I get that the variance basically indicates how much the possible outcomes for a random variable are spread or how much they may differ from the expected value. The higher the variance the more they may differ from the expected value. But what exactly is the meaning of a specific value for the variance? What does it mean if the variance is one?
2026-04-06 10:40:16.1775472016
What does the variance's value mean intuitively?
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Variance by itself does not convey how much a variable is spread. Variance may also be roughly regarded as a measure of fluctuation, or (in)consistency
Imagine a runner who is training for a month for olympics. If over the month her variance for the time taken to finish the practice race is 5 seconds, while the second athlete has 20 seconds, it does not mean first one is more consistent.
If the first runner is training for 400 metre race with the average around 45 seconds, and the second runner is practising for 5000 metres with the average closer to 15 minutes, second athlete has a better consistency.
Moral is, without knowing what the expectation is variance can not be a meaningful measure of fluctuation.
Of course two random variables with identical mean and variance can also be significantly different: the distribution function gives the full info on the nature of random variables.