Mathematical Indices

Chapter 3.6 – Tools for Analyzing Data

Mathematics of Data Management (Nelson)

MDM 4U

What is an Index?

An arbitrarily defined number that provides a measure of scale

Used to indicate a value so that we can make comparisons, but does not always represent an actual measurement or quantity

Interval Data (no meaningful starting point)

1) BMI – Body Mass Index

A mathematical formula created to determine whether a person’s mass puts them at risk for health problems

BMI = where m = mass in kg, h = height in m

Standard / Metric BMI Calculator http://nhlbisupport.com/bmi/bmicalc.htm

Underweight Below 18.5

Normal 18.5 - 24.9

Overweight 25.0 - 29.9

Obese 30.0 and Above

NOTE: BMI is not accurate for athletes and the elderly

2

m

h

2) Slugging Percentage

Baseball is the most statistically analyzed sport in the world A number of indices are used to measure the value of a

player Batting Average (AVG) measures a player’s ability to get on

base (hits / at bats) probability Slugging percentage (SLG) also takes into account the

number of bases that a player earns (total bases / at bats)

SLG = where TB = 1B + 2B×2 + 3B×3 + HR×4, OR

TB = H + 2B + 3B×2 + HR×3where 1B = singles, 2B = doubles,

3B = triples, HR = homeruns

TB

AB

Slugging Percentage Example

e.g. 3B/OF Jose Bautista, Toronto Blue Jays 2010 Statistics: 569 AB, 148 H, 35 2B, 3 3B, 54 HR NOTE: H (Hits) includes 1B as well as 2B, 3B and HR So

1B = H – (2B + 3B + HR) = 148 – (35 + 3 + 54) = 56

Slugging Percentage Example cont’dSLG = (1B + 2×2B + 3×3B+ 4×HR) / AB

= (56 + 2×35 + 3×3 + 4×54) / 569

= 351 / 569

= 0.617 This means Jose attained 0.617 bases per AB

Example 3: Moving Average Used when time-series data show a great deal of

fluctuation (e.g. stocks, currency exchange) Average of the previous n values e.g. 5-Day Moving Average

cannot calculate until the 5th day value for Day 5 is the average of Days 1-5 value for Day 6 is the average of Days 2-6

e.g. Look up a stock symbol at http://ca.finance.yahoo.com

Click Charts Interactive TECHNICAL INDICATORS SMA Useful for showing long term trends

Other Examples1) Consumer Price Index (CPI)

An indicator of changes in Canadian consumer prices Compares the cost of a fixed basket of commodities

through time Commodities are of unchanging or equivalent quantity

and quality reflecting only pure price change.

http://www.statcan.gc.ca/cgi-bin/imdb/p2SV.pl?Function=getSurvey&SDDS=2301&lang=en&db=imdb&adm=8&dis=2

What is included in the CPI? 8 major categories

FOOD AND BEVERAGES (breakfast cereal, milk, coffee, chicken, wine, full service meals, snacks)

HOUSING (rent of primary residence, owners' equivalent rent, fuel oil, bedroom furniture)

APPAREL (men's shirts and sweaters, women's dresses, jewelry) TRANSPORTATION (new vehicles, airline fares, gasoline, motor vehicle

insurance) MEDICAL CARE (prescription drugs and medical supplies, physicians'

services, eyeglasses and eye care, hospital services) RECREATION (televisions, toys, pets and pet products, sports

equipment, admissions); EDUCATION AND COMMUNICATION (college tuition, postage,

telephone services, computer software and accessories); OTHER GOODS AND SERVICES (tobacco and smoking products,

haircuts and other personal services, funeral expenses).

Other Examples cont’d2) NHL Fan Cost Index (FCI) Comprises the prices of:

four (4) average-price tickets two (2) small draft beers four (4) small soft drinks four (4) regular-size hot dogs parking for one (1) car two (2) game programs two (2) least-expensive, adult-size adjustable caps.

Other Examples cont’d2) NHL Fan Cost Index (FCI) Details Average ticket price represents a weighted average

of season ticket prices. Costs were determined by telephone calls with

representatives of the teams, venues and concessionaires. Identical questions were asked in all interviews.

All prices are converted to USD at the exchange rate of $1CAD=$.932418 USD.

MSIP / Homework

Read pp. 189-192 Complete pp. 193-195 #1a (odd), 2-3 ac, 4

(alt: calculate SLG for 3 players on your favourite team for 2010), 8, 9, 11

References

Halls, S. (2004). Body Mass Index Calculator. Retrieved October 12, 2004 from http://www.halls.md/body-mass-index/av.htm

Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page