# Margin of error

In statistics, it is common to estimate statistical characteristics of a population using a sample of the population. Since a sample cannot fully represent a population, estimations of population parameters based on samples always have some degree of error. These errors are referred to as sampling errors. The margin of error (MOE) is a statistic that indicates the amount of sampling error in the sample statistic, such as the mean. In a confidence interval, the margin of error is the range of values above and below the sample statistic. For example, given the following confidence interval (CI),

CI = 53 ± 4

the margin of error is 4. Generally, the higher the margin of error, the less confidence we have that a sample statistic accurately reflects the population parameter.

## Calculating the margin of error

The margin of error can be calculated using the following steps:

- Find the critical value, z* or t*. The critical value is dependent on whether the population standard deviation is known, as well as the sample size.
- For known population standard deviation and sample size n > 30, the critical value is the Z-score for the given confidence level. The table below shows the critical values for commonly used confidence levels:
Confidence level Critical value 90% 1.645 95% 1.96 98% 2.33 99% 2.575

- If the sample size is too small (n < 30), or the population standard deviation is unknown, the critical value is the t-statistic for the given confidence level with n-1 degrees of freedom.
- Determine the standard deviation if the population parameters are known. Otherwise estimate the standard error (SE) using the sample standard deviation (s), the sample size (n), and the formula
- Multiply the critical value by the standard deviation or the standard error. A Z-score is multiplied by the standard deviation while a t-statistic is multiplied by the standard error:

Margins of error are a key component of a confidence interval (CI), where a confidence interval is determined by adding and subtracting the margin of error from the sample mean:

CI = x ± MOE

Example

The average GPA of a sample of 75 seniors from a high school was 2.85. The standard deviation of the GPAs of the entire senior class is known to be 0.45. Find the margin of error and confidence interval for a confidence level of 90%.

Since the population standard deviation is known, the Z-score of the critical value can be determined using a Z-table. However, since 90% is a commonly used confidence level, we can reference the table above to find that the critical value for a 90% confidence level is 1.645. The sample size is 75 and the population standard deviation is 0.45, so:

Given the sample mean of 2.85, the confidence interval is expressed as:

2.85 ± 0.085

Based on the the above, we can be reasonably confident that there is a 90% chance that the population mean lies within the interval. In other words, we expect that the average GPA of all seniors at the high school lies between 2.765 and 2.935.

It is worth noting that as the sample size increases, the margin of error decreases, and the confidence interval becomes more precise. Assuming that the sample size in the above example was 200, and that all other variables stayed the same, the margin of error would instead be

and the confidence interval would be

2.85 ± 0.052

Thus, in this case, we would expect the average GPA of all seniors at the high school to lie between 2.798 and 2.902. This illustrates the effect of larger sample sizes on producing more precise, and generally more accurate results. This effect is not limited to margins of error; larger sample sizes typically produce more statistically robust results.