Survey Sample Size Calculator

Sample size determination controls for sampling error. It also controls for nonresponse that happens at random. It does not control for other nonsampling errors. A calculated sample size is a plan for sampling and expected nonresponse in a survey. It is not a guarantee against every kind of error. ¹

The formula needs four things. These are a target margin of error, a confidence level, an estimate of variability, and the size of the population. The confidence level sets a value called z. This value is used throughout the formula. For a percentage, the variability estimate is Pˆ times one minus Pˆ. Here, Pˆ is the percentage you expect to find. Finding the sample size also requires knowing the size of the population. ¹

The calculation runs in stages. First, you find an initial sample size from the margin of error and variability. Then you adjust it for the size of the population. Next, you adjust it for the design effect. Then you adjust it for the expected response rate. The formula for that initial sample size also holds when you give both the margin of error and the expected percentage in percentage points. ¹

First, you find the initial sample size. Then you adjust it down using the population size. To get the adjusted figure, multiply the initial sample size by the population. Then divide that by the population plus the initial sample size. This adjustment matters a lot for small populations. It matters somewhat for medium-size populations. It matters only a little for large populations. ¹

For very small populations, you often must survey a large share of the population. This lets you reach the precision you want. This is why, in practice, people sometimes conduct censuses for small populations. ¹

Consider a survey that found 15% of Ottawa residents attend religious services every week. That estimate came from a sample of 1,345 residents. It was considered accurate within plus or minus three percentage points, 19 times out of 20. This margin of error of three points gives a confidence interval of 12% to 18%. It means the true percentage is likely within three points of the 15% estimate. ²

The confidence level you choose sets the z used in the formula. A 90% confidence level corresponds to a z of 1.65. A 95% confidence level corresponds to a z of 1.96. A 99% confidence level corresponds to a z of 2.58. ²

When you estimate a percentage, a 50-50 split gives the most variability in the population. If you have no advance knowledge of that variability, it is often best to assume the greatest variability. For a variable with two possible values, you then assume a 50-50 split. If you have a good estimate of the percentage, use it instead. Assuming a 50-50 split when nothing is known gives the largest sample size, given the other assumptions. ¹

You need an estimate of variability before you can calculate sample size. The true variability is usually not known in advance. This estimate can be a percentage or a standard deviation. It can come from a past study on the same subject or from a pilot survey. This input is often the hardest one to get. People usually estimate it using past studies of a similar population. ¹

To reach your target number of completed responses, you must adjust the initial sample size for the response rate you expect. Not everyone you invite will respond. For example, an initial sample size of 400 with an expected response rate of 75% needs a selected sample of 400 divided by .75. That equals 533 people to invite. A lower expected response rate means more people must be invited for the same number of completed responses. ¹

Simply increasing the sample size does not fix nonresponse on its own. Potential bias can still result if the people who do not respond differ from those who do, with respect to what the survey measures. Dealing with nonresponse well means paying attention to who is not responding, not just how many. ¹

The formula on this page assumes a simple random sample. In this case, the design effect, called deff, equals 1. When the actual design is more complex than a simple random sample, you should multiply the required sample size by the design effect. For a highly clustered design, that design effect may be as high as 6 or 7. In some cases, a design effect of at least 2 might be used. ¹

A smaller margin of error always needs a larger sample size; it is not free. A margin of error of .05 can seem large next to an estimate expected to be around .05. In that case, a smaller margin of error, perhaps no greater than .01 or .02, should be specified instead. The best solution is not always the largest sample size that yields the smallest margin of error. Sometimes accepting a larger margin of error and using resources more efficiently gives suitably accurate results. ¹

Decide first what you'll do with each number before you decide how precise it needs to be. A sponsor of a national survey might need a margin of error of only ±3% for national estimates, but be fine with ±5% for provincial estimates and ±10% for subprovincial ones. ¹

Also check your margin of error against the size of the number you expect. A margin of error of ±.05 looks tight until you realize the estimate itself is expected to land around .05 — then it's actually huge. In a case like that, aim for something no larger than ±.01 or ±.02 instead. ¹

A percentage or an average from your sample is an estimate, not a fixed fact. It carries sampling variance: the amount it would shift if you drew the sample again. Precision and sample size are tied together for this reason — sampling variance shrinks as sample size grows, so tighter precision calls for a bigger sample. ¹

The standard error of that estimate is just the square root of its sampling variance. It's the quantity sitting underneath every margin of error you calculate. ²

Don't expect doubling your sample to halve your margin of error. The relationship isn't a straight line: going from a sample of 50 to a sample of 100 only tightens the margin of error on a proportion estimate from ±.139 to ±.098, not down to ±.070. ¹

Don't assume a bigger sample fixes nonresponse on its own. Simply increasing your sample size is inadequate to deal with total nonresponse — if the people who don't respond differ from the people who do on what you're measuring, that bias stays no matter how many extra people you invite. ¹

What does '±3 percentage points, 19 times out of 20' actually mean? Picture a survey of 1,345 residents finding that 15% attend religious services every week. The three points are the margin of error, and 12% to 18% is the confidence interval built from it. The '19 times out of 20' is the confidence statement — how often you'd expect that kind of interval to cover the true value if you ran the survey over and over. ²

Is a margin of error the same thing as a confidence interval? No. The margin of error is the plus-or-minus figure on its own; the confidence interval is the full range you get once you apply that figure to your estimate. ²

What it does say is that precision and sample size move together: the more precise you need your estimate to be, the larger the sample size you need. ¹

This page's formulas are described as covering an estimated average or a proportion. ¹

Everything above still applies if you're after a mean instead of a percentage. The same sample size formulas work for an estimated average just as they do for an estimated proportion. ¹

You still need to plug in an estimate of the population's variability before you can solve for n. The true variability is rarely known ahead of time, so get an estimate from a previous study on the same subject, or from a pilot survey. ¹

This is often the hardest input to pin down. It's usually approximated by looking at previous studies of a similar population. ¹

Here's what does bear on it: sample size math is built to control for sampling error, and for nonresponse that happens randomly. It is not built to control for other nonsampling errors. ¹

Simply increasing the sample size does not adequately deal with nonresponse, since bias could remain if nonrespondents differ from respondents. ¹