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### 1 Introduction

Suppose you want to measure something: your height, or the velocity of an airplane for example. You take repeated measurements x1,x2, . . . , xn and you would like to combine them into a final estimate. An obvious way of doing this is to use the sample mean (x1 + x2 + · · · + xn)/n. However, modern statisticians use many other estimators, such as the median or the trimmed mean (where you throw away the largest and smallest 10% of the measurements and take the average of what is left). Mathematical statistics helps us to decide when one estimate is preferable to another. For example, it is intuitively clear that throwing away a random half of the data and averaging the rest is foolish, ...

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