Every column has a shape, and so do averages.
Before modelling data, know its shape: a bell, flat, skewed or counts. Then the most useful fact in statistics: averages of samples form a bell, whatever the shape they come from, and you can say how wide.
- Recognise normal, uniform, skewed and count data
- Explain the central limit theorem in a sentence
- Tell the standard deviation from the standard error
You need: Nothing: this is where the path starts.
- 1Distributions and sampling
- 2Confidence intervals and tests
- 3Training, testing and overfitting
- 4Judging a classifier
- 5Trees and ensembles
- 6Clustering and PCA
- 7How a neural network learns
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Three shapes in one table
Predict first
The ML guide’s churn table has tenure, the monthly charge and complaints in 90 days. Which one is shaped like a bell?
Just show me
The monthly charge: a bell around ₹499, with a standard deviation of ₹150.
The table’s code draws each column the way real data often looks: the charge from a normal curve, tenure evenly from 1 to 72 months (a uniform distribution) and complaints as counts from a Poisson distribution, the shape of rare events in a period: 65% have none. The shape tells you which summaries, tests and models fit.
Monthly charges in ₹50 ranges, 4,755 customers (the ML guide’s churn table, synthetic)
Quick check
Orders an hour at a small shop: mostly 0, 1 or 2. Which distribution fits best?
Counts of independent events in a fixed time are the Poisson’s home ground.
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Random counts wobble
Predict first
A page converts 4% of its 1,000 visitors a day. How much do daily conversions vary?
Just show me
Mostly between 28 and 52. Conversions follow a binomial distribution: mean 40, standard deviation √(1,000 × 0.04 × 0.96) = 6.2, and about 95% of days land within two of them.
A day with 52 conversions isn’t a better day than one with 28: it’s the same 4% and chance. The binomial counts successes in a fixed number of tries; the Poisson counts events in a period; the normal describes sums and averages of many small effects. This everyday wobble is what an A/B test has to see past.
The A/B tests lesson, on the analyst path
Quick check
Daily sign-ups went from 40 to 49 after a change. Proof it worked?
With a mean of 40 and a spread of 6.2, days of 49 come by chance; you need a proper test.
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Samples, and the trouble with them
Predict first
You ask learners who finished a course whether it was useful. What’s wrong?
Just show me
The learners who quit, likelier to be unhappy, never answer: survivorship bias, a kind of sampling bias.
A sample stands in for everyone only if it’s drawn fairly. Simple random sampling gives everyone the same chance; stratified sampling draws from each group in proportion, as the ML guide’s train-and-test split does for churners with
stratify=y. A biased sample doesn’t improve with size: a million answers from the wrong people still mislead.Quick check
An app’s poll gets 50,000 votes. Is it representative of its users?
A self-selected poll reflects who votes, not everyone; size doesn’t fix that.
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Averages make a bell
Predict first
A die is flat: each face as likely. Average 30 rolls, and do it 1,000 times. What shape do the averages make?
Just show me
A bell around 3.5: the middle 95% of the 1,000 averages fall between 2.87 and 4.10.
That’s the central limit theorem: averages and sums of many independent draws pile into a normal curve, whatever the shape they’re drawn from, and the bell narrows as the samples grow. It’s why the normal curve is everywhere in statistics, and why confidence intervals and A/B tests work on data that isn’t normal at all.
1,000 averages of 30 rolls of a die, from the central limit theorem simulator’s own code (its seed)
Try other shapes in the central limit theorem simulator
Quick check
Delivery times are skewed. Is the average of 100 deliveries roughly normal?
With enough draws, averages come close to normal even when the data is skewed.
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Standard deviation or standard error
Predict first
Single rolls of a die have a standard deviation of 1.71. What’s the spread of averages of 30 rolls?
Just show me
0.30 in the simulation, against 1.71 ÷ √30 = 0.31 from the formula.
The standard deviation describes the data: how far single values sit from the mean. The standard error describes an estimate: how far a sample’s average would move from sample to sample, σ ÷ √n. Averages of 5 rolls spread 0.75 here, averages of 30 only 0.30; to halve the standard error, you need four times the data.
The standard error, symbol by symbol
Quick check
A report says the average order is ₹1,850, standard error ₹40. What does the ₹40 describe?
The standard error is the average’s precision, not the spread of the orders.
Lesson 1 done.
Next: Confidence intervals and tests. An estimate without its uncertainty is half an answer.
In the interview: distributions and sampling.
MediumNormal, binomial or Poisson: which distribution fits which data?
MediumWhat does the central limit theorem say, and why does it matter?
EasyWhat’s the difference between standard deviation and standard error?
EasyWhat sampling methods do you know, and what is sampling bias?
HardWhy is the standard error of the mean σ/√n? Prove that the variance of the sample mean is σ²/n.
EasyHow do left-skewed and right-skewed distributions differ?
Ask Prism about this lesson
Sources
- Diez, Çetinkaya-Rundel and Barr, “OpenIntro Statistics”, 4th edition, 2019 (CC BY-SA): distributions, sampling and the central limit theorem.
- The ML guide’s churn table, its code draws each column; the figures here computed with pandas and scikit-learn 1.8.0.
- Our central limit theorem simulator, its own code and seeds for the die.