Monday, December 28, 2020

I won an award! How to lose by winning

Company work anniversary awards

Sometimes, companies try and do a good thing but go about it so poorly, they end up doing something bad. 

A few years ago, I worked for a large company. I got to a work anniversary which triggered an award; a plastic slab I was supposed to display on my desk. How it was delivered was eye-opening.

(Winning a trophy like this would be meaningful. Image source: Wikimedia Commons. License: Public Domain.)

I was working at a different office from my manager, so the award was sent directly to me, including the written instructions to my manager on how to give me the award

How to do it wrong

The award was a tombstone-shaped piece of transparent plastic with some vaguely encouraging words embossed on it. Other than the company logo, there was no customization of any kind (not even the employee's name), it was completely generic. The instructions gave a formal pattern for how the plastic was to be awarded. They went something like this:
  • Allocate about 20 minutes for the award ceremony.
  • Gather the employee's colleagues together.
  • Thank the employee by name for their service to the company. Mention any noticeable successes. Be warm and encouraging. Use their name. Look them in the eye.
  • Hand over the award, being sure to note that it's a recognition of their service. Use their name.
  • State that you're looking forward to working with them in the future.
  • Start a round of applause.
I told my manager that this had happened and we both laughed. I told him I was going to have an award ceremony for myself and hand myself the award using the instructions in the box. He chuckled and told me to go for it. In other words, the whole thing meant nothing to either of us.

Obviously, the company's intention was to thank employees for not leaving. They'd thought it through sufficiently well enough to have a trophy that would be displayed on desks and that wouldn't cost very much. Of course, the goal of the ceremony was to celebrate the individual and make them feel special.

Unfortunately, the trophy wasn't meaningful to anyone - it didn't even look good. The instructions left a bad taste in my mouth. My guess is, the leadership was trying to reach managers who wouldn't normally celebrate individuals' contributions. By mandating the form of the ceremony, they were trying to introduce consistency and enforce meaning, but by describing the ceremony in detail, they undermined managers - this was a form of micro-managing and hinted at bigger issues with managers' people skills.

How to do it right

By contrast, I worked for another large organization that made a very big deal of work anniversaries. People who reached a significant anniversary were called into a big meeting and personally thanked by the CEO. There were meaningful gifts for reaching multiples of 5 years. Looking back on that experience, I believe the company, and the CEO were sincere - they put a lot of effort into thanking and recognizing people. The fact that the recognition was lead by the CEO made a huge difference.

Don't fake it

Employee recognition is a fraught topic and work anniversaries can be tricky. Do you celebrate or not and why? If you do celebrate, then it needs to be meaningful and focused on the person; you can't fake or mandate sincerity. If you're going to do it, do it well.

Monday, December 21, 2020

The $10 screwdriver: a cautionary management tale

Managers gone mild

I've told this story to friends several times. It's a simple story, but the lessons are complex and it touches on many different areas. See what you think.

I was a software developer for a large organization working on network-related software. For various reasons I won't go into, we had to frequently change network cards in our test computers and re-install drivers. My bosses' boss put a rule in place that we had to use IT Support to change cards and re-install drivers - we weren't to change the cards ourselves. No other team had a similar rule and there had been no incidents or injuries. Despite asking many times, he wouldn't explain why he put the rule in place.

At first, IT Support was OK with it. But as time wore on, we wanted to change cards twice a day or more. IT Support had a lot of demands on their time and got irritated with the constant requests. They wanted to know why we couldn't do it ourselves. One of the IT guys burned us a CD with the drivers on it and told us to get our own screwdrivers and change the cards ourselves. They started to de-prioritize our help requests because, quite rightly, they had other things to do and we could swap the cards ourselves. It got to the stage where we had to wait over two hours for someone to come, unscrew two screws, swap the card, and screw the two screws back in.

We were very sympathetic to IT Support, but the situation was becoming intolerable. My software development team complained to our management about the whole thing. My bosses' boss still wouldn't budge and insisted we call IT Support to change cards, he wouldn't explain why and he wouldn't escalate the de-prioritization of tickets. 

Excalibur the screwdriver

I got so fed up with the whole thing, I went out one lunchtime and bought a £7 ($10) screwdriver. It was a very nice screwdriver, it had multiple interchangeable heads, a ratchet action, and it was red. I gave it to the team. We used the screwdriver and stopped calling IT Support - much to their relief.

The blessed screwdriver

(This isn't the actual screwdriver I bought, but it looks a lot like it. Image source: Wikimedia Commons, Author: Klara Krieg, License: Creative Commons.)

