# Margin of Error: What is it and How to calculate it?

The margin of error tells the amount of the random difference underlying a survey’s results. This can be thought of a as measure of the difference one would see in reported percentages if the same poll were taken multiple times.

Margin of error is positive whenever a population is incompletely sampled and the result measure has positive difference.

The term margin of error is often used in non-survey contexts to specify observation error in reporting measured quantities.

A margin of error can be calculated for each figure produced from a sample survey, unless a nonprobability sample is used.

*See also: **Buying on Margin: Know More Tips, Benefits, and Risks** *

## Why do we use margin of error?

Every time we use a representative a sample to guess something about a full population, our guess will contain some doubt. Using our sample statistic, we have to infer the real statistic – and that inference will mean our guess will usually be somewhere nears the actual figure.

## Basic concept

Polls mostly include taking a sample from a certain population. The population of interest is the population of people who will vote. Because it is unfeasible to poll every person who will vote, pollsters take lesser samples that are planned to be representative, that is, a random sample of the population.

Given the size of the sample (1,013), probability theory enables the calculation of the chance that the poll reports 47 % for Kerry but is in fact 50 %, or is in fact 42 %. This theory and several Bayesian assumptions recommend that the “true” percentage will perhaps be very nearly 47 %. The more people that are sampled, the more confidence pollsters can be that the “true” percentage is closer and closer to the observed percentage. The margin of error is an uneven, poll-wide expression of that confidence.

Sampling theory provides systems for calculating the probability that the poll outcomes vary from certainty by more than a certain amount, simply because of chance. On the other hand, the margin of error only accounts for random sampling error, so it is blind to systematic errors that may be introduced by non-response or by interactions between the survey and subjects’ memory, motivation, communication and knowledge.

## Margin of Error Percentage

A margin of error expresses how many percentage points your outcomes will vary from the real population value. For example, a 95% confidence interval with a 4 percent margin of error means that your statistic will be in 4 percentage points of the real population value 95% of the time.

The Margin of Error can be calculated in two ways:

1. Margin of Error = Critical value X Standard deviation

2. Margin of Error = Critical value X Standard error of the statistic

## How to find the Critical Value

The critical value is a factor used to compute the margin of error. This section defines how to find the critical value, when the sampling distribution of the statistics is normal or nearly normal.

Once the sampling distribution is nearly normal, the critical value can be expressed as a t-score or as a z-score. To find the critical value, follow these steps.

- Compute alpha (α): α = 1 – (confidence level/100)
- Find the critical probability (p*): p* = 1 – α/2
- To express the critical value as a z-score, find the z-score having a cumulative probability equal to the critical probability (p*).

To express the critical statistic t statistic, follow these steps.

- Find the degrees of freedom (DF). When estimating a mean score or a proportion from a single simple, DF is equal to the sample size minus one. For other applications, the degrees of freedom may be calculated differently. We will describe those computations as they come up.
- The critical t statistic (t*) is the t statistic having degrees of freedom equal to DF and cumulative probability equal to the critical probability (p*).

## T-Score versus Z-Score

Should you express the critical value as a t statistic or as a z-score? One way to answer this question focuses on the population standard deviation.

- If the population standard deviation is identified, use the z-score.
- If the population standard deviation is unidentified, use the t statistic.

Another approach focuses on sample size.

- If the sample size is large, use the z-score. (The central limit theorem provides a useful basis for determining whether a sample is “large”.)
- If the sample size is small, use the t statistic.

In practice, researchers employ a mix of the above rules. On this site, we use z-scores when the population standard deviation is identified and the sample size is large. Otherwise, we use the t statistic, if the sample size is small and the underlying distribution is not normal.

Warning: If the sample size is small and the population distribution is not normal, we cannot be confident that the sampling distribution of the statistic will be normal. In this situation, neither the t statistic nor the z-score should be used to compute critical values.

## How to Calculate Margin of Error: Steps

Step 1: Find the critical value. The critical value is either a t-score or a z-score. In general, for small sample sizes (under 30) or when you don’t know the population standard deviation, use a t-score. Otherwise, use a z-score.

Step 2: Find the Standard Deviation or the Standard Error. These are essentially the same thing, only you must know your population parameters in order to calculate standard deviation. Otherwise, calculate the standard error.

Step 3: Multiply the critical value from Step 1 by the standard deviation or standard error from step 2. For example, if your CV is 1.95 and your SE is 9.019, then: 1.95*0.019 = 0.03705

## Margin of Error for a Proportion

Step 1: Find P-hat by dividing the number of people who responded positively. “Positively” in this sense doesn’t mean that they gave a “Yes” answer; It means that they answered according to the statement in the question. In this case, 380/1000 people (38%) responded positively.

Step 2: Find the z-score that goes with the given confidence interval. You’ll need to reference this chart of common critical values. A 90% confidence interval has z-score (a critical value) of 1.645.

Step 3: Insert the values into the formula and solve:

= 1.645 * 0.0153

= 0.0252

Step 4: Turn Step 3 into a percentage:

0.0252 = 2.52%

The margin of error is 2.52%.

## Maximum and specific margins of error

The margin of error usually reported in the media is a poll-wide figure that reflects the maximum sampling variation of any percentage based on all respondents from that poll the term margin of error also refers to the radius of the confidence interval of a particular statistic.

The margin of error for a particular individual percentage will usually be lesser than the maximum margin of error quoted for the survey. This maximum only applies once the observed percentage is 50%, and the margin of error shrinks as the percentage methods the extreme of 0% or 100%.

In the other words, the maximum margin of error is the radius of a 95% confidence interval for a reported percentage of 50%. If p moves away from 50%, the confidence interval for p will be shorter. Therefore, the maximum margin of error represents an upper bound to the uncertainty. One is at least 95% certain that the “true” percentage is within the maximum margin of error of a reported percentage for any reported percentage.

## Effect of population size

The formula above for the margin of error assume that there is an infinitely large population and thus do not rely on the size of the population of interest. According to sampling theory, this assumption is reasonable when the sampling when the sampling fraction is small. The margin of error for a specific sampling method is essentially the same despite whether the population of interest is the size of a school, city, state, or country, as long as the sampling fraction is less than 5%.

In cases where the sampling fraction exceeds 5%, analysts can adjust the margin of error using a “finite population correction” (FPC), to account for the added precious gained by sampling nearly a larger percentage of the population. FPC can be calculated using the formula.

To adjust for a large sampling fraction, the FPC factored into the calculation of the margin of error, this has the effect of narrowing the margin of error. It holds that the FPC approaches zero as the sample size (n) approaches the population size (N), which has the effect of removing the margin of error completely. This makes intuitive sense because when N = n, the sample becomes a census and sampling error becomes moot.

Analysts should be mindful that the samples stay truly random as the sampling fraction grows, lest sampling bias be introduced.

## Conclusion

Margin of error is typically defined as the “radius” (or the half of width) of a confidence interval for a specific statistic from a survey. The margin of error has been described as an “absolute” quantity, equal to a confidence interval radius for the statistic. In some cases, the margin of error is not expressed as an “absolute” quantity; rather it is expressed as a “relative” quantity.

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