On Confidence Intervals

As with practically every other Indian household, so with mine. Trudging back home after having written the math exam was never much fun.

It wasn’t fun because most of your answers wouldn’t tally with those of your friends. But it wasn’t fun most of all because you knew the conversation that waited for you at home. Damocles had it easy in comparison.

“How was the exam?”, would be the opening gambit from the other side.

And because Indian kids had very little choice but to become experts at this version of chess very early on in life, we all know what the safest response was.

“Not bad”.

Safe, you see. Non-committal, and just the right balance of being responsive without encouraging further questioning.

It never worked, of course, because there always were follow-up questions.

“So how much do you think you’ll get?”

There are, as any kid will tell you, two possible responses to this. One brings with it temporary relief, but payback can be hellish come the day of the results. This is the Blithely ConfidentTM method.

“Oh, it was awesome! I’ll easily get over 90!”

The other response involves a more difficult conversation at the present juncture, but as any experienced negotiator will tell you, expectations setting is key in the long run.

“Not sure, really.”

Inwardly, you’re praying for a phone call, a doorbell ring, the appearance of a lizard in the kitchen – anything, really, that will serve as a distraction. Alas, miracles occur all too rarely in real life.

“Well, ok”, the pater would say, “Give me a range, at least.”

We’ve all heard the joke where the kid goes “I’ll definitely get somewhere between 0 and 100!”.

Young readers, a word of advice: this never works in real life. Don’t try it, trust me.

But joke apart, there was a grain of truth in that statement. That was the range that I (and every other student) was most comfortable with.

Or, in the language of the statistician, the wider the confidence interval, the more confident you ought to be that the parameter will lie within it.1

What range should one go with? 0-100 is out unless you happen to like a stinging sensation on your cheek.

You’re reasonably confident that you’ll pass – it wasn’t that bad a paper. And if you’re lucky, and if your teacher is feeling benevolent, you might even inch up to 80. So, maybe 40-80?

“I’ll definitely pass, and if I’m lucky, could get around 60 or so”, you venture.

“Hmmm,” the pater goes, ever the contemplative thinker. “So around 60, you’re saying?”

“Well yeah, around that”, you say, hoping against hope that this conversation is approaching the home stretch now.

“Around could mean anything!”, is the response. “Between 50 and 70, or between 40 and 80?! Which is it?!”

And that, my friends, is the intuition behind confidence intervals. Your parents are optimizing for accurate estimates (a narrower range), and you want to tell them that sure, you can have a narrower range – but the price they must pay is lesser confidence on your part.

And if they say, well, no, we want you to be more confident about your answer, you want to tell them that sure, I can be more confident – but the price they must pay is lower accuracy (a broader range).

And sorry, you can’t have both.

(Weird how parents get to say that all the time, but children, never!)

But be careful! This little story helps you get the intuition only. The truth is a little more subtle, alas:

The confidence interval can be expressed in terms of samples (or repeated samples): “Were this procedure to be repeated on numerous samples, the fraction of calculated confidence intervals (which would differ for each sample) that encompass the true population parameter would tend toward 90%


Or, in the case of our little story, this is what an Indian kid could tell their parents:

Were I to give the math exam a hundred times over, I would score somewhere between 50 and 70 about ninety times. And I would score between 40 and 80 about 95 times.

Now, if you ask where we get those specific sets of numbers from ( [50-70, {90}] , [40-80, {95}] ) , that takes us into the world of computation and calculation. Time to whip out the textbook and the calculator.

But if you are clear about why broader intervals imply higher confidence, and narrow intervals imply lower confidence, then you are now comfortable about the intuition.

And I hope you are clear, because that was my attempt in this blogpost.

Kids, trust me. Never try this at home.

But please, do read the Wikipedia article.

  1. Statisticians reading this, I know, I know. Let it slide for the moment. Please.[]