Estimation Strategies and Tricks

There are a few common areas that trip candidates up when answering estimation interview questions. In this lesson we’ll cover strategies to help you:

• Effectively estimate unknown quantities
• Prevent basic math errors
• Gut-check your estimates as you go

Estimating unknowns

It’s likely that you’ll have to make an estimate for some quantity you know little or nothing about. Here are some tips for moving past unknowns quickly and effectively.

Estimate via proxy

Estimation via a proxy allows you to substitute a complete unknown for something familiar to you. Instead of guessing, choose a related concept and make adjustments up or down as needed.

Let’s take the example question “How many cars are there in Seattle?”

It’s likely you won’t know offhand. However, you might hypothesize that the number of cars is directly related to the number of families in the area. You might further hypothesize that the city of Seattle is about as densely-packed and perhaps twice as large as San Francisco. Estimating the number of families in Seattle is easier to estimate directly than number of cars as you can draw from personal experience.

If you live in a large, densely-packed city, Seattle may have a similar number of families. If you live in the countryside, Seattle might contain 3x the number of families per square mile or more. Make an estimate based on your judgment, and then adjust it as needed.

For example, you may arrive at a first estimate of 300,000 households in Seattle. Returning specifically to the number of cars, you might further assume that each household has, on average, two cars.

Therefore, your estimate would be 300,000 households x 2 cars = 600,000 cars in Seattle

Segment large groups

It’s difficult to reason about large, diverse groups.

For example, if you were asked “How many photos does the average iPhone user take a week?” finding an initial estimate might feel tricky. Is it 20? Or one? Or is it 100? We recommend breaking up large groups into smaller segments, estimating for those segments specifically, and summing the results.

To do this, come up with a segmentation scheme that makes sense given your question. State assumptions about each segment to your interviewer and briefly discuss the size of each group and any important behaviors.

Returning to the iPhone question, you might segment users according to their iPhone use. Let's say you come up with three categories: infrequent, standard, and power users. You might state the following assumptions about the size of each group and how often they take photos.

• Infrequent: “I’m defining infrequent users as relatively non-tech-savvy users or people who don’t care about photos. They may still occasionally take photos, say to share information with someone. I estimate this group represents 10% of users, and they take four photos per week.”
• Standard: “I define standard users as those who sometimes share photos with friends, say to document notable events, and who sometimes post to social media. They likely take a low, consistent number of photos, with infrequent spikes representing activities where they’ll take many photos. I’ll say this represents 80% of users, and they take 20 photos per week.”
• Power: “I define power users as those who actively document their life, communicate frequently using photos, or have a professional reason to take photos. I estimate this group to be 10% of users taking upwards of 150 photos per week.”

If you take the weighted average of these, you’d end up with ~30 photos/week.

Use personal references

In essence, estimation is the process of relating an unknown quantity to some known quantity. An effective last-ditch strategy is to relate the unknown quantity to some known quantity in your life — with the important caveat that you must account for bias in your reference.

Returning to the question “How many cars are there in Seattle?” you might calculate that 50% of your friends have cars. This is a helpful starting point, but it would be naive to assume that 50% of the general population of Seattle has cars too. For instance, if most of your friends live in especially dense city neighborhoods with great public transportation, you may need to adjust your Seattle estimate upwards, as your friends are probably less likely to own cars than the average Seattle resident.

One way to mitigate the risk that a biased reference introduces is to combine the personal reference strategy with segmentation. For instance, It may be safer to estimate that 40–50% of Seattle is around the same age as your friend group. The other 50% of the population might have more or fewer vehicles (use your PM judgment to make an assumption).

If you assume that the population of Seattle is somewhere around 1 million, this would give you at least 1 million x 50% of the population same age as friends x 50% of your friends have cars = 250,000 cars. You’d estimate that the segment of the Seattle population that is similar to your friend group, or ~50%, accounts for 250,000 cars in Seattle.

Now, you only have to make estimates for the remaining 50% of Seattle dwellers, and you’ve got a solid reference number to gut-check against.

Define upper and lower bounds

One final strategy for estimating unknowns is to state upper and lower limits on what’s reasonable. What are the realistic upper and lower limits on the quantity you’re estimating? Setting boundaries upfront will keep your estimate from spiraling out of control, and you can continue to refine your bounds through the course of the interview.

For example, continuing the “cars in Seattle” question, you know that the extreme lower limit on cars in Seattle is zero. The extreme upper limit is likely equivalent to the population of Seattle, assuming each resident owns a car. This puts the upper limit around one million based on your earlier estimate.

By taking the average of the two numbers, you’d get 500,000 cars in Seattle. The actual number is closer to 400,000, but, in the interview, arriving at a correct answer is less important than demonstrating thoughtful problem-solving and estimation skills.

