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1. Introduction.
Statistical procedures are often given in mathematical notation. Sec
tions 2 to 8 below introduce a notation based on high school algebra,
and sections 9 to 20 show you how to use the algebra implicit in the
notation.
2. Notation for lists of numbers.
Data comes in many forms. One of the simplest is a list of numbers:
6
3
1
8
2
Usually there is a label at the top of the list–e.g: “Income (thousands
of dollars)” , “Time (hours)”, “Number of Visits”, etc. For statistical
formulas, it is traditional to use letters of the alphabet as labels:
x
6
3
1
8
2
y
7
4
9
5
The letters make it easy to talk about lists; for example:
The largest number on the list x is 8.
The smallest number on y is 5.
The sum of y is 25.
Informal language is not always clear. The list x has ±
ve numbers on
it; but what about the list below?
z
6
7
6
6
7
Does z have ±
ve numbers on it—or two? To avoid this possibility for
confusion, the word “entry” is often used. Then z is said to have ±
ve
Some Statistical Notation.
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entries: the f
rst one is 6, the second is 7, and so on.
3. Putting lists together.
The two lists:
u
2
8
5
14
v
6
1
7
can be combined into one list by writing the entries oF v below the
entries oF u:
w
2
8
5
14
6
1
7
Example 1 in the next section involves a list built up in this way.
There is no widely accepted term For the operation oF putting lists
together like this, even though it is what happens when data From
separate sources is compiled into one source. Programmers might
reFer to w as the result oF
appending
v to u, but the term will not be
used here.
4. The sum and average oF a list oF numbers.
The sum oF a list oF numbers is the result oF adding up all the entries
on the list, and the average is the result oF dividing that sum by the
number oF entries on the list.
A straightForward notation For the sum and average oF a list is:
sum(x), av(x)
±or average, statisticians oFten use the more symbolic notation:
x
which is read “xbar”. In that notation, the def
nition oF average
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becomes:
x
=
number of entries of x
sum(x)
Or, more brieﬂ
y:
x
=
n
sum(x)
where n is the number of entries of x. Multiplying both sides by n
leads to the following:
sum(x) = nx
The sum of a list is the number of entries times the average.
Simple as it is, this little equation is useful.
Example 1.
An income study involves 300 men and 200 women; the average
income of the men is $38,000 and the average income of the women
is $33,000. Find the average income of the 500 people. (Try to do this
before reading further.)
Hint and answer.
You are asked to ±
nd the average of a list of 500 incomes. The ±
rst
step is to get its sum. Think of the list as the result of putting togeth
er two lists:
the list of 300 incomes of the men in the study,
the list of 200 incomes of the women in the study
So, if you can get the sum of each of these lists, you can get the sum of
the 500 incomes. To get the two sums, use the above equation twice.
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This note was uploaded on 10/13/2008 for the course STAT 21 taught by Professor Anderes during the Fall '08 term at University of California, Berkeley.
 Fall '08
 anderes

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