Intuitive explanation of "invariance"

...assuming that I'm able to augment their knowledge about variance in an intuitive fashion Understanding "variance" intuitively and about covariance How would you explain covariance to someone who understands only the mean?

Surprisingly, this wasn't asked before – at least I didn't find anything besides some vaguely related questions and no better definitions over the internet.

So, What is invariance and how would one explain it in simple layman terms?

• The highlighted hit in a Google search is a good place to begin. ("In statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties of different estimators for the same quantity. It is a way of formalising the idea that an estimator should have certain intuitively appealing qualities." -- Wikipedia)
– whuber
Jun 18, 2020 at 13:46
• – whuber
Jun 22, 2020 at 16:02

1 Answer

Invariance is a relative property not an absolute one. Something is described as invariant under some manipulation. For a non-mathematical (and hence non-statistical) example we consider the egg.

The shape of an egg is invariant under boiling (it remains egg-shaped).

The shape of an egg is not invariant under poaching, frying, scrambling, making and omelette.

The separate integrity of the yolk and the white is invariant under boiling, poaching and careful frying.

The separate integrity of the yolk and the white is not invariant under scrambling, making an omelette, clumsy frying.

Note that in the above we assume a standard hen's egg with a single yolk and the operation as usually performed in cookery in the south east of England. There seem to be different rules for preparing eggs, especially fried ones, in some other countries.