Margin of ErrorHow accurate are statistics in the media?

What is margin of error?

You may have heard the term margin of error used during elections in the media. Media outlets may say that a candidate has 34% of the votes with a 3% margin of error.

Why can’t they say exactly what percent of the votes the candidate has?

Because they are using a sample to make an inference about the population.

Definition

Margin of error is the possible error made when using your sample to make inferences about your population.

Because you are not sampling every single unit in your population, some error is possible.

How can we determine the margin of error?

How big is our margin of error?

The margin of error is dependent on two things: the sample standard deviation, and the confidence level.

The standard deviation shows how spread out our data points are.

The confidence level gives us a multiplier based on how accurate we would like to be with our estimate.

Margin of Error in the Media

Elections are the most common places that margins of error are reported by the media.

Presidential elections garner constant updates on who has how many percentage points of the vote, and which candidate is winning. But it is important to know what the margin of error is before you get to excited about your candidate being ahead.

Hypothetical Election

Suppose there is an election between two candidates. Candidate A has 48% of the vote, and Candidate B has 52% of the vote. But since we only used a small sample of the population, there is some error. Specifically, our margin of error is 5%.

Is there a statistical difference between our two candidates?

Example continued

With a margin of error of 5%, that means the true percentage of people who will vote for Candidate A may be 5% above the sample percentage of 48%, or 5% below it. So the true percentage is between 43% and 53%.

Similarly, the true percentage for Candidate B may be 5% above the sample percentage of 52%, or 5% below it. So the true percentage is between 47% and 57%.

Is Candidate B ahead?

Since there is error involved, we can not definitively state that candidate B is ahead. His percentage may be as low as 47%, whereas his opponents may be as high as 53%. When an election is this close, it is said to be a statistical tie, since, within the methods of sampling and inference, there is no way to tell who is ahead.

Confidence Intervals

A confidence interval is an interval around a statistic that will capture the parameter most of the time. How often depends on your confidence level.

The most common confidence level is 95%. A 95% confidence interval will contain the parameter 95% of the time. Or if you calculate 100 confidence intervals for the population, roughly 95 of them will capture the parameter.

How does a confidence interval relate to margin of

error?The margin of error is half the width of a confidence level.

When dealing with means, the equation for a confidence interval is as follows:

x-bar ± multiplier * standard error

Or: x-bar ± (margin of error)

Since we are adding and subtracting the margin of error to and from the mean, the width of our interval is 2*(margin of error)

Confidence Intervals for Candidates A and B

• Going back to our hypothetical political example, the confidence interval for candidate A would be:

• 48% ± 5% = [43%, 53%]

• For candidate B, the confidence interval would be:

• 52% ± 5% = [47%, 57%]

• Since there is overlap between these two intervals, we can conclude that there is no statistical difference between the two candidates’ percentages.