Data Practice and Strategy · Module 1
Mathematical foundations of data systems
Maths in data systems describes patterns, uncertainty, and change.
Previously
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Join up data architecture, streaming, governance, and product thinking for real systems.
This module
Mathematical foundations of data systems
Maths in data systems describes patterns, uncertainty, and change.
Next
Data models and abstraction at scale
Models are simplified representations of reality.
Why this matters
Definitions:
What you will be able to do
- 1 Explain mathematical foundations of data systems in your own words and apply it to a realistic scenario.
- 2 Uncertainty enters through measurement, sampling, and modelling. You manage it, not eliminate it.
- 3 Check the assumption "Probability is a model" and explain what changes if it is false.
- 4 Check the assumption "Assumptions are stated" and explain what changes if it is false.
Before you begin
- Comfort with earlier modules in this track
- Ability to explain trade-offs and risks without jargon
Common ways people get this wrong
- False precision. Extra decimals can hide weak data. Precision should match evidence quality.
- Ignoring bias. Bias is not only ethical. It is also statistical, and it breaks inference.
Main idea at a glance
From values to uncertainty-aware decision
How I turn raw observations into decisions that account for what I don't know
Stage 1
Observed values
The measurements I collect from the system or experiment.
I never trust a single observation. I always ask what else might be true.
Maths in data systems describes patterns, uncertainty, and change. Abstraction turns messy reality into numbers we can reason about. At small scale numbers feel friendly. At scale, tiny errors compound and variation matters more than a single “best guess”.
My opinion: people do not fear maths because it is hard. They fear it because it is often introduced without kindness. If I introduce a symbol, I will tell you what it means, and I will show you a concrete example before I move on.
A vector is a list of measurements, like . A matrix is a grid of numbers, often used to represent many vectors at once, like rows of customers and columns of attributes. Probability is bookkeeping for uncertainty. It tells us how unsure we are, not what will happen.
Mean (average)
Definitions:
Mean notation
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$x_i$
The i-th value in the dataset.
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$n$
Total number of values.
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$\bar{x}$
Mean of the values.
In words, add all values and divide by how many there are. Example: values give mean .
Variance (spread)
Definitions:
Variance notation
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$\sigma^2$
Variance, measuring spread around the mean.
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$x_i - \bar{x}$
Deviation of each value from the mean.
In words, measure how far each value is from the mean, square it, and average it. Example: values with mean give variance .
Standard deviation is the square root of variance. It puts spread back into the same units as the data. Example: .
Probability distributions describe how likely different values are, which is why a fair coin is often described as 0.5 heads and 0.5 tails. Real data distributions are rarely symmetrical, so noticing the shape stops us trusting averages when the average is hiding the story.
Foundations. Ratios, units, and “does this number even make sense”
Expert data work starts with units. A number without a unit is a rumour. If one system logs energy in kWh and another in MWh, the data can be perfectly stored and perfectly wrong. Simple check: write the unit beside the value and ask if the magnitude is plausible.
Next step. Dot product (a simple way to combine measurements)
If and , the dot product is:
Intuition: it combines two lists of measurements into one number. In modelling, dot products appear everywhere (for example, linear models).
Dot product interpretation
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Pair and multiply
Multiply each aligned pair of values from vectors a and b.
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Add the products
Sum all pairwise products to obtain one scalar score.
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Use the score operationally
In linear models, this score contributes directly to prediction output.
Dot product workflow
How I combine two vectors into a single prediction score
Stage 1
Vector a
A list of values, like feature measurements for one example.
I think of vectors as aligned observations waiting to be combined.
Check yourself
What is the main decision or explanation this module gives you about mathematical foundations of data systems?
Maths in data systems describes patterns, uncertainty, and change.
Artefact and reflection
Artefact
A concise design or governance brief that can be reviewed by a team
Reflection
Where in your work would explain mathematical foundations of data systems in your own words and apply it to a realistic scenario. change a decision, and what evidence would make you trust that change?
Optional practice
Solve a complex scenario with explicit assumptions and constraints