The Confidence Equation: Can You Use Math to Decide Your Future?
Is there a secret formula for getting rich? For going viral? For deciding how long to stick with your current job, Netflix series, or even relationship? Turns out there is — the Confidence Equation.
How can you be confident you are doing the right thing? How can you know if you are in the right job? If your partner really is the love of your life? You might just be wondering whether or not your vacation plans will turn out as you hoped. Or you might be more concerned about your social interactions with others. Do they treat you fairly? Do you treat everyone equally? What gives you confidence that you are the type of person you would like to be?
These questions aren’t limited to your own life. How do we know if a company is guilty of sexual discrimination when hiring? How do we decide if our legal and health-care systems are fair to everyone? When a Black person applies for a job and a white person is hired, how can we be confident that this is racism? Amazingly, a single equation answers all these questions and many more. Voilà, the Confidence Equation.
Here’s how you can put it to use in your own life:
Jess isn’t sure about her career choice. She has a job at a human-rights organization. It is definitely a worthwhile cause, but her boss is horrible. She rings Jess up all hours of the day and makes unreasonable demands. Meanwhile, Jess’s friend Steve has been with Kenny for six months. Their relationship is volatile; one minute it’s hot, the next it’s cold. The arguments are terrible, but when it works, it is wonderful.
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The confidence equation can offer guidelines to both Jess and Steve—for exactly how many days she should stay in her job, and how long he should give his relationship before he gives up on Kenny. How? To see, let’s first detour into a smaller-scale problem.
You are looking for a hotel on TripAdvisor. You are happy to stay at a place that gets 4-star reviews or better, but are skeptical about anywhere with 3.5 stars or less.
Star reviews vary a bit on TripAdvisor. There are always a few enthusiasts who give straight 5s and a few disgruntled individuals who dole out single stars. To deal with all the variation in the reviews we start by calculating the standard deviation. This is done by summing the square differences between the average review and each of the individual reviews, then taking the square root of the sum and dividing by one minus the number of reviews. For example, if there are three reviews that are 3, 4 and 5 then the standard deviation, σ, is
σ=(3-4)2+(4-4)2+(5-4)23-1
If you don’t have time to do the math yourself, you can just put the numbers in an online standard deviation calculator. You will find that, in this case, σ=1.
Our question is how many observations we need to be certain that the true ranking of the hotel lies within a certain interval. To work this out, we use the Confidence Equation.
h± 2× σ /n
In this equation, h denotes the average score. Let’s say this is h=4, so that the average review is 4 stars. Let’s further assume that the standard deviation of the reviews is σ=1. Finally, n is the number of reviews. Imagine we have only had four reviews and n=4. Putting these numbers in to the confidence equation gives
4 ±2 ×1/4
which is
4 ±1
This means that the true ranking of the hotel might be as high as 4+1=5, but it could also be as low as 4-1=3. This is what the symbol denotes, we can be reasonably confident that the hotel has a rating between 4 ‘plus or minus’ 1.
How many reviews do we need to read to be confident that the hotel is at least a 3.5? The answer is n = 16. To see why, we substitute n = 16 in to the confidence equation to get,
4 ±2 ×116=4±0.5
We are now confident that the hotel has a rating between 3.5 and 4.5. If you don’t want to do the calculation every time, sixteen should be your rule of thumb.
Instead of looking at the average of all the hundreds of reviews for a hotel spanning many years, pick out the latest sixteen and take the average. This gives you both up-to-date and reliable information.
Returning to Steve and Jess, it turns out it isn’t just hotels that can be rated in stars. To figure out if and when they should change their respective situations, the first thing they need to do is identify the relevant time intervals. They decide to rate every day with 0 to 5 stars. They then plan to meet up regularly to evaluate their respective situations.
On the Friday night of the first week, Steve has a massive argument with Kenny, because he refuses to go out with Steve’s friends. Steve rings Jess to cry over the phone. He has given three days in his week 1 star each. She reminds him that they agreed not to draw conclusions too quickly. After all, n = 7. They can’t yet find the signal in the noise. Jess has had an OK week at work, mainly because her pain-in- the- ass boss is away on a trip, so she has collected 3 and 4 stars.
After a month, n = 30, they meet up for lunch. They are starting to get a better idea of how things are working out. Steve has had a good few weeks with Kenny. Last weekend, the couple went to the Hamptons for the weekend and, with the inclusion of a few nice dinners, they have had a wonderful time. Steve has rows of 5-star days. For Jess, it has been the opposite. When her boss came back, she was angry all the time, shouting and losing her patience over the smallest of mistakes. Jess’s days are turning into 2, 1, and some 0 stars.
After a little over two months, n = 64, and their level of confidence is now three times higher than that of the first week. For Steve, the good days outweigh the bad days, but there are still small arguments now and again: 3-and 4-star weeks. Jess’s boss is a real problem, but Jess has been working on a worthwhile project that she had always wanted to focus on. At best, she has a few 3 and 4 stars, but otherwise mostly 1 star and 2 stars.
Although each week offers new observations, the square root of n rule in the confidence equation means that Jess and Steve aren’t gaining information as rapidly as they did when they started meeting. The returns gained by observations diminish. They decide that they are going to set a deadline for their weekly deliberations. After just less than three and a half months (after 100 days), they should be confident enough to decide their futures for good.
It is the big day: n = 100. They look back, not just over the last couple of weeks but over everything that has happened during that time. For Steve and Kenny, the fights have become less frequent. They have started going to cooking lessons together and enjoy nights in making food, often inviting friends over. Life is good. Steve makes a confidence interval for his relationship: 4.3 ± 0.2, a solid average, confidently over 4 stars. Steve decides to stop moaning about Kenny; he is reassured he has found a partner in life. Things aren’t going that well for Jess. Her confidence interval is 2.1 ± 0.1. Basically, Jess has a 2-star job. She has already started looking for new positions, and on Monday, she is going to hand in her notice.
To read more about the confidence equation, check out The Ten Equations that Rule the World by David Sumpter.
Copyright 2021, reprinted with permission from Flatiron Books.
