σ² · Variance & Standard Deviation
The second fact the mean hides — variance.
Assets A and B both average 5% over 10 years — are they the same investment?
Why you can drown crossing a river that averages just one meter deep.
Open the immersive version →Why does this question matter?
Two assets averaging 5% — one drifts between 4 and 6%, the other swings from −20% to +30%. The single number 'average' erases this difference completely.
What's needed is a second number measuring 'how far from the mean'. But deviations sum to zero, so a detour was invented: square them, then average — variance.
Squaring breaks the units (dollars²?), so we take the root back: the standard deviation. Financial risk, factory quality, standardized test scores — the common language wherever wobble matters.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — both assets below average 5% over 10 years. Are they the same investment?
Eyeball the yearly returns
Asset A (x̄ = 5%)
Asset B (x̄ = 5%)
Assemble the wobble-meter (using Asset B)
Read the idea from another angle
The mean tells you the location; the variance tells you the wobble.
Two numbers answering two questions: 'whereabouts?' (mean) and 'how much does it swing?' (variance). Summarizing a distribution takes at least these two.
Squaring is a penalty — big misses count far more.
A deviation of 2 counts as 4; a deviation of 10 counts as 100. Why variance is sensitive to extremes, and why one big loss shakes a financial risk metric so hard.
Now read the definitions with their conditions
The experiment's three buttons (deviations → squares → average) become the formula verbatim.
Variance
Deviations, squared, then averaged. Squaring kills the signs and simultaneously penalizes large misses.
Standard deviation
The root brings variance back to the original units, enabling intuitive phrases like 'mean ± σ'.
Deviations sum to zero
The mean is by definition the point where deviations cancel. Which is why wobble can't be measured without squaring.
Make one more distinction
Just average the deviations (value − mean) and you get the wobble.
Deviations always sum to exactly 0 — overshoots cancel undershoots. So we square them first to kill the signs, then average. That's variance.
High variance is bad.
Variance is neither good nor bad — it's the size of variation. In investing it's risk but also the width of opportunity; in quality control it's defects; in ecology it's diversity. Interpretation belongs to context.
Where else can this idea help?
Investment risk
The standard deviation of returns IS finance's 'volatility'. Equal average returns? The lower-σ asset earns the same money more comfortably.
Factory quality control
Small σ in part dimensions means no out-of-spec defects. The name Six Sigma (6σ) is literally the standard deviation.
Standardized test scores
Converting raw scores via (score−mean)/σ gives z-scores — fair comparison across tests of different difficulty.
Forecast uncertainty
'20±3 degrees' — the scatter (σ) of the prediction models sets the width of the confidence band.
Explain it in your own words
Try another question
Q1Compute the mean and standard deviation of [3, 5, 7] by hand.Show answer ▾
Mean 5. Deviations [−2, 0, 2] → squares [4, 0, 4] → average 8/3 ≈ 2.67 = variance. σ = √2.67 ≈ 1.63. Three steps.
Q2Add 10 to every value — what happens to σ? Double every value?Show answer ▾
+10 leaves σ unchanged (everything shifts together; wobble intact); ×2 doubles σ (gaps double). Location shifts and scale changes affect variance differently.
Q3What does a dataset with σ = 0 look like?Show answer ▾
Every value exactly equals the mean — all identical. The only case with zero wobble.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
Previous concept
Mean, Median & Mode
Know what the mean summarizes and discards, and the tool that measures the discarded part (the wobble) makes sense.
← Mean, Median & Mode labLeads to next
Normal Distribution
A magical distribution completely determined by just two numbers — mean and σ. The bell curve with its 68-95-99.7 rule awaits.
Go to the Normal Distribution lab →Related lab
Expected Value
If EV is the scale's reading, variance is the scale's wobble — the two numbers complete a decision together.
Go to the Expected Value lab →