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Notation


In order to explain anything useful about statistics we have to use some sort of algebraic notation. This does not imply you have to be able to 'do' algebra - you just need some idea of what the various symbols are about - and how they tend to go together. Unfortunately, despite what mathematicians may tell you, these symbols are anything but standardized - and all manner of abbreviated notation is in common use. To avoid a great deal of unnecessary suffering - let us 'bite the bullet' and describe one of the most common types of notation using a very familiar statistic.

The mean () (pronounced 'y bar') is commonly written as [ΣY]/n. This is the sum (Σ) of a set of observations (Y), divided by the number of observations in that set (n). (Many standard statistical texts use X instead of Y, and some authors use , or instead of .) Given that even this simple notation may appear in various forms, we would be wise to examine the notation in a little more detail.

To keep us in touch with reality, let us define variable Y as milligrams of active ingredient (a.i.) per ml of a dewormer - assayed in our laboratory. Let's say we take seven samples of the dewormer. Each of these can be considered as a separate observation, or replicate:

Sample number: 1234567
mg a.i. dewormer 121104111107104100102

Then we could describe each observation of variable Y like this:

Observation: Y1Y2Y3Y4Y5Y6Y7
mg a.i. dewormer 121104111107104100102

To save space we can abbreviate this list of observations to Y1, Y2, Y3, ... Yn
where n is the number of observations, and Yn is the nth observation of the variable Y

So the 'sum of Y' is Y1 + Y2 + Y3 + ... Yn
or, to save space, we can use Y to indicate the entire set of observations.

To save space, instead of writing 'the sum of Y', this may be written as Σ[Y] - or, more briefly, as ΣY - pronounced 'Sigma Y'.

Alternately, if you wish to be explicit, you could write this as:      Σ(Y1, Y2, Y3, ... Yn)

Which can be abbreviated to:        i=n Yi 
 Σ
i=1

Where i stands for observation numbers 1 to 7, in our current list of (n) measurements. The subscript (the expression below Σ) and the superscript (the expression above sigma) indicate the first and last items to be included in the sum.

Y1-7 =        121104111107104100 102

i =

1 23456 7

We can abbreviate the above formula in various ways.

     n     
Such as:        Σ Yi    or    Σ Yi    or    Σ Yi    or    Σ(Y)
  i=1     i        

Let us use the simplest of these notations.

In this case Σ Y   =   749    for n = 7 observations

So our mean    =    Σ Y    =    749   =   101 mg per ml
  n   7