Many times political polls and other applications of statistics state their results with a margin of error. It is not uncommon to see that an opinion poll states that there is support for an issue or candidate at a certain percentage of respondents, plus and minus a certain percentage. It is this plus and minus term that is the margin of error. But how is the margin of error calculated? For a simple random sample of a sufficiently large population, the margin or error is really just a restatement of the size of the sample and the level of confidence being used.

### The Formula for the Margin of Error

In what follows we will utilize the formula for the margin of error. We will plan for the worst case possible, in which we have no idea what the true level of support is the issues in our poll. If we did have some idea about this number, possibly through previous polling data, we would end up with a smaller margin of error.

The formula we will use is: *E* = *z*_{α/2}/(2√ n)

### The Level of Confidence

The first piece of information we need to calculate the margin of error is to determine what level of confidence we desire. This number can be any percentage less than 100%, but the most common levels of confidence are 90%, 95%, and 99%. Of these three the 95% level is used most frequently.

If we subtract the level of confidence from one, then we will obtain the value of alpha, written as α, needed for the formula.

### The Critical Value

The next step in calculating the margin or error is to find the appropriate critical value.

This is indicated by the term *z*_{α/2} in the above formula. Since we have assumed a simple random sample of a large population, we can use the standard normal distribution of *z*-scores.

Suppose that we are working with a 95% level of confidence. We want to look up the *z*-score *z**for which the area between -z* and z* is 0.95.

From the table, we see that this critical value is 1.96.

We could have also found the critical value in the following way. If we think in terms of α/2, since α = 1 - 0.95 = 0.05, we see that α/2 = 0.025. We now search the table to find the *z*-score with an area of 0.025 to its right. We would end up with the same critical value of 1.96.

Other levels of confidence will give us different critical values. The greater the level of confidence, the higher the critical value will be. The critical value for a 90% level of confidence, with corresponding α value of 0.10, is 1.64. The critical value for a 99% level of confidence, with corresponding α value of 0.01, is 2.54.

### Sample Size

The only other number that we need to use the formula to calculate the margin of error is the sample size, denoted by *n* in the formula. We then take the square root of this number.

Due to the location of this number in the above formula, the larger the sample size that we use, the smaller the margin of error will be. Large samples are therefore preferable to smaller ones. However, since statistical sampling requires resources of time and money, there are constraints to how much we can increase the sample size. The presence of the square root in the formula means that quadrupling the sample size will only half the margin of error.

### A Few Examples

To make sense of the formula, let’s look at a couple of examples.

- What is the margin of error for a simple random sample of 900 people at a 95% level of confidence?
By use of the table we have a critical value of 1.96, and so the margin of error is 1.96/(2 √ 900 = 0.03267, or about 3.3%.

- What is the margin of error for a simple random sample of 1600 people at a 95% level of confidence?
At the same level of confidence as the first example, increasing the sample size to 1600 gives us a margin of error of 0.0245 or about 2.5%.