How do I determine sample size for an unknown standard deviation?

Updated February 21, 2017

In elementary statistics or Advanced Placement statistics, you'll be asked to calculate various statistics for samples. A sample is a small segment of a population that is used to make generalisation about the population as a whole. How big a sample you take depends upon the population characteristics: a population of 1 million will require a different sample size than a population of 100, and a population with a known standard deviation will require a different sample size than one with an unknown standard deviation.

Divide your confidence interval by 2. A confidence level represents an interval that your data will fall into and is represented by a percentage, usually 98, 95 or 90 per cent. For example, if you wanted a 90 per cent confidence interval then .90 / 2 = 0.45.

Look up your answer from Step 1 in the middle of the z-table. If you can't find the exact number, get as close as you can. The closest value to 0.45 in the z-table is 0.4495, which falls at the intersection of row 1.6 and column 0.04.

Add the row and column values from Step 2 to get the z-score: 1.6 + 0.04 = 1.64.

Divide the interval for your data by 2. The interval represents a spread of data. For example, if you had a 6 per cent confidence interval, then 0.0 6 / 2 = 0.03.

Divide Step 3 by Step 4: 1.64 / 0.03 = 54.67.

Square Step 5: 54.67^2 = 2988.81.

Subtract the given percentage in your sample (the question might state something like "51 per cent of voters" or "19 per cent of students") from 1, then multiply the percentage by the number you just calculated.

For example, 41 per cent is 1 - 0.41 = 0.59

0.59 X 0.41 = 0.2419.

Multiple the answer from Step 6 by the answer from Step 7.

2988.81. x 0.2419 = 722.993139.

This is your sample size.


There are many calculators that can calculate sample sizes for you, like graphing calculators and mathematical software.

Things You'll Need

  • Sample data with confidence interval and width
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About the Author

Stephanie Ellen teaches mathematics and statistics at the university and college level. She coauthored a statistics textbook published by Houghton-Mifflin. She has been writing professionally since 2008. Ellen holds a Bachelor of Science in health science from State University New York, a master's degree in math education from Jacksonville University and a Master of Arts in creative writing from National University.