Statistics
Notation Conventions

This is an overview of how some basic mathematical symbols and processes are represented in these Internet pages. You need to have the Symbol font on your computer to read this and the other pages correctly.

Standard Symbols

These standard conventions and symbols have their familiar meaning:

• ab = (a)(b) = "a times b"
• ´ = "times;" a ´ b = (a)(b) = ab
• > = greater than
  • ³ "equal to or greater than; not less than"
• < = less than
  • £ "equal to or less than; not more than"
• ~ "approximately equal to"
  • compare: 1 ~ k = from 1 to k inclusive
  • ~ is also used here as a quasi-infinity sign, meaning "a very large number"
• ¹ "not equal to"
• ± "plus or minus"
• log(x) = logarithm of x
• ln(x) = natural logarithm of x, log to the base "e" of x.
• e = the constant 2.71828 . . . [that's a Greek "letter e"]
• p = 3.14159 . . .
• f(x) = function of x
• n! = n factorial = (n)(n - 1)( n - 2) . . . (2)(1)

To the last of these items, we add the following finesse:

• 2:n! = the first 2 terms of n! = (n)(n - 1)
• 3:n! = the first 3 terms, or (n)(n - 1)(n - 2)

For refresher explanations of key terms, consult the Math section.

Exponents and Roots

Only three exponents, x¹, x², and x³, can be represented as such in the HTML convention. We will sometimes use those familiar forms, but we also need a general convention for representing any exponent. We will use an asterisk (since its raised position suggests superscript placement), followed by the exponent:

• n*k = n to the kth power
  • thus, n*2 = n to the 2nd power = n²
  • or for complex exponents, n*(2x - y) = n to the power (2x - y)

For roots, the basic symbol Ö will normally mean "square root." If it should be necessary to indicate the index of a root, the above * marker will be used:

• Ök = ²Ö(k) the [square] root of k
  • thus, 5Ök = 5 times the square root of k
• *3Ö(k) = ³Ö(k) = the cube [etc] root of k
  • thus, 5*3Ök or more fully 5(*3Ök)= 5 times the cube root of k
  • and 5*3Ö(n - 1) or 5[*3Ö(n - 1)] = 5 times the cube root of (n - 1)

Remember that exponents in division formulas subtract: x³ /x² = x¹. It follows that x² / x³ = x*(-1). These negative exponents are mathematically elegant, but offputting to the ordinary reader. We will generally avoid them by leaving those expressions "below the line" in the division formula; in other words, by expressing them as reciprocals. That is, we will usually represent x*(-2) by 1/x².

Sums and Infinities

We represent sums by a capital sigma in the Symbol font (your screen may show a capital S), thus

• S(n) = sum of all values of n

When precision is required, the range can be explicitly specified, thus:

• S(n)[1 ~ k] = sum of the 1st through kth values of n, inclusive

Infinity is a place that none of us will ever get to. For the infinity symbol, we substitute the practice of simply leaving the endpoint unspecified, thus:

• S(n)[1 ~ ] = sum of the 1st value of n, and indefinitely beyond that
• S(n)[ ~ ] = sum from a very large minus to a very large plus value of n

or as many values of n, in the indicated directions, as you may have the taste and time to compute.

The following bracket convention can also be used as a general substitute for subscripts, thus

• n[i] = The ith value of n

A New Concept

As a convenient counterpart to the factorial n! we venture to introduce

• n# = summarial n, the sum of all numbers n + (n-1) + . . . 2 + 1.
  • 4# = 4 + 3 + 2 + 1 = 10

In parallel to the partial factorial, ad defined above, we may define the partial summarial:

• 2:n# = the first 2 terms of n# = n + (n-1)
  • 3:5# = the first 3 terms of 5# = 5 + 4 + 3 = 12

Summarials underlie the relationships embodied in the Arithmetical Triangle, and thus, ultimately, many things which are conventionally expressed by factorials or in other ways.

Statistics is Copyright © 2001- by E Bruce Brooks 

14 Sept 2005 / Contact The Project / Exit to Statistics Page