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Statistics (Descriptive, analysis of data, experiments) (top of page)

Basic Definitions . . . . (10/98)
Our data set consists of n values, or scores; x1, x2, . . . , xn.
These might be the ages xk of the n people in your class, for example.

The mean, xbar , of the scores is (I can't do an overscore here so imagine an x with a bar over it!)
xbar = (x1 + x2 + . . . + xn) / n .
This is the average value of the n scores.

Another thing to know is how spread out the scores are. This involves the variance
xvar, the average squared distance from the mean:
xvar = [ (xbar - x1)^2 + (xbar - x2)^2 + . . . + (xbar - xn)^2 ] / n

The standard deviation xsd is the square root of the variance; that gives the average
distance from the mean, with outliers (scores far from the mean) having the most influence.
xsd = (xvar).

Note: Technically these are the population variance and population standard deviation.
Sometimes the sample variance is used, where the denominator is n - 1 instead of n. The
sample standard deviation is the square root of the sample variance. The idea here is to
increase the variance a bit because in taking a sample you are less likely to include a rare outlier,
so the variance is too small with an n down there.

Example 1: Three burrito shops charge \$2, \$3, and \$7 for a bean & cheese.
What are the average burrito price and the standard deviation?

Solution: There are n = 3 values x1 = 2, x2 = 3, x3 = 7.
So the mean (avg) is xbar = (2 + 3 + 7) / 3 = 12 / 3 = \$4.
If you randomly visit the three shops, then in the long run (no tasteless burrito jokes!)
you'll spend \$4 per burrito (even though none of the places charges \$4!).
The variance is
[ (2 - 4)^2 + (3 - 4)^2 + (7 - 4)^2 ] / 3 = [4 + 1 + 9] / 3 = 14 / 3 = 4.667 approx.
So the standard deviation is Sqrt[14/3] = Sqrt[4.667] = \$2.16 approx.
This means that in the long run, the average price difference from the mean is \$2.16.

Random Sampling and Surveys (7/00) (top of page)
Ask several people to "pick a random number between 1 and 100." The distribution
you'll get is far from random; lots of people will pick 50, some will pick 99, others will
try to be very random and say 37. Each of these three numbers comes up much more
than 1% of the time! People rarely say things like "80" because it doesn't sound very
offbeat. Human nature, I guess.

When doing a random experiment, such as polling a sample of voters, it's crucial that
these biases are not introduced. Randomizing tools like flipping coins, spinning a wheel,
or rolling dice will do the trick.

I've heard of surveys that say "57.1% of the voters polled favored Clinton" when it turns
out they called 7 of their friends and 4 of them were Clinton supporters. Even with a
large sample size, things like "touch-tone phones only, 95 cents per call" will only allow
people with telephones, TV's, and enough money or a strong enough opinion to call in.
I like the "no opinion" part; would you call in, wait on the line, and pay 95c, just to say
you have no opinion?

There are tables of random digits in stat books, and most calculators have a "Ran#"
button; that's how I select the 'lucky' people from the waiting lists that will get into my
class! Another method is to use the decimal digits of pi = 3.1415926535897932384626...
which is 'normal' in the sense that, in the long run, each digit occurs 1/10 of the time,
each pair of digits occurs 1/100 of the time, etc.

Number Theory (expanded 8/01) (top of page)
Prime Numbers
Prime Factorization
Number and Sum of Divisors
Supercomposite Numbers
Other Integer Functions

Basic Definitions

The natural numbers are the counting numbers:
1, 2, 3, 4, . . . .
The whole numbers are the natural numbers with 0 thrown in.
0, 1, 2, 3, . . .
The integers consist of the whole numbers and their negatives:
. . . , -3, -2, -1, 0, 1, 2, 3, . . .

A natural number bigger than 1 is called prime if there are only two numbers that go into it
evenly; itself and 1.

Examples: 3 is prime because only 1 and 3 go into it. But 77 is not prime because 7 and 11

The GCD or greatest common divisor of two or more numbers is the largest number that
goes into both (or all) of the numbers: GCD(12, 18, 28) = 2 and GCD(385, 1001) = 77.
Two numbers are relatively prime if their GCD is 1; they have no (other) common factors.
So for example 6 and 25 are rel. prime even though neither is a prime number.

