Posted September 3, 2013

**What** is **margin of error**? Margin of error is an interval estimate—a pair of percentages surrounding a guess about some attribute of the full population based on a random sample from that population. “Margin of error allows us to feel confident a certain percentage of the time, within a range above or below the ideal guess, represented by a margin we believe is least in error” (Statistics Solutions, 2013a, para. 5).

**Why** do we use margin of error? Whenever we use a representative sample to guess something about a full population, our guess will contain some uncertainty. Using our sample statistic, we have to *infer* the real statistic—and that inference will mean our guess will usually be somewhere near the actual figure (a bit too low or a bit too high, Statistics Solutions, 2013b).

**How** does margin of error work? Let’s say we conduct a survey of college students at four year institutions asking whether they prefer physical text books or electronic books (eBooks). According to the U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics (2012) there are about 13,494,131 students at four year institutions. We can’t realistically survey 13,494,131 students, so we gather a random sample from 2,500 that are representative of the full population. Let’s say our data show that 1,875 out of 2,500 prefer eBooks (1875 / 2500 = .75 or 75%). Our margin of error, at a 95% confidence level, would be ±2% (*M* = 75, 95% CI [73.0, 77.0]). But how did we get the ±2% figure for margin of error (our confidence interval)? According to Sullivan (2006), our basic formula for a dichotomous outcome is:

*margin of error = critical value * standard error*

The resulting margin of error is what we will add or subtract from our guess to create our confidence interval. What is the *critical value*? The critical value is a cut-off value that tells us how far from the sample mean we can vary and remain confident—usually one *standard deviation* from the mean. We usually look it up in a *z* table or *t* table, although we can also compute it. For our example, for a large sample size of 2,500 and a 95% level of confidence, our critical value would be ±1.96.

What do we mean by *standard error*? Well, standard error is to a sample what standard deviation is to a population. To compute it, per Smith (2009), we estimate the *population proportion* (a number between 0 and 1). Our statistic is 75%, so as a proportion that would be 75 / 100 = .75. We might call this *p* but we don’t know *p* for sure, so we use *p̂* (pronounced *p-hat*). Now we’re ready to calculate standard error using our statistic (.75). The formula is just:

*SE* = sqrt (*p̂ *(1 – *p̂*) / *n*)

Where: *SE* stands for *standard error*, *p̂* is our estimated population proportion, and *n* is our sample size. Substituting our values we get:

*SE* = sqrt (.75 (1 – .75) / 2500) = .0087

Now let’s use our complete formula (*margin of error = critical value * standard error*)

*E* = 1.96 * sqrt (*p̂ *(1 – *p̂*) / *n*) = (1.96 * .0087) = **.017**

To get our interval, all we need to do is subtract from or add to our guess (while rounding to 2 decimal places):

= (*p̂* – (1.96 * *SE*), *p̂* + (1.96 * *SE*) ) = (.75 – .017), (.75 + .017) = (**.73, .77**)

So, our margin of error, at a 95% confidence level, would be *M* = 75, 95% CI [73.0, 77.0] (based on Smith, 2009; Sullivan, 2006).

**References**

Smith, R. L. (2009). *Point and interval estimates*. Available at http://www.unc.edu/~rls/s151-09/class19.pdf

Statistics Solutions. (2013a). *Margin of error*. Available at https://www.statisticssolutions.com/margin-of-error/

Statistics Solutions. (2013b). *Why do we have margin of error in statistics?* Available at https://www.statisticssolutions.com/why-do-we-have-margin-of-error-in-statistics/

Sullivan, L. M. (2006). Estimation from samples. *Circulation, 114*(5), 445-449. doi:10.1161/CIRCULATIONAHA.105.600189

U.S. Department of Education. Institute of Education Sciences, National Center for Education Statistics. (2012). Table 223: Total fall enrollment in degree-granting institutions, by control and level of institution: 1970 through 2011. *In U.S. Department of Education, National Center for Education Statistics* (Ed.), Digest of Education Statistics (2012 ed.). Retrieved from http://nces.ed.gov/programs/digest/d12/tables/dt12_223.asp