Learning from Data
Statistics is the study of how data can be collected, summarised and interpreted. Real-world data is often messy, incomplete and noisy. Statistics helps extract useful information, but it must be used carefully.
- Data reflects reality imperfectly
- Interpretation matters as much as calculation does
- Statistics helps identify trends and patterns
Why We Use Statistics
Modern society generates a large amount of data, from exam results to economic indicators (statistics that measure economic activity).
Statistics allows us to organise this data and make sense of it.
However, statistical conclusions always involve a level of uncertainty and assumptions.
- Helps manage large datasets
- Used in science, economics and policy
- Results depend on data quality
- No statistical conclusion is perfectly certain
Describing Data
Descriptive statistics summarise the key features of a dataset. Measures of central tendency, such as the mean, median and mode, describe typical values, but relying on averages can give a misleading story.
- Mean vs median:
- Median: the value exactly in the middle of a dataset when the data points are arranged in order from lowest to highest. This can differ from the mean.
- Mean: the value you get by adding up all the values and dividing by the number of values.
- Mode: the most frequent value, showing common occurrences.
- Averages can be distorted by extreme values.
- Central measures must be interpreted alongside "spread" (see below).
Measures of Spread
Measures of spread describe how widely data values are distributed.
They show how much variation exists within a dataset, and help reveal the inequality, reliability and uncertainty that averages alone can "hide".
Example: Measures of Spread
Imagine you and your friends are playing a game where you all try to build the tallest tower with toy blocks.
The "average" (central tendency):
- First, you can work out the average height of all the towers. This is like finding the typical height for your group.
- But knowing only the average doesn't tell you if everyone's tower was about the same height, or if some were super short and one was super tall.
The "measures of spread" (variability):
This is where measures of spread come in. They tell us how much the tower heights varied from that average. Think of them as "spread detectives".
- Were all the towers very close to the average height? If so, the "spread" is small. Everyone was pretty consistent.
- Were some towers tiny and some gigantic, even if the average was the same? If so, the "spread" is big. The results were all over the place!
Distributions and the Shape of Data
Distributions show how values are spread across a range.
While many natural measurements follow a bell-shaped distribution (bell curve), real-world data is uneven in many cases, especially in social and economic categories.
- Bell curves describe "traditional variation"
- Uneven distributions are very common in income and wealth
- Extreme values can heavily influence averages
Correlation is Not the Same as Causation
Correlation: a situation where two variables change together, either increasing or decreasing at the same time.
Causation: a change in one variable directly produces a change in another variable.
Although correlated variables appear connected at first, correlation alone does not prove that one variable caused the other. Two variables may be linked by the influence of another variable.
Example of Correlation
Just because two things happen at the same time doesn't mean one caused the other.
Sometimes:
- They are both caused by something else, or
- They line up by chance, even though they aren't actually connected.
Simple example: when it's hot,
- Ice cream sales go up
- People go swimming more
These two things are correlated (they increase together), but ice cream does not cause swimming.
The real cause is a third factor: hot weather.
Hidden Variables and Coincidence
Some correlations exist because of hidden variables or pure coincidence. A third factor may influence both variables, or the relationship may be random rather than meaningful.
- Hidden (confounding) variables can create false patterns
- Coincidental correlations occur by chance
- Not all patterns reflect real relationships
- Context and reasoning are essential
Why Critical Thinking Matters
Because data can be misleading, critical thinking is essential when analysing statistics.
Interpreting data responsibly means questioning assumptions, considering alternative explanations and recognising uncertainty.
- Always question where data comes from
- Avoid jumping from correlation to conclusions
- Combine data with reasoning and context
- Statistics inform decisions, but do not replace judgement