Mathematical Indices
Chapter 3.6 – Tools for Analyzing Data
Mathematics of Data Management (Nelson)
MDM 4U
What is an Index?
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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
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A mathematical formula created to determine whether a
person’s mass puts them at risk for health problems
BMI = m where m = mass in kg, h = height in m
h
2
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
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2) Slugging Percentage
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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 = TB where TB = 1B + 2B×2 + 3B×3 + HR×4, OR
AB
TB = H + 2B + 3B×2 + HR×3
where 1B = singles, 2B = doubles,
3B = triples, HR = homeruns
Slugging Percentage Example
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e.g. 3B/OF Jose Bautista, Toronto Blue Jays
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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
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1B = H – (2B + 3B + HR)
= 148 – (35 + 3 + 54)
= 56
Slugging Percentage Example cont’d
SLG = (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
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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
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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
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Click Charts  Interactive
TECHNICAL INDICATORS  SMA
Useful for showing long term trends
Other Examples
1) Consumer Price Index (CPI)
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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/cgibin/imdb/p2SV.pl?Function=getSurvey&SDDS=2301&lang
=en&db=imdb&adm=8&dis=2
What is included in the CPI?
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8 major categories
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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’d
2) NHL Fan Cost Index (FCI)
 Comprises the prices of:
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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’d
2) 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
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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
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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