The consequences

I then made a big mistake. I put in an expenses claim for the screwdriver.

It went to my boss, who didn't have the authority to sign it off. It then went to his boss, who wasn't sure if he could sign it off. It then went to his boss, who did have the authority but wanted to know more. He called a meeting (my boss, my bosses' boss, my bosses' bosses' boss) to discuss my expenses claim. I heard they talked about whether it was necessary or not and whether I had bought a screwdriver that was too expensive when a cheaper one would have done. They decided to allow my expenses claim this one time.

I was called into a meeting with my bosses' bosses' boss and told not to put in a claim like that again. I was called into a meeting with my bosses' boss who told me not to put in an expenses claim like that again and that I should have used IT Support every single time and if I were to do it again to buy a cheaper screwdriver. I was then called into a meeting with my boss who told me it was all ridiculous but next time I should just eat the cost. Despite asking, no-one ever explained why there had been a 'rule'. Once the screwdriver existed, we were expected to use it and not call IT Support.

Of course, the team all knew what was going on and there was incredulity about the company's behavior. The team lost a lot of respect for our leadership. The screwdriver was considered a holy relic to be treasured and kept safe.

What happened next

Subsequent to these events, I left and got another job. In my new job, I ended up buying thousands of pounds worth of equipment with no-one blinking an eye (my new boss told me not to bother him with pre-approval for anything under £1,000). All the other technical people in the group left not long after me. A competitor had been making headway in the market while I was there and really started to break through by the time I left. To respond to the competitive challenge, new management came in to make the company more dynamic and they replaced my entire management chain.

What I learned

Here's what I learned from all this. I should have eaten the cost of the screwdriver and avoided a conflict with my management chain, at the same time, I should have been looking for another job. The issue was a mismatch of goals: I wanted to build good things quickly, my management team didn't want to rock the boat. Ultimately, you can't bridge a gap this big. Buying the screwdriver was a subversion of the system and not a good thing to do unless there was a payoff, which there wasn't. 

I promised myself I would never behave like the management I experienced, and I never have. With my teams now, I'm careful to explain the why behind rules; it feels more respectful and brings people on side more. I listen to people and I've reversed course if they can make a good case. I've told people to be wise about expenses, to minimize what they spend, but when something needs to be bought, they need to buy it.

What do you think?

If you liked this post, you might also like these ones...

Sunday, December 13, 2020

What's a probability distribution?

Why should you care about probability distributions?

Using the wrong probability distribution can be extremely expensive for businesses:

  • for businesses using machinery (factories, vehicles, aircraft, etc.), it can lead to parts being changed too frequently or too infrequently
  • for business relying on returning customers, it can lead to substantial under or over-estimates of revenue and/or targeting the wrong customers with promotions
  • for businesses forecasting future sales by territory and/or product, it can lead to poor territory allocation or poor product resource allocation.

Given that it's so important, what is a probability distribution, and what are some examples?

What's a probability distribution?

At its simplest, a probability distribution tells you how likely an outcome is given some input. For example, how is sales probability distributed by price, or how likely is a component to fail in the next month? 

If something is certain to occur, the probability is 1, if it's certain not to occur, the probability is zero.  Let's imagine a component lasts a maximum of 6 months before failure. Our probability distribution might show the probability of failure on days 1 to 180. The sum of all failure probabilities for all days must sum to 1.

In the real world, data is noisy and we don't expect real data to exactly follow theoretical distributions, but given enough data, the match should be close enough for us to reason about what's going on.

Uniform distribution - gambling and manufacturing

If the probability is the same for all input values, the distribution is uniform.

Let's imagine we're manufacturing candy, and we want to have equal numbers of red, blue, green, black, and white sweets in a packet. In theory, here's what we should observe.

But in reality, there's random noise so we might see something like this below. We can quantify the difference between the expected distribution and the actual distribution, which tells us something about the variability in the manufacturing process. 

The uniform distribution also occurs in gambling, for example, lotteries or dice games.

Reading more

Uniform distribution description by NIST

Binomial distribution - pass/fail and conversion

Each customer who comes into a store or who visits a website will either buy or not buy, which we can turn into a conversion rate. We can model these kinds of pass/fail processes using the binomial distribution. Here's the probability distribution.

The binomial distribution shows us the probability of measuring different results given an underlying 'truth'. Let's imagine the 'true' conversion rate was 0.04, we might not measure 0.04 due to sampling error, instead, we might measure 0.045 or 0.055, depending on how many samples we take. It's important to understand what this means: 

  • There is uncertainty in our measurement. 
  • The smaller the sample, the bigger the uncertainty.

