Statistical estimates are often given along with a "plus or minus" value. For instance, it may be stated that "Candidate A has 40% support and Candidate B has 38%, with a margin of error of 2%". This article will explain the terms used, how to calculate margin of error, and what the implications of the results are.
What Are Confidence Interval and Margin of Error?
Confidence intervals are a range of values within which the "true" population value is likely to be, based on an estimate from a sample. The value quoted for confidence interval is usually stated at the 95% confidence value. For example
"Candidate A has 40% support, with a margin of error of 2%"
It is not possible to ask every voter how they will vote, so a sample is taken. A confidence level of 95% means that the pollster is 95% certain that the "true" value of support for candidate A (if every voter could be asked) is 40% ± 2%, or 38% - 42%. The margin of error is the ± 2% part of the data.
Why Are Confidence Limits and Margins of Error Important?
Confidence limits and margins of error are important because they show how uncertain the result is. If candidate A has 40% support and candidate B has 38%, and each has a margin of error of ± 2%, then candidate A is not far enough ahead to be confident about winning the election: he knows that his poll result is 95% sure to be in the range 38% - 42%, and that candidate B is 95% certain of polling 36% - 40%. Since the upper limit for candidate B is higher than the lower limit for candidate A, it is still quite possible for candidate B to win.
What Factors affect Confidence Limits and Error Margins?
There are several factors that affect the confidence interval, or margin of error:
- Sample size - larger sample sizes give smaller margins of error, and tighter confidence intervals
- Confidence Level - How sure the user wants to be about the result. Higher confidence levels require wider confidence intervals, but 95% confidence intervals are the most commonly used.
- Distribution Type - the "shape" of the distribution needs to be known. The formula to calculate margin of error for means is different to that for proportions, for example.
- Population Spread - usually stated as a standard deviation. As the amount of spread in the data increases, so do the confidence intervals and margins of error.
How Are Confidence Intervals Calculated For Means and Proportions?
The formula to calculate the confidence interval for means is shown in Figure 1. The upper limit is the mean plus the error margin, and the lower limit is the mean minus the error margin. To calculate the error margin in Excel, for 95% confidence and a sample size of 1000, and using s = 1, use:
=TINV(0.025,1000-1) * 1 / sqrt(1000)
This gives a margin of error of ± 7%.
The formula for proportions is similar. For proportions (the type of distribution type used with the poll for candidate A), the standard deviation is equal to
√[p × (1 - p)], so the Excel formula for 95% confidence and 1,000 sample size is
=TINV(0.025, 1000-1) * sqrt(0.4 * (1 - 0.4)/1000)
= 0.035, or ± 3.5%
In the candidate A example, it means that if the sample of 1,000 and proportion of 40% (i.e. p = 0.4) were found, then the 95% confidence interval is
40% ± 3.5%, or
36.5% to 43.5%.
Confidence Intervals Summary
Confidence intervals are important because they show how much potential error there is in a sample, when trying to estimate the population. The Excel formulas for means and proportions are included for convenience. Care must be taken when applying any formula - there is a presumption that data are normally (Gaussian) distributed, either naturally or as a result of the Central Limit Theorem.
Confidence Intervals References
Smithson, Dr Michael J. Confidence Intervals (Quantitative Applications in the Social Sciences) (Paperback)
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
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