem 7: Calculate the z-value if the sample size is 250 and the margin of error is 3.2%.
Solution:
We have,
n = 250
m = 3.2% = 0.032
p = 0.5
Using the formula we have,
n = Z2p(1 – p)/m2
=> Z2 = nm2/(p(1 – p))
=> Z2 = 250 × (0.032)2 / (0.5 × 0.5)
=> Z2 = 0.256/0.025
=> Z2 = 10.24
=> Z = 3.2
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