7.5: Estimating a Confidence Interval

Statistics means never having to say you’re certain – Unknown origin

Up to this point in this chapter, we’ve outlined the basics of sampling theory which statisticians rely on to make guesses about population parameters on the basis of a sample of data. As this discussion illustrates, one of the reasons we need this sampling theory is that every data set leaves us with some uncertainty, so our estimates are never going to be perfectly accurate. The thing that has been missing from this discussion is an attempt to quantify the amount of uncertainty that attaches to our estimate. It’s not enough to be able to guess that, say, the mean IQ of undergraduate psychology students is 115 (a probably fictional number to be sure). We also want to be able to say something that expresses the degree of certainty (or uncertainty, as the case may be) that we have in our guess. For example, it would be nice to be able to say that there is a 95% chance that the true mean lies between 109 and 121. The name for this is a confidence interval for the mean.

We'll begin with a short Crash Course video about confidence intervals:

Armed with an understanding of sampling distributions, constructing a confidence interval for the mean is actually pretty easy. Here’s how it works. Suppose the true population mean is μ and the standard deviation is σ. We’ve just finished running a study that has N participants, and the mean IQ among those participants is \(\bar{X}\). We know from our discussion of the central limit theorem (Section 7.3.3 that the sampling distribution of the mean is approximately normal. We also know from our discussion of the normal distribution Section 4.5 that there is a 95% chance that a normally-distributed quantity will fall within two standard deviations of the true mean. To be more precise, we can use SPSS to compute the 2.5th and 97.5th percentiles of the normal distribution:

 

Okay, so I lied earlier on. The more correct answer is that 95% chance that a normally-distributed quantity will fall within 1.96 standard deviations of the true mean. Next, recall that the standard deviation of the sampling distribution is referred to as the standard error, and the standard error of the mean is written as SEM. When we put all these pieces together, we learn that there is a 95% probability that the sample mean \(\bar{X}\) that we have actually observed lies within 1.96 standard errors of the population mean. Mathematically, we write this as:

μ−(1.96×SEM) ≤ \(\bar{X}\) ≤ μ+(1.96×SEM)

where the SEM is equal to σ/\(\sqrt{N}\), and we can be 95% confident that this is true. However, that’s not answering the question that we’re actually interested in. The equation above tells us what we should expect about the sample mean, given that we know what the population parameters are. What we want is to have this work the other way around: we want to know what we should believe about the population parameters, given that we have observed a particular sample. However, it’s not too difficult to do this. Using a little high school algebra, a sneaky way to rewrite our equation is like this:

\(\bar{X}\)−(1.96×SEM) ≤ μ ≤ \(\bar{X}\)+(1.96×SEM)

What this is telling us is that the range of values has a 95% probability of containing the population mean μ. We refer to this range as a 95% confidence interval, denoted CI₉₅. In short, as long as N is sufficiently large – large enough for us to believe that the sampling distribution of the mean is normal – then we can write this as our formula for the 95% confidence interval:

\(\mathrm{CI}_{95}=\bar{X} \pm\left(1.96 \times \frac{\sigma}{\sqrt{N}}\right)\)

Of course, there’s nothing special about the number 1.96: it just happens to be the multiplier you need to use if you want a 95% confidence interval. If I’d wanted a 70% confidence interval, I could have used the qnorm() function to calculate the 15th and 85th quantiles:

 

and so the formula for CI₇₀ would be the same as the formula for CI₉₅ except that we’d use 1.04 as our magic number rather than 1.96.

A Slight Mistake in the Formula

The formula given above for the 95% confidence interval is approximately correct, but we glossed over an important detail in the discussion. Notice the formula requires you to use the standard error of the mean, SEM, which in turn requires you to use the true population standard deviation σ. Yet, in Section @ref(pointestimates we stressed the fact that we don’t actually know the true population parameters. Because we don’t know the true value of σ, we have to use an estimate of the population standard deviation \(\hat{σ}\) instead. This is pretty straightforward to do, but this has the consequence that we need to use the quantiles of the t-distribution rather than the normal distribution to calculate our magic number, and the answer depends on the sample size. When N is very large, say 10000, we get pretty much the same value using the t-distribution (tScore10000) as we did for the normal distribution (ZScore). However, when our sample size (N) is small, say 10, we get a very much different and higher value using the t-distribution (tScore10):

There’s nothing too mysterious about what’s happening here. Bigger values mean that the confidence interval is wider, indicating that we’re more uncertain about what the true value of μ actually is. When we use the t-distribution instead of the normal distribution, we get bigger numbers, indicating that we have more uncertainty. And why do we have that extra uncertainty? Well, because our estimate of the population standard deviation \(\hat{σ}\) might be wrong! If it’s wrong, it implies that we’re a bit less sure about what our sampling distribution of the mean actually looks like… and this uncertainty ends up getting reflected in a wider confidence interval.

