# How to Determine Sample Size With Mean & Standard Deviation

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The right sample size is an important consideration for those that conduct surveys. If the sample size is too small, the sample data obtained will not be an accurate reflection of the data that is representative of the population. If the sample size is too large, the survey will be too expensive and time-consuming to complete. For instance, if your survey goal was to find the mean age of women in the United States, it would be impractical to ask every woman her age.

The determination of the sample size requires that you define the confidence level you want and the level of error you will tolerate, and that you either know or have an estimate of the standard deviation of the population parameter that you are trying to determine.

Define the level of error you will tolerate. Pick a value that will give a result that is less than 5 percent of the population parameter that you are trying to estimate. Consider that the higher the error level tolerated the less significant your survey results are.

Consider a situation where you would need to find the mean age of women (the population parameter) in the United States. First make an estimate of the mean age of women. For that estimate use a previous study and then multiply that number by 0.05 to find the error.

If a study is not available, roughly estimate the average age of women yourself. For that estimate, obtain data with 10 different surveys of your own that have a sample size of 31 women each. For each survey, calculate the mean age for the 31 women. Then calculate the mean of the means for all the surveys. Use this number as the estimate of the mean age for women. Then multiply that number by 0.05 to obtain the error. If the mean of the means obtained for your surveys was 40, multiply 0.05 (5 percent) times 40 to obtain 2. So, select the error you will tolerate to be within two years.

Write this number down; you will use it to calculate the sample size. If you use 2 for the error for your sample calculation, your survey will produce a result that is accurate within two years of the actual mean age of women in the population. Remember that the smaller the error is, the larger the sample size will be.

Define the confidence level you want to use. Pick a confidence level of 90, 95 or 99 percent. Use a higher confidence level if you want to increase the probability that the results from your sample survey will be within the error tolerance you calculated in the previous step. Remember that the higher the confidence level you choose, the larger the sample size will be.

Determine the critical value for the given confidence interval. For a confidence level of 90 percent, use a critical value of 1.645. For a 90 percent confidence interval, use a critical value of 1.960, and for a confidence level of 99 percent, use a critical value of 2.575. Write this number down; you will use it to calculate the sample size.

Next find out the standard deviation for the population parameter you're trying to estimate with your survey. Use the standard deviation of the population parameter given in the problem or estimate the standard deviation. If it is not given, use the standard deviation from a similar study. If neither is available, roughly estimate a standard deviation such that it will be approximately 34 percent of the population.

For the example stated in Step 1, assume that 20 years is one standard deviation. For an average age of 40, this would mean that 68 percent of women in the population are estimated to be between 20 years old and 60 years old.

Calculate the sample size. First multiply the critical value by the standard deviation. Then divide this result by the error from Step 1. Now square this result. This result is the sample size.

For a problem that uses a confidence interval of 90 percent (a critical value of 1.645), specifies an error within two years, and gives a population standard deviation of 20 years, first multiply 1.645 by 20 to obtain 32.9. Divide 32.9 by 2 to obtain 16.45. Square 16.45 to obtain 270.6. Round up to the next highest integer to obtain a sample size of 271.

State the conditions for your survey results. For the example in Step 1, with a sample size of 271, you can be 90 percent confident that the mean of the sample of 271 women will be within two years of the actual mean of the total women’s population. So if your survey resulted in a mean age of 43 years, you can ascertain that there is a 90 percent chance that the mean age of the population of women in the United States will be between 42 and 44.

#### Things You'll Need

• Pencil
• Paper
• Calculator
• Statistics tables
• Elementary statistics book