This essay is inspired by the assertion that hard work is more important for success than being smart. There are, of course, many meta-objections one could have against such an assertion, especially in the context in which I heard it. For example, one could notice that the speaker, being American, might be culturally obligated to say something like this. Or one could observe that speaking to an audience of people who wish to be told how to behave in order to be successful forces suggesting something the audience will think they can actually do. Or one could reflect on who benefits from a general milieu that encourages working hard. I am not concerned with these meta-objections, though.
In support of this assertion, the particular speaker adduced the empirical observation that most successful people work hard rather than are smart; and also that plenty of smart people do not work hard and are not successful. This sounds compelling on the surface, but is still open to objection. First, of course, the data being from a survey, the results will be much biased by whether successful people feel obligated to downplay their intelligence (and privilege) in favor of the more culturally acceptable hard work; but there is also the issue that even if all information transmission mechanisms are perfect, looking only at successful people cannot provide adequate information about the factors that matter to success, because it neglects to account for the prevalence of those factors in failure. This essay consists of a worked example illustrating this point.
Let us take a somewhat fanciful little model, and see what features it has. Suppose success is driven by both talent and hard work, and also chance, in the following way: Let all individuals be born with some normally-distributed level of talent. Suppose that there is some threshold of talent, say 3 standard deviations out, that is required for success—that is, people whose talent exceeds the mean by 3\(\sigma\) or more may be successful, and others will not. Suppose society calls these people “smart”. But let there be two wrinkles: suppose also that any individual can choose to work hard, which imposes some cost but increases their effective talent by one \(\sigma\); and suppose furthermore that among all the people whose effective talent exceeds 3\(\sigma\), 10% succeed at random and the rest fail. In equations:
\[\begin{eqnarray*} talent & = & N(0, \sigma)\\ talent_{eff} & = & \begin{cases} talent & \mbox{if slacking off}\\ talent + \sigma & \mbox{if working hard} \end{cases} \\ P(\mbox{success}) & = & \begin{cases} 0.1 & \mbox{if } talent_{eff} > 3\sigma\\ 0 & \mbox{otherwise}\end{cases}\\ \end{eqnarray*}\]
If we furthermore suppose that i) everyone wants to succeed, ii) no one wants to work hard if it will have no effect on their success, and iii) everyone knows their endowment (and all the rules), then the following things will happen:
- 0.13% of people will be called “smart” (talent > 3\(\sigma\)), but none of them will work hard (why bother?)
- 2.15% of people will have 2\(\sigma\) < talent < 3\(\sigma\), so will work hard and be eligible to succeed.
- The remaining 97.72% of people will not work hard and will not succeed.
After the randomness of success is taken into account, the distribution of results will look like this (as percentages of the overall population):
| Succeeded | Failed | |
|---|---|---|
| Smart and worked hard | 0 | 0 |
| Smart and didn’t work hard | 0.013% | 0.117% |
| Not smart and worked hard | 0.215% | 1.935% |
| Not smart and didn’t work hard | 0 | 97.720% |
| Total | 0.228% | 99.772% |
But we are tempted to examine the data selectively. If we look just at people who succeeded, it looks like this:
| Succeeded | |
|---|---|
| Smart | 5.7% |
| Not smart but worked hard | 94.3% |
| Total | 100% |
That’s a pretty compelling-looking case for hard work. What if we do some homework and include people who obviously could have succeeded but didn’t (that is, those who were smart to begin with)?
| Succeeded | Failed | |
|---|---|---|
| Smart | 3.8% | 33.9% |
| Not smart but worked hard | 62.3% | no data |
| Total | 66.1% | 33.9% |
If anything, that makes the case for hard work seem even stronger. But what’s the real story? If you weren’t born into this world yet, would you rather be born smart or commit to working hard? 1
\[\begin{eqnarray*} P(\mbox{success}|\mbox{smart}) & = & P(\mbox{success}|talent_{eff} > 3\sigma) \\ & = & 0.1 = 10\% \\ P(\mbox{success}|\mbox{work}) & = & P(talent > 2\sigma) \cdot P(\mbox{success}|talent_{eff} > 3\sigma) \\ & = & .0228 \cdot 0.1 = 0.228\% \end{eqnarray*}\]
I know which of those chances I would rather take. 2 3
How did our examination of the data lead to such poor intuitions about what actually happens in this world? Because those examinations were constrained by a pretty harsh filter: looking at only the successful, or only those who were smart or successful, shows us only 0.228% or 0.345% of this population, respectively. Since the factors we are trying to study affect individuals’ membership in the group we observe, the selection grossly distorts our statistics.
This is not to say that I am encouraging my dear readers to slack off—on the contrary, working hard at the proper kind of work can be its own reward. This is just a cautionary tale about statistical arguments, and when one ought to find them convincing.
Notes
Note that there is a fine point in the last question: in this model, the only people who actually do work hard are the ones for whom it makes a difference, so working hard is not independent of being at least a little smart. But we need to artificially separate these two variables to judge the value of hard work as such, so we need to assume a hypothetical person who always works hard regardless of their endowment.↩
Note that even though in the abstract, smarts are much more important to success in this model than hard work, hard work is still the only thing an individual can affect, so “work hard” is still good advice—only the justification “it matters more than talent” is wrong.
Actually, whether it’s good advice or not can be a mathematical question too, if we specify how obnoxious hard work is. If I am giving advice to a completely random individual from this world, then as far as I know, P(2\(\sigma\) < their talent < 3\(\sigma\)) is 2.15%; so if succeeding is at least 500 times better than working hard is bad, my expectation of their well-being increases if they decide to work hard. That ratio goes down significantly if we consider that maybe they wouldn’t have asked me for advice if they didn’t sense that they at least had some chance of becoming successful; etc.↩
What about being born lucky? That is, imagine an individual to whom success will come automatically if they manage to get their effective talent above 3\(\sigma\). For them, if they work hard when appropriate instead of leaning exclusively on their luck,
\[ P(\mbox{success}|\mbox{luck}) = P(talent > 2\sigma) = 2.28\% \] so in this particular world, it’s better to be smart than lucky too. But that’s very dependent on the choice of parameter values.↩