Determining Sample Size: How to Ensure You Get the Correct Sample SizeApril 08, 2013How many responses do you really need? This simple question is a never-ending quandary for researchers. A larger sample can yield more accurate results but excessive responses can be pricey.Consequential research requires an understanding of the statistics that drive sample size decisions. A simple equation will help you put the migraine pills away and sample confidently.Before you can calculate a sample size, you need to determine a few things about the target population and the sample you need:1. Population Size How many total people fit your demographic? For instance, if you want to know about mothers living in the US, your population size would be the total number of mothers living in the US. Dont worry if you are unsure about this number. It is common for the population to be unknown or approximated.2. Margin of Error (Confidence Interval) No sample will be perfect, so you need to decide how much error to allow. The confidence interval determines how much higher or lower than the population mean you are willing to let your sample mean fall. If youve ever seen a political poll on the news, youve seen a confidence interval. It will look something like this: 68% of voters said yes to Proposition Z, with a margin of error of +/- 5%.3. Confidence Level How confident do you want to be that the actual mean falls within your confidence interval? The most common confidence intervals are 90% confident, 95% confident, and 99% confident.4. Standard of Deviation How much variance do you expect in your responses? Since we havent actually administered our survey yet, the safe decision is to use .5 this is the most forgiving number and ensures that your sample will be large enough.Okay, now that we have these values defined, we cancalculate our needed sample size.Your confidence level corresponds to a Z-score. This is a constant value needed for this equation. Here are the z-scores for the most common confidence levels: 90% Z Score = 1.645 95% Z Score = 1.96 99% Z Score = 2.576If you choose a different confidence level, use thisZ-score table* to find your score.Next, plug in your Z-score, Standard of Deviation, and confidence interval into this equation:**Necessary Sample Size = (Z-score) * StdDev*(1-StdDev) / (margin of error)Here is how the math works assuming you chose a 95% confidence level, .5 standard deviation, and a margin of error (confidence interval) of +/- 5%.((1.96) x .5(.5)) / (.05)(3.8416 x .25) / .0025.9604 / .0025384.16385 respondents are neededVoila!Youve just determined yoursample size.

Howto Determine Sample Size, Determining Sample SizeIn order to prove that a process has been improved, you must measure the process capability before and after improvements are implemented. This allows you to quantify the process improvement (e.g., defect reduction or productivity increase) and translate the effects into an estimated financial result something business leaders can understand and appreciate. If data is not readily available for the process, how many members of the population should be selected to ensure that the population is properly represented? If data has been collected, how do you determine if you have enough data?Determining sample size is a very important issue because samples that are too large may waste time, resources and money, while samples that are too small may lead to inaccurate results. In many cases, we can easily determine the minimum sample size needed to estimate a process parameter, such as the population mean.When sample data is collected and the sample meanis calculated, that sample mean is typically different from the population mean. This difference between the sample and population means can be thought of as an error. The margin of erroris the maximum difference between the observed sample meanand the true value of the population mean:

where:is known as the critical value, the positivevalue that is at the vertical boundary for the area ofin the right tail of the standard normal distribution.is the population standard deviation.is the sample size.

Rearranging this formula, we can solve for the sample size necessary to produce results accurate to a specified confidence and margin of error.

This formula can be used when you knowand want to determine the sample size necessary to establish, with a confidence of, the mean valueto within. You can still use this formula if you dont know your population standard deviationand you have a small sample size. Although its unlikely that you knowwhen the population mean is not known, you may be able to determinefrom a similar process or from a pilot test/simulation.Lets put all this statistical mumbo-jumbo to work. Take for example that we would like to start an Internet service provider (ISP) and need to estimate the average Internet usage of households in one week for our business plan and model.SampleSize Calculation ExampleProblemWe would like to start an ISP and need to estimate the average Internet usage of households in one week for our business plan and model. How many households must we randomly select to be 95 percentsure that the sample mean is within 1 minute of the population mean. Assume that a previous survey of household usage has shown= 6.95 minutes.SolutionWe are solving for the sample size.A 95% degree confidence corresponds to= 0.05. Each of the shaded tails in the following figure has an area of= 0.025. The region to the left ofand to the right of= 0 is 0.5 0.025, or 0.475. In the table of the standard normal () distribution, an area of 0.475 corresponds to avalue of 1.96. The critical value is therefore= 1.96.

