The formula above is used to calculate the margin of error of a sample mean, provided that we 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.
The Level of Confidence
The symbol α is the Greek letter alpha. It is used to denote the level of confidence that we are working with. Any percentage less than 100% is possible here, 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.
The 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 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 are of 0.95 from -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.
The 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.
The 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.
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.
Analysis of the Formula
There are a few features of the formula that deserve note:
- A somewhat surprising feature about the formula is that other than the basic assumptions being made about the population, the formula for the margin of error does not rely upon the size of the population.
- Since the margin of error is inversely related to the square root of the sample size, the larger the sample, the smaller the margin of error.
- The presence of the square root means that we must dramatically increase the sample size in order to have any effect on the margin of error. If we have a particular margin of error of and want to cut this is half, then at the same confidence level we will need to quadruple the sample size.
- In order to keep the margin of error at a given value while increasing our confidence level will require us to increase the sample size.