The LCM is the least common multiple, the smallest thing that they both (or all) go into.
Here LCM(10, 14) = 70.

Challenge: Prove, for any numbers m and n, we have GCD(m,n) * LCM(m,n) = m*n.

Unique Prime Factorization

The Fundamental Theorem of Arithmetic states that every natural number greater than 1
can be written as a product of prime numbers, and that up to rearrangement of the factors,
this product is unique. This is called the prime factorization (or PF for short) of the number.

Example: 36 = 6 x 6 = 9 x 4 = 12 x 3 = 18 x 2 , but all are equal to 2 x 2 x 3 x 3.
This is the PF of 36, often written with exponents: 36 = 2^2 * 3^2. You can use these PFs
to figure out GCDs, LCMs, and the number (and sum) of divisors of n.

Not all sets of numbers have this 'UF' or unique factorization property. For example,
9 = 3 * 3 = (7 + 2 sqrt[10])(7 - 2 sqrt[10]) but none of these can be broken up further using
numbers of the form a + b sqrt[10], so the "ring of integers" Z[sqrt[10]] is not a UFD or
'unique factorization domain.'

Divisors (counting and adding them) (See Prob Of Week #150)

Example 1: How many numbers go into 36 evenly (with no remainder)?

There are 9 divisors of 36 : {1, 2, 3, 4, 6, 9, 12, 18, 36}.
These can be arranged into a 3 x 3 rectangle, called a divisor chart:

 1 2 4 3 6 12 9 18 36

Notice from above that 36 = 2^2 * 3^2 ; the exponents are 2 and 2.
There is one more column than powers of 2, and one more row than powers of 3,
so there are 3 x 3 = 9 divisors. Our function notation is d(36) = 9.

Example 2: How many divisors does 21 million have?

21,000,000 = 3 * 7 * 10^6 = 2^6 * 3^1 * 5^6 * 7^1 ;

the exponents are 6, 1, 6, and 1 ; so the number has 7 * 2 * 7 * 2 = 196 divisors :
They are {1, 2, 3, 4, 5, 6, 7, 8, 10, . . . , 10500000, 21000000} (I left a few out.)

Example 3: What do the divisors in the 36-chart add up to?
See below that the actual sum is 91 ; but also notice that the column sum and row sum
that start with 1 are 1+3+9 = 13 , 1+2+4 = 7 , and that 13*7 = 91. Coincidence? Hmm . . .

Arithmetic Functions (pronounced "a-rith-MEH-tic") 8/01

Let's look at several integer functions using function notation, inputting only integers:
d(n) = number of (positive) divisors of n (including 1 and n)
s(n) = sum of divisors of n (incl. 1 and n; usually 'sigma(n)')
ph(n) = number of nos. less than n with no common factors with n (usu. 'phi of n').
A(n) = abundancy index of n, defined as the ratio s(n) / n .

For example if n = 36 we have nine divisors ; so
d(36) = #{1, 2, 3, 4, 6, 9, 12, 18, 36} = 9 ; s(36) = 1+2+3+4+6+9+12+18+36 = 91 ;
ph(36) = #{1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35} = 12 ; A(36) = 91 / 36 ~ 2.528.

One special case: if n = p prime, then d(p) = 2, s(p) = p+1, ph(p) = 1, A(p) = 1+(1/p).
The n is perfect if s(n) = 2n, or A(n) = 2 ; deficient if A(n) < 2 , and abundant if A(n) > 2.
For example, 36 is abundant, 35 is deficient, and 28 is perfect. Check it yourself!

For a good source of super-abundant numbers see the supercomposites page.
Challenge: Try to prove or at least verify with examples that d, s, and ph are
"multiplicative" functions, meaning that if m and n are relatively prime then
d(mn) = d(m) d(n) and so on (for s and ph)..
What about the case where m and n have a common factor k? What if there
are more than two numbers? What about the A function, is it multiplicative?

Well, that's it for now. Check back often for new stuff!
Click below for other topics, or visit the ask dan page!

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