The margin of error= 1 and the standard deviation= 6.95. Using the formula for sample size, we can calculate:

So we will need to sample at least 186 (rounded up) randomly selected households. With this sample we will be 95 percent confident that the sample meanwill be within 1 minute of the true population of Internet usage.

This formula can be used when you knowand want to determine the sample size necessary to establish, with a confidence of, the mean valueto within. You can still use this formula if you dont know your population standard deviationand you have a small sample size. Although it is unlikely that you knowwhen the population mean is not known, you may be able to determinefrom a similar process or from a pilot test/simulation.

Sample Size CalculatorThis Sample Size Calculator is presented as a public service of Creative Research Systemssurvey software. You can use it to determine how many people you need to interview in order to get results that reflect the target population as precisely as needed. You can also find the level of precision you have in an existing sample.Before using the sample size calculator, there are two terms that you need to know. These are:confidence intervalandconfidence level. If you are not familiar with these terms,click here. To learn more about the factors that affect the size of confidence intervals,click here.Enter your choices in a calculator below to find the sample size you need or the confidence interval you have. Leave the Population box blank, if the population is very large or unknown.Top of FormDetermine Sample Size

Confidence Level:95%99%

Confidence Interval:

Population:

Sample size needed:

Bottom of FormTop of FormFind Confidence Interval

Confidence Level:95%99%

Sample Size:

Population:

Percentage:

Confidence Interval:

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Sample Size Calculator Terms: Confidence Interval & Confidence LevelTheconfidence interval(also called margin of error) is the plus-or-minus figure usually reported in newspaper or television opinion poll results. For example, if you use a confidence interval of 4 and 47% percent of your sample picks an answer you can be "sure" that if you had asked the question of the entire relevant population between 43% (47-4) and 51% (47+4) would have picked that answer.Theconfidence leveltells you how sure you can be. It is expressed as a percentage and represents how often the true percentage of the population who would pick an answer lies within the confidence interval. The 95% confidence level means you can be 95% certain; the 99% confidence level means you can be 99% certain. Most researchers use the 95% confidence level.When you put the confidence level and the confidence interval together, you can say that you are 95% sure that the true percentage of the population is between 43% and 51%. The wider the confidence interval you are willing to accept, the more certain you can be that the whole population answers would be within that range.For example, if you asked a sample of 1000 people in a city which brand of cola they preferred, and 60% said Brand A, you can be very certain that between 40 and 80% of all the people in the city actually do prefer that brand, but you cannot be so sure that between 59 and 61% of the people in the city prefer the brand.Factors that Affect Confidence IntervalsThere are three factors that determine the size of the confidence interval for a given confidence level: Sample size Percentage Population sizeSample SizeThe larger your sample size, the more sure you can be that their answers truly reflect the population. This indicates that for a given confidence level, the larger your sample size, the smaller your confidence interval. However, the relationship is not linear (i.e., doubling the sample size does not halve the confidence interval).PercentageYour accuracy also depends on the percentage of your sample that picks a particular answer. If 99% of your sample said "Yes" and 1% said "No," the chances of error are remote, irrespective of sample size. However, if the percentages are 51% and 49% the chances of error are much greater. It is easier to be sure of extreme answers than of middle-of-the-road ones.When determining the sample size needed for a given level of accuracy you must use the worst case percentage (50%). You should also use this percentage if you want to determine a general level of accuracy for a sample you already have. To determine the confidence interval for a specific answer your sample has given, you can use the percentage picking that answer and get a smaller interval.Population SizeHow many people are there in the group your sample represents? This may be the number of people in a city you are studying, the number of people who buy new cars, etc. Often you may not know the exact population size. This is not a problem. The mathematics of probability proves the size of the population is irrelevant unless the size of the sample exceeds a few percent of the total population you are examining. This means that a sample of 500 people is equally useful in examining the opinions of a state of 15,000,000 as it would a city of 100,000. For this reason, The Survey System ignores the population size when it is "large" or unknown. Population size is only likely to be a factor when you work with a relatively small and known group of people (e.g., the members of an association).The confidence interval calculations assume you have a genuine random sample of the relevant population. If your sample is not truly random, you cannot rely on the intervals. Non-random samples usually result from some flaw in the sampling procedure. An example of such a flaw is to only call people during the day and miss almost everyone who works. For most purposes, the non-working population cannot be assumed to accurately represent the entire (working and non-working) population.Stats: Sample Size Determination

