One way to think about quantum operators is as Questions that are asked of a quantum system.

- Identity operator =
*“Who are you?”*
- Energy operator =
*“How much do you weigh?”*
*“What is your spin along the *`z`

axis?”
- and so on.

Statistical moments, letter values, and other verbs that are often just called “statistics” can be thought of the same way: asking questions of a data set.

For example, after you run the **∑/n**

operation to get the mean happiness in Europe (**2.0 / 3.0**) versus the mean happiness in the US (**1.2 / 2.0**), you naturally would want to ask things like:

- What about the least happy people? Are there more people answering near 0.0 in the US or Europe?
- What’s the variance
**√∑²/n**

?
- What’s the skewness? (Blanchflower & Oswald’s data survey 45,000 Americans and 400,000 Europeans — enough degrees of freedom to meaningfully measure skew.)
- What’s the conditional value-at-risk at the 10% level? (average of the bottom 10% unhappiness.)
- Apply a smoothing kernel to pick up which country has the more least-happy people without choosing a particular cutoff. (And maybe a second kernel to deal with the different scales: should we assume
`US1.0 = EUR1.5`

? Or maybe count from the top, to `US1.8 = EUR2.8`

?)

Running these operators on the dataset will tell you an answer to one question, just like in English.

One difference is that classical statistical operators typically spit out *two* numbers in reply to your question: an answer, and a confidence level in that answer. The confidence in the answer is computed based on experimental assumptions by people with names like Pearson, Fisher, and Chisquare.

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