Three shapes cover most of what you track
A distribution is just a picture of how often each outcome shows up - where the values pile up and how far they spread. For Lumen Skincare, the brand we follow through this course, three shapes do almost all the work. Order values pile into a lopsided hill. The count of conversions out of a batch of sessions has its own shape. The count of orders per day, or support tickets per hour, has another. Learn to recognise these three and you can look at any metric and know what "normal" and "surprising" should look like - before anyone hands you a number.
The normal curve and the 68-95-99.7 rule 8 min live
The bell curve is the most recognised shape in statistics for a reason: enormous numbers of things cluster symmetrically around a center, with rarer values trailing off evenly on both sides. Its real gift to a leader is a built-in ruler for "how unusual is this?" - the 68-95-99.7 rule - and a single portable number, the z-score, that says how far from normal any value sits.
LiveThe bell, the rule, and the z-score4 min▶
Three things to carry out of the normal curve:
- The bell shape. Symmetric, single peak at the mean, thinning tails on both sides. Mean and median sit together in the middle. Heights, measurement noise, and - crucially for later - averages of samples all lean this way.
- The 68-95-99.7 rule. About 68% of values land within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. That is your instant sense of "normal range" versus "worth a look".
- The z-score - "how many SDs from the mean is this value?" A z of 0 is dead average; a z of +2 is out near the unusual edge; a z beyond 3 is genuinely rare. It lets you compare wildly different metrics on one honest scale.
Lumen's order values are roughly bell-ish but right-skewed - a callback to Session 1, where the mean ($74) sat above the median ($63) because a tail of big gift-set orders pulls the average right. So the plain 68-95-99.7 rule is a decent rough guide near the middle but understates the fat right tail: genuinely large orders show up more often than a perfect bell would predict. The shape is close to normal, not exactly normal, and that gap is the whole reason we care about shape.
The binomial and the Poisson 6 min live
The bell is for continuous measurements. But a lot of what a business tracks is a count - how many of something happened. Two shapes cover most counts: the binomial (how many successes out of a fixed number of tries) and the Poisson (how many events in a fixed stretch of time or space). Match the metric to the shape and the "expected" number falls right out.
LiveBinomial: counting successes out of N tries3 min▶
The binomial shape shows up whenever you count how many "yes" outcomes you get out of a fixed number of independent yes/no tries, each with the same chance of yes.
- Ingredients: a number of tries N, and a per-try chance p. The expected count is simply N x p.
- For Lumen: out of N = 1000 checkout sessions at p = 3.2%, you expect about 32 conversions - but with normal wobble around that, exactly the clumping we saw in Session 2.
- Bigger N makes the count more predictable in relative terms - the reason small campaigns swing wildly and big ones settle down.
LivePoisson: counting events per period3 min▶
The Poisson shape shows up when you count how many events land in a fixed window of time (or space), when events happen independently at some steady average rate.
- One ingredient: the average count per window, called lambda. That average is also the peak of the shape.
- For Lumen: at ~4,000 sessions a day converting at 3.2%, you get about 128 orders per day (lambda = 128). Support tickets arrive at about lambda = 5 per hour.
- The Poisson tells you the natural spread of those counts - so you know whether "9 tickets this hour" is a normal busy patch or a genuine spike worth staffing for.
Lumen's support lead sees 9 tickets in one hour against a usual 5 and wants to pull someone off another queue. But with lambda = 5, an hour of 9 is well inside normal Poisson swing - busy hours and quiet hours are baked into a random arrival process. Staffing to every busy hour would over-hire; staffing to the average with a little slack for the real spikes is the honest read. Knowing the shape turns "that felt like a lot" into "that is within normal".
Self-studySkew, and why real money and time data is never a perfect bell4 min read▶
The bell is a beautiful idealisation, but most real business data is skewed - lopsided, with a long tail on one side.
- Money is floored at zero and open-ended above. Order value, revenue per account, and deal size cannot go negative but can go very large, so they pile up low and trail off high - a right skew, mean above median.
- Time is the same. Response times and session lengths have a hard floor and a long tail of slow cases, which is why "average latency" hides the painful p95.
- The practical consequence: the 68-95-99.7 rule is a rough guide near the middle of skewed data but underestimates how often big values appear. Use it as a sanity check, not a promise, on money and time.
Play with the three shapes 3 min live
Do not take the shapes on faith - move them around. Flip the sampler between Normal, Binomial, and Poisson, then drag the parameters: push the mean and SD on the bell, change N and p on the binomial, slide lambda on the Poisson. Watch how the peak moves and how the spread widens or tightens. A minute of dragging builds more intuition than a page of description.
This week ◐ 25 min total
- Name the shape. Pick three metrics your org watches and label each: a measurement (normal-ish), a count out of tries (binomial), or a count per period (Poisson). Notice how the label sets your expectation for the typical value.
- Apply the 68-95-99.7 rule. Find a roughly bell-shaped metric with a known average and spread. Sketch the range you would call "normal" (within 2 SD) and the value that would make you look twice (beyond 3 SD).
- Check a "spike". Take a per-period count you track (tickets/hour, sign-ups/day). Estimate its average, then judge whether a recent "busy" reading is really outside normal Poisson swing or just a busy patch.
- Optional: find a money or time metric someone described with the average and note where the skew likely hides a fat tail the average glosses over.
Three questions before you go 🎯 ◐ 90 seconds
1 · A metric is roughly bell-shaped with mean 100 and SD 10. Roughly what share of values land between 80 and 120?
80 to 120 is the mean plus or minus 2 SD (2 x 10 = 20). The 68-95-99.7 rule puts about 95% of values within 2 SD of the mean.
2 · You want to model how many conversions Lumen gets out of 1000 sessions at a 3.2% rate. Which shape fits, and what is the expected count?
"How many successes out of a fixed N yes/no tries" is the binomial. The expected count is N x p = 1000 x 0.032 = 32, with normal wobble around it.
3 · Support tickets arrive at about 5 per hour. One hour brings 9. What shape describes this count, and how should you read the 9?
Counting events in a fixed window is Poisson, with lambda = 5. Poisson counts swing naturally, so an occasional 9 against an average of 5 is a busy patch, not proof the rate changed.
What this session covers
This leader session teaches how to recognise the three distributions and read them - the ~80% you need to interpret any metric's shape. The hands-on computation (fitting and simulating these in code) lives in Builder Session 4; the sources below go deeper.