In statistics, margin of error is a measurement concept that expresses the level of likelihood and confidence that results obtained from a sample will be similar to a report wherein an entire population was successfully surveyed. To an extent, the margin of error is one of the foundations of practical statistics.

Aside from its use in statistics, the margin of error concept can also be applied scientifically to disciplines that demand observation and the reporting of quantities. The influence of factors that can produce errors in observation such as the human condition, acts of God and atmospheric events, can be measured in terms of margin of error to improve understanding.

Although there are various formulas that can be used to calculate margin of error, the most commonly used by statisticians involves multiplying the sample proportions times each other, dividing this mathematical product by the sample size, taking the square root of this result, and multiplying it by the confidence level.

**The formula is:**

## Margin of error = z * √ p (1-p) / n

**The operands of the formula above are as follows:**

z = confidence level

p = sample proportion

n = sample size

**The value of the confidence levels is derived from accepted statistic levels based on the following percentages:**

80 = 1.28

90 = 1.645

95 = 1.96

98 = 2.33

99 = 2.58

To exemplify the formula, think about a poll that surveyed the opinion of 1,000 Americans about the future of the United States economy. The results of this hypothetical poll are:

* 520 respondents believe that the future is bright and positive for the American economy

* 480 respondents believe that the future is gloomy and negative for the American economy

The sample proportion is first calculated by dividing the survey results by the sample size; thus:

p = 520 / 1000 and 480 / 1000

The statisticians conducting this survey have assumed a 95 percent confidence level based on historical data resulting from similar polls; thus, the confidence level value that will be used is 1.96. The formula will look as follows:

Margin of error = 1.96 * √ (0.52 * 0.48) / 1000

The result for the above is 0.0310. Since survey results and their margin of error are expressed as percentages, a correct statement for the example survey would be:

52 percent of Americans feel positive about the future of the economy; this figure was obatined with a 95 percent confidence level, plus or minus 3.1 percent margin of error.

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### Author: Jay White

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