- dansmath > lessons >
**other stuff** - (c) 1997-2001 Dan Bach and B & L Math Enterprises; all rights reserved. Download for personal use only.
**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
**x****k**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**x****bar**= (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)- 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- go into it (as opposed to going into the 7-Eleven!). Click here for more info about primes and factors.
- 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 36evenly (with no remainder)?

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

124361291836

- 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 functionsusing function notation, inputting only integers:d(n) = number of(positive)divisorsof n (including 1 and n)s(n) = sum of divisorsof n (incl. 1 and n; usually 'sigma(n)')ph(n) = number of nos. less than n withno common factorswith n (usu. 'phi of n').A(n) = abundancy indexof n, defined as the ratios(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
perfectif s(n) = 2n, or A(n) = 2;deficientif A(n) < 2 , andabundantif 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![ number theory | geometry | top of page | lessons index]+ Basic Skills + Arithmetic, Prealgebra, Beginning Algebra + Precalculus + Intermed.Algebra, Functions & graphs, Trigonometry + Calculus + Limits, Differential Calc, Integral Calc, Vector Calc + Beyond Calculus + Linear Algebra, Diff Equations, Math Major stuff + Other Stuff + You are hereStatistics, Number Theory, Geometry, Other requests?

[ home | info | meet dan | ask dan | dvc ]

This site maintained by B & L Web Design, a division of B & L Math Enterprises.