The sample size determination formulas come from the formulas for the maximum error of the estimates. The formula is solved for n. Be sure to round the answer obtainedupto the next whole number, not off to the nearest whole number. If you round off, then you will exceed your maximum error of the estimate in some cases. By rounding up, you will have a smaller maximum error of the estimate than allowed, but this is better than having a larger one than desired.Population MeanHere is the formula for the sample size which is obtained by solving the maximum error of the estimate formula for the population mean for n.

Population ProportionHere is the formula for the sample size which is obtained by solving the maximum error of the estimate formula for the population proportion for n. Some texts use p hat and q hat, but since the sample hasn't been taken, there is no value for the sample proportion. p and q are taken from a previous study, if one is available. If there is no previous study or estimate available, then use 0.5 for p and q, as these are the values which will give the largest sample size, and it is better to have too large of a sample size and come under the maximum error of the estimate than to have too small of a sample size and exceed the maximum error of the estimate.

Different Ways to Set Sample Size In an AuditByMaire LoughranfromAuditing For DummiesYou can use several methods to determine the size of an audit sample. You can set the audit sample size based on tolerable and expected error or the previous years policy. You can use tables and software to set the sample size, or you can adjust the size based on your analysis.Using tolerable and expected errorTolerable error refers to the maximum number of client errors in a sample size that youre prepared to accept and still conclude that youve achieved the audit objectives.Expected erroris the amount of error in your sample size that you plan for and expect. The following sections explain what tools you can use and adjustments you can make to reach an unqualified report.Following last years auditAudit firms use the same types of criteria to set an audits confidence level (firm policy, population size, results of analytical review, and other known facts about the client and its business environment) as they do to set tolerable and expected error figures.If the audit client is new, your firms policy will dictate the tolerable and expected error figures.If youre working with a repeat client, last years tolerable and expected error figures will be your baseline. However, you may determine that those figures should be changed. Keep reading, if so.Using tables or software to set sample sizeYou can use many different methods to calculate sample size. They are based on statistics and probability so you can measure results. The method you use will be a function of your firms policy. Most auditors use one of two tools to determine sample size: Attribute-sampling tables:Attribute sampling means that an item being sampled either will or wont possess certain qualities. If your firm uses an attribute-sampling table, tolerable error is normally shown horizontally, and expected error is shown vertically.The following table shows how to determine sample size using attribute sampling. To use the table, you identify the column for tolerable error (for example, 5 percent) and read down that column until you reach the row for your expected error (for example, 3 percent). The figure at the intersection (91) is the number of items youll have in your sample.Expected Error RateTolerable Error

2%3%4%5%

0.00%124786658

1.001531038877

2.0018112710381

3.0020815010991

Audit command language (ACL) software programs:ACL software performs the same calculation as an attribute-sampling table. You open a client workbook within the software and choose a command such as calculate sample size. The software asks for the same qualifiers used for the attribute-sampling table (tolerable and expected error) to produce the sample size.Adjusting sample size based on your analysisDuring the audit, you may notice significant discrepancies between the company youre auditing and other companies in the same industry. If so, you may need to adjust the sample size to get a clear picture. For example, say youre working with a repeat client. Your analytical procedures show that your clients inventory turnover rate has sharply declined in the last year, while companies in the same industry have shown only a slight decline.Inventory turnoveris represented by a ratio: the cost of goods sold divided by the inventory value. An unusual decline in this ratio can indicate that inventory is overstated, which would artificially increase net income.