The formula below is used to calculate the margin of error for an confidence interval of a population mean. The conditions that are necessary to use this formula is that we must have a sample from a population that is normally distributed and know the population standard deviation. The symbol E denotes the margin of error of the unknown population mean. An explanation for each of the variable follows.

01 of 06 Level of Confidence The symbol α is the Greek letter alpha. It is related to the level of confidence that we are working with for our confidence interval. Any percentage less than 100% is possible for a level of confidence, but in order to have meaningful results, we need to use numbers close to 100%. Common levels of confidence are 90%, 95% and 99%. The value of α is determined by subtracting our level of confidence from one, and writing the result as a decimal. So a 95% level of confidence would correspond to a value of α = 1 - 0.95 = 0.05.

02 of 06 Critical Value The critical value for our margin of error formula is denoted by zα/2. This is the point z* on the standard normal distribution table of z-scores for which an area of α/2 lies above z*. Alternately is is the point on the bell curve for which an area of 1 - α lies between -z* and z*. At a 95% level of confidence we have a value of α = 0.05. The z-score z* = 1.96 has an area of 0.05/2 = 0.025 to its right. It is also true that there is a total area of 0.95 between the z-scores of -1.96 to 1.96. The following are critical values for common levels of confidence. Other levels of confidence can be determined by the process outlined above. A 90% level of confidence has α = 0.10 and critical value of zα/2 = 1.64.

A 95% level of confidence has α = 0.05 and critical value of zα/2 = 1.96.

A 99% level of confidence has α = 0.01 and critical value of zα/2 = 2.58.

A 99.5% level of confidence has α = 0.005 and critical value of zα/2 = 2.81.

03 of 06 Standard Deviation The Greek letter sigma, expressed as σ, is the standard deviation of the population that we are studying. In using this formula we are assuming that we know what this standard deviation is. In practice we may not necessarily know for certain what the population standard deviation really is. Fortunately there are some ways around this, such as using a different type of confidence interval.

04 of 06 Sample Size The sample size is denoted in the formula by n. The denominator of our formula consists of the square root of the sample size.

05 of 06 Order of Operations Since there are multiple steps with different arithmetic steps, the order of operations is very important in calculating the margin of error E. After determining the appropriate value of zα/2, multiply by the standard deviation. Calculate the denominator of the fraction by first finding the square root of n then dividing by this number.