Although many technical people understand this, most non-technical people do not, which can lead to tension.

Reading more

Yale stats

Poisson distribution - waiting in line

Imagine you're a bank serving customers with ATMs at a location. ATMs are expensive, but you don't want to keep people waiting in long lines to do their transactions, it's bad for business. So how do you balance the cost of an ATM against its use? By modeling how many people are using the ATM over a time period.

It turns out, the number of people who visit an ATM over a time period can be modeled using the Poisson distribution, which I've shown below. This gives us a way of assessing how much variation there might be in usage and therefore how many machines we might want to install. 

The Poisson distribution is often used to model counting processes. It's very attractive because it has an unusual feature, the standard deviation for the distribution is \(\sqrt{\gamma}\) where \(\gamma\) is the mean. Unfortunately, this property makes it a little too attractive; it's sometimes used when it shouldn't be.

Reading more

The Poisson Distribution and Poisson Process Explained

Exponential distribution

How long does a car battery last? How long do phone calls last? When will the next earthquake occur? These durations typically follow the exponential distribution (which is strongly related to the Poisson distribution). I've shown this distribution below.

Reading more

The exponential distribution

Power law distribution - finding fraud

How are incomes distributed in a population? How might you find fraud in the pattern of digits in expenses? It turns out, the distribution of the first digits in invoices follows a power-law distribution. The chart below shows a generic power-law distribution - for fraud detection, it's 'flipped'.

Reading more

Power law distribution

Normal distribution - almost everywhere, but not quite

What's the probability distribution for male soldiers' chest measurements? How are the results of A/B tests distributed? What about the distribution of measurement errors? All these, and many, many more follow the normal distribution, which is also called the Gaussian distribution or the bell curve. If you only learn one distribution, this is the one to learn. 

The properties of this distribution are extremely well-known, and every student of statistics and probability theory will know them. It's ubiquitous because of something called the Central Limit Theorem, which, simplifying a great deal, says that the sum of samples from any distribution follows a normal distribution.

Because it's everywhere, for some people, it's the only distribution they know. Like the old saying goes, if you only have a hammer, every problem is a nail. This distribution can be over-used, with bad consequences.

Here's the distribution. It ought to look familiar.

Reading more

The normal distribution

Lognormal distribution

How long do visitors spend on web pages? What about the distribution of internet traffic? Or the distribution of city sizes? These all follow a log-normal distribution that looks like the example below. The lognormal distribution is quite common in business.


Note the 'fat tail' or 'long tail' on the right-hand side. Many businesses have been caught out because they assumed sales or market risk followed a normal distribution when in fact they followed a lognormal distribution.

There's a variation of the Central Limit Theorem that yields log-normal distributions instead of normal distributions.

Reading more

Other distributions

There are lots and lots of different distributions. I saw a list of 90 the other day. Almost all of them are esoteric and apply in a very limited set of cases. You don't have to know all of them but you should be aware that choosing the right distribution is important to make the correct estimates. The distributions I've listed in this blog post are probably the most important, and you should know them and their properties.

As you asked nicely, here is a list of some distributions.

Alpha Distribution
Anglit Distribution
Arcsine Distribution
Beta Distribution
Beta Prime Distribution
Bradford Distribution
Burr Distribution
Burr12 Distribution
Cauchy Distribution
Chi Distribution
Chi-squared Distribution
Cosine Distribution
Double Gamma Distribution
Double Weibull Distribution
Erlang Distribution
Exponential Distribution
Exponentiated Weibull Distribution
Exponential Power Distribution
Fatigue Life (Birnbaum-Saunders) Distribution
Fisk (Log Logistic) Distribution
Folded Cauchy Distribution
Folded Normal Distribution
Fratio (or F) Distribution
Gamma Distribution
Generalized Logistic Distribution
Generalized Pareto Distribution
Generalized Exponential Distribution
Generalized Extreme Value Distribution
Generalized Gamma Distribution
Generalized Half-Logistic Distribution
Generalized Inverse Gaussian Distribution
Generalized Normal Distribution
Gilbrat Distribution
Gompertz (Truncated Gumbel) Distribution
Gumbel (LogWeibull, Fisher-Tippetts, Type I Extreme Value) Distribution
Gumbel Left-skewed (for minimum order statistic) Distribution
HalfCauchy Distribution
HalfNormal Distribution
Half-Logistic Distribution
Hyperbolic Secant Distribution
Gauss Hypergeometric Distribution
Inverted Gamma Distribution
Inverse Normal (Inverse Gaussian) Distribution
Inverted Weibull Distribution
Johnson SB Distribution
Johnson SU Distribution
KSone Distribution
KStwo Distribution
KStwobign Distribution
Laplace (Double Exponential, Bilateral Exponential) Distribution
Left-skewed Lévy Distribution
Lévy Distribution
Logistic (Sech-squared) Distribution
Log Double Exponential (Log-Laplace) Distribution
Log Gamma Distribution
Log Normal (Cobb-Douglass) Distribution
Log-Uniform Distribution
Maxwell Distribution
Mielke’s Beta-Kappa Distribution
Nakagami Distribution
Noncentral chi-squared Distribution
Noncentral F Distribution
Noncentral t Distribution
Normal Distribution
Normal Inverse Gaussian Distribution
Pareto Distribution
Pareto Second Kind (Lomax) Distribution
Power Log Normal Distribution
Power Normal Distribution
Power-function Distribution
R-distribution Distribution
Rayleigh Distribution
Rice Distribution
Reciprocal Inverse Gaussian Distribution
Semicircular Distribution
Student t Distribution
Trapezoidal Distribution
Triangular Distribution
Truncated Exponential Distribution
Truncated Normal Distribution
Tukey-Lambda Distribution
Uniform Distribution
Von Mises Distribution
Wald Distribution
Weibull Maximum Extreme Value Distribution
Weibull Minimum Extreme Value Distribution
Wrapped Cauchy Distribution

