Calculating Margin of Error
A common question to ask during the analysis of survey results is what the margin of error is of
the research findings. Margin of error is a measure of how precise the data are that you collected
in your survey. Another common term for margin of error is “confidence interval.” However,
there are two main components to margin of error: confidence interval and level of confidence.
The level of confidence is a number that you will set. The confidence interval is a number you
will calculate based on the chosen level of confidence. Both values are important to accurately
state a margin of error for your data.
Level of Confidence
Another name for level of confidence is level of significance. This is essentially the same idea as
the level of significance discussed in association with the Statistical Significance Testing
tutorial. The most commonly used levels of confidence for this class and in practice are 90% or
95%. For example, if a 90% level of confidence were chosen, this would mean that if an
identical survey were given to ten different samples from the same population, nine out of the ten
surveys would have results that fall inside of the confidence interval. The 10th survey would
have results that could be anywhere. A 90% level of confidence is generally the lowest level of
confidence that is considered to be statistically significant.
Confidence Interval
You are probably familiar with the concept of confidence interval from polls conducted in
association with election forecasts. For example, a poll reported in the news may say that 53%
of voters intend to vote for a political candidate, plus or minus 5%. This means that the actual
number of voters who intend to vote for the candidate may be anywhere from 48% to 58%. A
confidence interval can be calculated based on the chosen level of confidence using the
following formula:
Interval 
p  1  p 
Z
n
This equation has three different variables:
n. This is the actual number of people who responded to the survey.
Z. This is the Z-score associated with the level of confidence chosen. For this class the level
of confidence you will use is either 90% or 95%. The Z-score associated with a 90%
level of confidence is 1.645. This can be looked up in a table of Z-scores (such as the
one in the back of the textbook) by dividing the level of confidence number by two and
looking up the row and column of the Z-score table associated with this value. For
example, to look up the Z-score for a 90% confidence interval, first divide 0.9 by two, or
0.45. Looking up 0.45 in the Z-score table shows it is in row 1.6 and halfway between
columns .04 and .05, or .045. Therefore, the Z-score for a 90% confidence interval is 1.6
+ 0.45 = 1.645. Similarly, the Z-score for a 95% confidence interval is 1.96.
p. This is an estimate of the percentage of respondents answering a particular survey
question. As a general rule of thumb, you will typically use 50% (or .50) for this number.
That is because this will give you the most conservative estimate of your confidence
interval. In other words, using p = .50 will ensure that you are using the largest possible
confidence interval based on the input for the other two variable.
Example
Suppose you conduct a survey that asks respondents if they would purchase a certain product at a
particular price. Out of 80 people that respond to the survey, 46 of the people (or 57.5%) say
that they would purchase the product. The client wants to know the margin of error of this
survey at a 90% level of confidence. Based on this example you would enter into the above
formula: n = 80, p = .50, and Z = 1.645. The formula returns the value of 0.092 or 9.2%. This
means you are 90% confident that 57.5% of the population, plus or minus 9.2%, would purchase
this product at this price. The margin of error for this survey is 9.2% at a 90% level of
confidence.
One final item to note is that this formula only works for large sample sizes (30 or more
respondents) and for random samples. If there is sampling error in your data (which may result
from choosing a sample of convenience rather than a true random sample of the target
population) the actual confidence interval will be larger than the interval calculated from this
formula.