Interpreting a Confidence Interval

The hardest thing about confidence intervals is understanding what they mean. Whenever people first encounter confidence intervals, the first instinct is almost always to say that “there is a 95% probabaility that the true mean lies inside the confidence interval”. It’s simple, and it seems to capture the common sense idea of what it means to say that I am “95% confident”. Unfortunately, it’s not quite right. The intuitive definition relies very heavily on your own personal beliefs about the value of the population mean. I say that I am 95% confident because those are my beliefs. In everyday life that’s perfectly okay, but if you remember back to Section 9.2, you’ll notice that talking about personal belief and confidence is a Bayesian idea. Personally (speaking as a Bayesian) I have no problem with the idea that the phrase “95% probability” is allowed to refer to a personal belief. However, confidence intervals are not Bayesian tools. Like everything else in this chapter, confidence intervals are frequentist tools, and if you are going to use frequentist methods then it’s not appropriate to attach a Bayesian interpretation to them. If you use frequentist methods, you must adopt frequentist interpretations!

Okay, so if that’s not the right answer, what is? Remember what we said about frequentist probability: the only way we are allowed to make “probability statements” is to talk about a sequence of events, and to count up the frequencies of different kinds of events. From that perspective, the interpretation of a 95% confidence interval must have something to do with replication. Specifically: if we replicated the experiment over and over again and computed a 95% confidence interval for each replication, then 95% of those intervals would contain the true mean. More generally, 95% of all confidence intervals constructed using this procedure should contain the true population mean. This idea is illustrated in Figure 7.13, which shows 50 confidence intervals constructed for a “measure 10 IQ scores” experiment (top panel) and another 50 confidence intervals for a “measure 25 IQ scores” experiment (bottom panel). A bit fortuitously, across the 100 replications that I simulated, it turned out that exactly 95 of them contained the true mean.

Figure 7.13: 95% confidence intervals. The top (panel a) shows 50 simulated replications of an experiment in which we measure the IQs of 10 people. The dot marks the location of the sample mean, and the line shows the 95% confidence interval. In total 47 of the 50 confidence intervals do contain the true mean (i.e., 100), but the three intervals marked with asterisks do not. The lower graph (panel b) shows a similar simulation, but this time we simulate replications of an experiment that measures the IQs of 25 people.

The critical difference here is that the Bayesian claim makes a probability statement about the population mean (i.e., it refers to our uncertainty about the population mean), which is not allowed under the frequentist interpretation of probability because you can’t “replicate” a population! In the frequentist claim, the population mean is fixed and no probabilistic claims can be made about it. Confidence intervals, however, are repeatable so we can replicate experiments. Therefore a frequentist is allowed to talk about the probability that the confidence interval (a random variable) contains the true mean; but is not allowed to talk about the probability that the true population mean (not a repeatable event) falls within the confidence interval.

I know that this seems a little pedantic, but it does matter. It matters because the difference in interpretation leads to a difference in the mathematics. There is a Bayesian alternative to confidence intervals, known as credible intervals. In most situations credible intervals are quite similar to confidence intervals, but in other cases they are drastically different. As promised, though, I’ll talk more about the Bayesian perspective in Chapter 17.

Plotting Confidence Intervals in SPSS

There are several different ways you can draw graphs that show confidence intervals as error bars. Two of those ways are shown here. Let's assume that you want to draw a plot showing the means and confidence intervals for one variable, broken down by different levels of a second variable. For instance, in our MLBGL2021.sav data, we might be interested in plotting the average attendance by home team. Here's how we might do that.

First, open the MLBGL2021.sav data file in SPSS. Harkening back to chapter ???, click on Graphs and then Chart Builder.... Once open, click on Bar in the gallery and then double-click on the Simple Bar icon to create a simple bar graph (obviously!). Then, from the list of variables on the upper-left, drag Home Team to the x-axis area and Attendance to the y-axis. On the far right, check the box that says Display Error Bars. That's it! The chart builder window should look something like this:

Click on OK and SPSS will produce the following bar graph:

Notice that the 95% confidence intervals are plotted at the top of each bar which represents the mean attendance of each game at each home baseball stadium. If you're more into line graphs, you can do that too:

which produces the following output in SPSS:

Bar charts are probably the most commonly used for displaying the confidence intervals, but, hey, at least you have some creative control.