Continuous or discrete - shaken or stirred?

Some quantities are discrete and some are continuous. A discrete quantity is something like a sales territory (e.g. Germany, Ireland, Spain) or customer count (you can't have 0.5 of a customer). A continuous quantity can take any value, for example, speed can be 45.2 kph, 120.01 kph, and so on. Some distributions apply to both continuous and discrete, and some apply only to continuous or discrete. To muddy the waters, sometimes continuous distributions are used to approximately model discrete quantities.

Business examples

Vehicles

Imagine you're running a delivery vehicle fleet. You need to keep your vehicles on the road, but you need to keep an eye on maintenance costs. You decide to use math to guide your decisions, so you work out the average lifetime for different components. You have two components A and B with the same lifetimes in miles. If either component fails, you have to tow the vehicle, which is very expensive.

  • Component A. Lifetime is 150,000 miles.
  • Component B. Lifetime is 150,000 miles.

A vehicle comes in for maintenance with 149,000 miles on the odometer. Should you replace components A and B?

As you might expect, there's a gotcha. Without knowing the probability distribution for failures, we can't make these decisions. For example, a windshield might have a uniform failure rate distribution, with the probability of failure for miles 1-100 the same as the probability of failure for miles 100,000-100,100. A clutch may have a failure rate that increases with mileage, the probability of failure at miles 100,000-100,100 being much higher than the probability of failure at miles 0-100. Because we know what a clutch and a windshield are, we might decide to replace the clutch and leave the windshield. But what if A and B were a serpentine belt and a heat shield?

The only way to make rational decisions is to understand what distribution the probability of failure follows, which may well be very different for different components (e.g. car seats vs. tires).

Marketing

A new analyst is studying the market for luxury goods in Germany. They have partial data for the fraction of the population that have a certain income. Using what they have, they assume their data is normally distributed and they make a forecast for the fraction of the population that will have an income high enough to afford luxury items. Do you think their forecast will be too low, just right, or too high?

Incomes are usually log-normally distributed, so the analyst, in this case, has chosen the wrong distribution. Because the lognormal has a very long right tail, the analyst's estimate is likely to be an underestimate and may be substantially out. A competitor might not make the same mistake.

Takeaways

I've interviewed people who claim data science on their resumes, but only know the normal distribution. If you assume your data is normal, when in reality it's log-normal or Poisson, things are going to go badly wrong for you. Any analyst in business needs to be very comfortable with different distributions and needs to know which may be applicable and when.

Monday, December 7, 2020

How to (maybe) win at dice

Does God play dice with the universe?

Imagine I gave you an ordinary die, not special in any way, and I asked you to throw the die and record your results (how many 1s, how many 2s, etc.). What would you expect the results to be? Do you think you could win by choosing some numbers rather than others? Are you sure?

(Image source: Wikimedia Commons. Author: Diacritica. License: Creative Commons.)

What you might expect

Let's say you thew the die 12,000 times, you might expect a probability distribution something like this. This is a uniform distribution where all results are equally likely.

You know you'll never get an absolutely perfect distribution, so in reality, your results might look something like this for 12,000 throws. 

