Welcome back to class! Today we are tackling Section 6.3: The Statistics of Polling. In a world filled with news headlines, election forecasts, and product surveys, understanding how polling works is an essential skill for quantitative literacy. Have you ever wondered how surveying just 1,000 people can tell us what millions of Americans are thinking? The answer lies in the math.

Key Concepts: Can We Believe the Polls?

To interpret a poll correctly, you need to look beyond the headline number. There are three interconnected terms you need to master:

  • Margin of Error: This expresses how close to the "true" result (the result for the whole population) the poll result is expected to lie. It represents the "wiggle room" due to sampling.
  • Confidence Interval: This is the range found by adding and subtracting the margin of error from the poll's result.
  • Confidence Level: This tells us the percentage of such polls in which the confidence interval actually includes the true result. In this class (and most standard statistics), we typically use a 95% confidence level.

The Mathematics of Margin of Error

For a standard 95% level of confidence, there is a direct relationship between the number of people surveyed ($n$) and the margin of error. We can estimate the margin of error using this approximation formula:

$$ \text{Margin of Error} \approx \frac{100}{\sqrt{n}} \% $$

For example, if a survey asks 900 people a question (so $n = 900$), the math looks like this:

$$ \frac{100}{\sqrt{900}} = \frac{100}{30} \approx 3.3\% $$

This means we can be 95% confident that the true population opinion is within 3.3 percentage points of the poll's finding.

Calculating Sample Size

Conversely, if you are designing a study and you know how accurate you need to be, you can calculate how many people you need to interview. To find the sample size ($n$) needed to achieve a specific margin of error ($m$), we rearrange the formula:

$$ \text{Sample Size } (n) \approx \left(\frac{100}{m}\right)^2 $$

Example: Suppose you want a very precise poll with a margin of error of just 2% ($m=2$). How many people do you need to call?

$$ n \approx \left(\frac{100}{2}\right)^2 = (50)^2 = 2500 $$

You would need to survey approximately 2,500 people. Notice that to cut the error in half (from roughly 4% to 2%), you have to quadruple the sample size!

Interpreting the Results

When analyzing poll data, always look for the interval. If a poll says 52% of people support a measure with a margin of error of 3%, the confidence interval is $49\%$ to $55\%$. Because the low end of that interval drops below 50%, we cannot conclude with certainty that a majority exists.

Review the attached slides for more examples, including the "Oricon Fashion Survey" and data regarding Hurricane Katrina residents. Stay curious and keep crunching those numbers!