Avoiding math errors

Mental math is difficult under pressure. You are always free to simplify numbers and calculations as needed — though you should state your simplifications to your interviewer. Here are some tips for making calculations easier.

Round difficult numbers

If you have an inconvenient number, round it.

For example, the current US population is 334,944,277. If you need to use this figure in your calculations, you can make it more usable by rounding it down to 300 million.

Other examples include:

• 7.6 billion becomes 8 billion (or 10 billion, if it makes sense to over rather than underestimate)
• 237 million becomes 200 million
• 84 becomes 80

Another strategy is to round to numbers that make the math easier rather than rounding up or down based on the figure itself. For instance, to find the average spend of your customers, assuming $85K in revenue with 9K customers, you could round your revenue down to $81K, which divides evenly into 9K for an average spend of $9 per customer. Alternatively, you could round up to $90K in revenue for an average spend of $10 per customer.

Split multiplications

Even after rounding, multiplying big numbers can be difficult.

One trick is to split the multiplications. First, separate out the powers of ten from your numbers, multiply the remaining numbers, then bring the powers of ten back in. Here’s a demonstration:

70,000,000 x 40

= (7 x 10,000,000) x (4 x 10)

= (7 x 4) x (10,000,000 x 10)

= (28 x 100,000,000)

= 2.8 billion

You can also split tricky multiplication problems into a sum of easier multiplications. For example:

17 x 120

= 17 x (100 + 20)

= (17 x 100) + (17 x 20)

= 1700 + 340

= 2040

This method works on percentages as well. Let’s say you needed to take 15% of 70.

15% of 70

= (10% + 5%) of 70

= 10% of 70 + 5% of 70

= (0.1 x 70) + (0.05 x 70)= 7 + 3.5

= 10.5

The easiest numbers to multiply or divide by are ten and two, so it can be helpful to round your numbers such that they’re all multiples of these.

Simplify powers of ten

There’s a handy trick for multiplying by a power of ten — simply move the decimal place over.

You’ll move the decimal to the right when multiplying by a power of ten, and you’ll move the decimal to the left when dividing by a power of ten. For example, to multiply 6 x 1,000,000, simply move the decimal in 6.0 six places to the right. Therefore, 6 x 1,000,000 = 6,000,000.

Let’s work through another example. Assume you need to multiply 50 by 1,000.

1) Add a decimal to the number you’re multiplying or dividing by. For example, 50 becomes 50.0.

2) Count the number of zeros in the power of 10. In the above example, there are three zeroes in 1,000.

2) Move the decimal to the right or left by that many spaces, depending on the operation. Moving the decimal in 50.0 three places to the right yields a total of 50,000.

Another trick is to treat big powers of ten like units when multiplying and dividing. Calculate the non-power-of-10 numbers first, then calculate the powers of ten at the end. For example:

12 million x 8 thousand

= (12 x 8) million x one thousand

= 96 million x one thousand

= 96 billion

Increasing the accuracy of your answer

You should always check your estimates before giving a final answer. In addition to checking any math, you want to ensure that your answer makes logical sense and “seems right.” Interviewers like to see candidates point out unrealistic numbers upon review and backtrack to identify the factors that might have led to errors.

Here are a few tips for ensuring your answer is as accurate as possible.

Gut-check estimates

Often, you’ll know if a number is too high or too low simply by taking a second look at it. If you think an answer is unreasonable, don’t be afraid to say so. Interviewers are looking to see that you can recognize unrealistic situations. From there, go back and try to pinpoint what you might have inaccurately estimated.

For example, say that you estimated that each household in Seattle has three cars. You’d have ended up with an estimate of 1.5 million cars in Seattle. This number seems too high.

Upon second glance, you might realize that your estimate of three cars per household is too high based on what you might know about big city transportation. You could then go back and point out the inflated cars-per-household figure, stating your initial assumptions and what may have led you astray.

Put the answer in context

Some estimates are difficult to gut-check. You may be unfamiliar with the situation you’ve been asked about, or you might be dealing with extremely large values that are difficult to visualize.

For example, if you estimate that Google spends $100M a year running a specific product, it’s likely you won’t know whether that’s reasonable unless you have specific experience in corporate finance or strategy.

It can be helpful to put the number in a different context. If you previously estimated that the Google product in question has 20M users, your $100M estimate would result in a cost of $5/user. This should be easier to gut-check based on your sense of the product and its users.

Check against competitors

Another simple way to check estimates is to make use of competitive benchmarks. For example, if you were estimating Dropbox’s monthly users, you could use prior knowledge about Google Drive/Google Cloud users to make sure your Dropbox estimates aren’t too far off. Of course, this strategy hinges on prior knowledge.