The deviations from the expected values are random noise which we can quantify. Further, we know that by adding more dice throws, random noise gets less and less and we approach the ideal uniform distribution more closely. 

I've simulated dice throws in the plots below, the top chart is 12,000 throws and the chart on the bottom is 120,000 throws. The blue bars represent the actual results, the black circle represents the expected value, and the black line is the 95% confidence interval. Note how the results for 120,000 throws are closer to the ideal than the results from 12,000 throws.


What happened in reality - not what you expect

My results are simulations, but what happens when you throw dice thousands of times in the real world?

There's a short history of probability theorists and statisticians throwing dice and recording the results.
  • Weldon threw 12 dice 26,306 times by hand and sent the results to his friend Francis Galton.
  • Iversen ran an experiment where 219 dice were rolled 20,000 times.
Weldon's data set is widely used to illustrate statistical concepts, especially after Pearson used it to explain his \(\chi^2\) technique in 1900. 

Despite the excitement you see at the craps tables in Las Vegas, throwing dice thousands of times is dull and is, therefore, an ideal job for a computer. In 2009, Zachariah Labby created apparatus for throwing dice and recording the scores using a camera and image processing. You can read more about his apparatus and experimental setup here. He 'threw' 12 dice 26,306 times and his machine recorded the results. 

In the chart below, the blue bars are his results, the black circle is the expected result, and the black line is the 95% confidence interval. I've taken the results from all 12 dice, so my throw count is \(12 \times 26,306\).
This doesn't look like a uniform distribution. To state the obvious, 1 and 6 occurred more frequently than theory would suggest - the deviation from the uniform distribution is statistically significant. The dice he used were not special dice, they were off-the-shelf standard unbiased dice. What's going on?

Unbiased dice are biased

Take a very close look at a normal die, the type pictured at the start of this post which are the kind of die you buy in shops.

By convention, opposite faces on dice sum to 7, so 1 is opposite 6, 3 is opposite to 4, and so on. Now look very closely again at the picture at the start of the post. Look at the dots on the face of the dice. Notice how they're indented. Each hole is the same size, but obviously, the number of holes on each face is different. Let's think of this in terms of weight. Imagine we could weigh each face of the dice. Let's pair up the faces, each side is paired with the face opposite it. Now let's weigh the faces and compare them.

The greatest imbalance in weights is the 1-6 combination. This imbalance is what's causing the bias.

Obviously, the bias is small, but if you roll the die enough times, even a small bias becomes obvious. 

Vegas here I come - or not...

So we know for dice bought in shops that 1 and 6 are ever so slightly more likely to occur than theory suggests. Now you know this, why aren't you booking your flight to Las Vegas? You could spend a week at the craps tables and make a little money.

Not so fast.

Let's look at the dice they use in Vegas.

(Image source: Wikimedia Commons. Author: Alper Atmaca License: Creative Commons.)

Notice that the dots are not indented. They're filled with colored material that's the same density as the rest of the dice. In other words, there's no imbalance, Vegas dice will give a uniform distribution, and 1 and 6 will occur as often as 2, 3, 4, or 5. You're going to have to keep punching the clock.

Some theory

Things are going to get mathematical from here on in. There won't be any new stories about dice or Vegas.

How did I get the expected count and error bars for each dice score? Let's say I threw the dice \(x\) times, it seems obvious we would get an expected count of \(\frac{x}{6}\) for each score, but why? What about the standard error?

Let's re-think the dice as a Bernoulli trial. Let's choose a score, say 1. If we throw the dice and it shows a 1, we consider that a success. If it shows anything else, we consider it a failure. Because we have a Bernoulli trial, we can use the binomial distribution to model the results.

  • \(n\) is the number of throws
  • \(p\) is the probability of getting a 1, which is \(\frac{1}{6}\)
  • \(q = 1- p\) is the probability of getting 2-6, which is \(\frac{5}{6}\)
So, again using Wikipedia's handy summary, for \(n\) throws:
  • The mean is \(np = 12 \times 26,306 \times \frac{1}{6} = 52,612\)
  • The standard deviation is \(\sqrt{npq} = \sqrt{12 \times 26,306 \times \frac{1}{6} \times \frac{5}{6}} = 209.388\) 
  • The 95% confidence interval is \(52,202 \) to \(53,022\) (standard deviation by 1.96).

Publications

Academics live or die by publications and by citations of their publications. Labby's work has rightly been widely cited on the internet. I keep hoping that some academic will be inspired by Labby and use modern robotic technology and image recognition to do huge (million-plus) classical experiments, like tossing coins or selecting balls from an urn. It seems like an easy win to be widely cited!