How to Calculate Sample Size

Last Updated: March 28, 2023 References Approved

Scientific studies often rely on surveys distributed among a sample of some total population. Your sample will need to include a certain number of people, however, if you want it to accurately reflect the conditions of the overall population it's meant to represent. To calculate your necessary sample size, you'll need to determine several set values and plug them into an appropriate formula.

Part 1

Part 1 of 4:

Part One: Determining Key Values

1
Know your population size. Population size refers to the total number of people within your demographic. For larger studies, you can use an approximated value instead of the precise number.
- Precision has a greater statistical impact when you work with a smaller group. For instance, if you wish to perform a survey among members of a local organization or employees of a small business, the population size should be accurate within a dozen or so people.^{[1]
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- Larger surveys allow for a greater deviance in the actual population. For example, if your demographic includes everyone living in the United States, you could estimate the size to roughly 320 million people, even though the actual value may vary by hundreds of thousands.
2
Determine your margin of error. Margin of error, also referred to as "confidence interval," refers to the amount of error you wish to allow in your results.^{[2]
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- The margin of error is a percentage the indicates how close your sample results will be to the true value of the overall population discussed in your study.
- Smaller margin of errors will result in more accurate answers, but choosing a smaller margin of error will also require a larger sample.
- When the results of a survey are presented, the margin of error usually appears as a plus or minus percentage. For example: "35% of people agree with option A, with a margin of error of +/- 5%"
  - In this example, the margin of error essentially indicates that, if the entire population were asked the same poll question, you are “confident” that somewhere between 30% (35 - 5) and 40% (35 + 5) would agree with option A.
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3
Set your confidence level. Confidence level is closely related to confidence interval (margin of error). This value measures your degree of certainty regarding how well a sample represents the overall population within your chosen margin of error.^{[3]
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- In other words, choosing a confidence level of 95% allows you to claim that you 95% certain that your results accurately fall within your chosen margin of error.
- A larger confidence level indicates a greater degree of accuracy, but it will also require a larger sample. The most common confidence levels are 90% confident, 95% confident, and 99% confident.
- Setting a confidence level of 95% for the example stated in the margin of error step would mean that you are 95% certain that 30% to 40% of the total concerned population would agree with option A of your survey.
4
Specify your standard of deviation. The standard of deviation indicates how much variation you expect among your responses.
- Extreme answers are more likely to be accurate than moderate results.
  - Plainly stated, if 99% of your survey responses answer "Yes" and only 1% answer "No," the sample probably represents the overall population very accurately.
  - On the other hand, if 45% answer "Yes" and 55% answer "No," there is a greater chance of error.
- Since this value is difficult to determine you give the actual survey, most researchers set this value at 0.5 (50%). This is the worst case scenario percentage, so sticking with this value will guarantee that your calculated sample size is large enough to accurately represent the overall population within your confidence interval and confidence level.
5
Find your Z-score. The Z-score is a constant value automatically set based on your confidence level. It indicates the "standard normal score," or the number of standard deviations between any selected value and the average/mean of the population.
- You can calculate z-scores by hand, look for an online calculator, or find your z-score on a z-score table. Each of these methods can be fairly complex, however.
- Since confidence levels are fairly standardized, most researchers simply memorize the necessary z-score for the most common confidence levels:
  - 80% confidence => 1.28 z-score
  - 85% confidence => 1.44 z-score
  - 90% confidence => 1.65 z-score
  - 95% confidence => 1.96 z-score
  - 99% confidence => 2.58 z-score

Part 2

Part 2 of 4:

Part Two: Using the Standard Formula

1
Look at the equation.^{[4]
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Research source} If you have a small to moderate population and know all of the key values, you should use the standard formula. The standard formula for sample size is:
- Sample Size = ^{[z² * p(1-p)] / e²} / _{1 + [z² * p(1-p)] / e² * N}]
  - N = population size
  - z = z-score
  - e = margin of error
  - p = standard of deviation
2
Plug in your values. Replace the variable placeholders with the numerical values that actually apply to your specific survey.
- Example: Determine the ideal survey size for a population size of 425 people. Use a 99% confidence level, a 50% standard of deviation, and a 5% margin of error.
- For 99% confidence, you would have a z-score of 2.58.
- This means that:
  - N = 425
  - z = 2.58
  - e = 0.05
  - p = 0.5
3
Do the math. Solve the equation using the newly inserted numerical values. The solution represents your necessary sample size.
- Example: Sample Size = ^{[z² * p(1-p)] / e²} / _{1 + [z² * p(1-p)] / e² * N]}
  - = ^{[2.58² * 0.5(1-0.5)] / 0.05²} / _{1 + [2.58² * 0.5(1-0.5)] / 0.05² * 425]}
  - = ^{[6.6564 * 0.25] / 0.0025} / _{1 + [6.6564 * 0.25] / 1.0625]}
  - = 665 / 2.5663
  - = 259.39(final answer)

Part 3

Part 3 of 4:

Part Three: Creating a Formula for Unknown or Very Large Populations

1
Examine the formula.^{[5]
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Research source} If you have a very large population or an unknown one, you'll need to use a secondary formula. If you still have values for the remainder of the variables, use the equation:
- Sample Size = [z² * p(1-p)] / e²
  - z = z-score
  - e = margin of error
  - p = standard of deviation
- Note that this equation is merely the top half of the full formula.
2
Plug your values into the equation. Replace each variable placeholder with the numerical values chosen for your survey.
- Example: Determine the necessary survey size for an unknown population with a 90% confidence level, 50% standard of deviation, a 3% margin of error.
- For 90% confidence, use the z-score would be 1.65.
- This means that:
  - z = 1.65
  - e = 0.03
  - p = 0.5
3
Do the math. After plugging you numbers into the formula, solve the equation. Your answer will indicate your necessary sample size.
- Example: Sample Size = [z² * p(1-p)] / e²
  - = [1.65² * 0.5(1-0.5)] / 0.03²
  - = [2.7225 * 0.25] / 0.0009
  - = 0.6806 / 0.0009
  - = 756.22 (final answer)

Part 4

Part 4 of 4:

Part Four: Using Slovin's Formula

1
Look at the formula.^{[6]
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Research source} Slovin's formula is a very general equation used when you can estimate the population but have no idea about how a certain population behaves. The formula is described as:
- Sample Size = N / (1 + N*e²)
  - N = population size
  - e = margin of error
- Note that this is the least accurate formula and, as such, the least ideal. You should only use this if circumstances prevent you from determining an appropriate standard of deviation and/or confidence level (thereby preventing you from determining your z-score, as well).
2
Plug in the numbers. Replace each variable placeholder with the numerical values that apply specifically to your survey.
- Example: Calculate the necessary survey size for a population of 240, allowing for a 4% margin of error.
- This means that:
  - N = 240
  - e = 0.04
3
Do the math. Solve the equation using your survey-specific numbers. The answer you arrive at should be your necessary survey size.^{[7]
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- Example: Sample Size = N / (1 + N*e²)
  - = 240 / (1 + 240 * 0.04²)
  - = 240 / (1 + 240 * 0.0016)
  - = 240 / (1 + 0.384}
  - = 240 / (1.384)
  - = 173.41 (final answer)

Community Q&A

Question

If the total population size is not given in the problem, what formula will apply?

Community Answer

If the population size is not given, then a t-distribution formula is applicable.