(2.24) |

where is the number of enrolled eighteen year-olds, et cetera. Suppose that we were to pick a student at random and then ask ``What is the probability of this student being eighteen?'' From what we have already discussed, this probability is defined

(2.25) |

where is the total number of enrolled students. We can now see that the average age takes the form

(2.26) |

There is nothing special about the age distribution of students at UT Austin. So, for a general variable , which can take on any one of possible values, , , with corresponding probabilities , , the mean, or average, value of , which is denoted , is defined as

Suppose that is some function of . For each of the possible values of , there is a corresponding value of that occurs with the same probability. Thus, corresponds to , and occurs with the probability , and so on. It follows from our previous definition that the mean value of is given by

(2.28) |

Suppose that and are two general functions of . It follows that

(2.29) |

so that

(2.30) |

Finally, if is a general constant then it is clear that

(2.31) |

We now know how to define the mean value of the general variable . But, how can we characterize the scatter around the mean value? We could investigate the deviation of from its mean value, , which is denoted

(2.32) |

In fact, this is not a particularly interesting quantity, because its average is obviously zero:

(2.33) |

This is another way of saying that the average deviation from the mean vanishes. A more interesting quantity is the square of the deviation. The average value of this quantity,

(2.34) |

is usually called the

(2.35) |

giving

The variance of is proportional to the square of the scatter of around its mean value. A more useful measure of the scatter is given by the square root of the variance,

which is usually called the