Warm up

In a class where x  74,  8

State the interval containing the following % of
marks:

a) 68%
b) 95%
c) 99.7%



Answers:



a) 66 – 82
b) 58 – 90
c) 50 – 98
Applying the Normal
Distribution: Z-Scores
Chapter 3.5 – Tools for Analyzing Data
Mathematics of Data Management (Nelson)
MDM 4U
Comparing Data


Consider the following two students:
Student 1



MDM 4U, Mr. Lieff, Semester 1, 2004-2005
Mark = 84%, x  74,  8
Student 2
MDM 4U, Mr. Lieff, Semester 2, 2005-2006
 Mark = 83%, x  70,   9.8

Can we compare the two students fairly when the
mark distributions are different?
Mark Distributions for Each Class
Semester 1, 2004-05
50
58
66
74
82
90
Semester 2, 2005-06
98
40.6 50.4 60.2
70
79.8 89.6
99.4
Comparing Distributions


It is difficult to compare
two distributions when
they have different
characteristics
For example, the two
histograms have
different means and
standard deviations
z-scores allow us to
make the comparison
Histogram
4
2
4 5 6 7 8 9 10 11
b
Collection 1
6
5
4
3
2
1
Histogram
Count

Count
Collection 1
6
1 2 3 4 5 6 7 8
a
The Standard Normal Distribution




A distribution with a mean of zero and a standard
deviation of one
X~N(0,1²)
Each element of any normal distribution can be
translated to the same place on the Standard Normal
Distribution using the z-score of the element
the z-score is the number of standard deviations the
piece of data is below or above the mean
If the z-score is positive, the data lies above the
mean, if negative, below
z
xx

Standardizing


The process of mapping a normal
distribution to the standard normal distribution
N(0,12) is called standardizing
The Standardized normal distribution has a
mean of 0 and a standard deviation of 1
Example 1


For the distribution X~N(10,2²), determine the number
of standard deviations each value lies above or below
the mean:
a.
x=7
z = 7 – 10
2
z = -1.5


b.
x = 18.5
z = 18.5 – 10
2
z = 4.25
7 is 1.5 standard deviations below the mean
18.5 is 4.25 standard deviations above the mean
(anything beyond 3 is an outlier)
Example continued…
99.7%
95%
34%
34%
13.5%
13.5%
2.35%
2.35%
6
8
7
10
12
14
16
18.5
Example 2:





The class mean is 68.0 and the std.dev. is 10.9.
If your mark is 64, what % of the class has a
mark below yours? Above yours?
z = (64 – 68.0)/10.9 = -0.37
(using the z-score table on page 398)
We get 0.3557 or 35.6%
So 35.6% of the class has a mark less than or
equal to yours
Therefore 100 – 35.6 = 64.4% of the class has
a mark above yours.
Example 3: Percentiles




The kth percentile is the data value that is
greater than k% of the population
If another student has a mark of 75, what
percentile is this student in?
z = (75 - 68)/10.9 = 0.64
From the table on page 398 we get 0.7389 or
73.9%, so the student is in the 74th percentile
– their mark is greater than 74% of the others
Example 3: Percentiles cont’d



NOTE: Always round UP for percentiles
i.e. If 65.2% of the data is below a particular
value, then that value is in the 66th
percentile.
It is analogous to your age – the day after your
15th birthday, you are in your 16th year of life
– round UP!!!
(The textbook uses standard rounding rules which, by the
definition of a percentile, is not correct)
Do now

The mean of an online IQ test is 110 with a standard
deviation of 8. If you scored 120:
 a. What is your z-score? What does it mean?
 b. What % of the population are you ‘smarter than’?
(use table on p. 398)
 c. What percentile are you at?
a. z = 120 – 110 = 1.25 so you are 1.25 std.dev. above
the mean
8
b. 0.8944 or 89.44%
c. 90th percentile
Example 4: Ranges






Find the percent of data between a mark of 60 and 80
For 60:
 z = (60 – 68)/10.9 = -0.73
gives 23.3%
For 80:
 z = (80 – 68)/10.9 = 1.10
gives 86.4%
86.4% - 23.3% = 63.1%
So 63.1% of the class is between a mark of 60 and 80
See
http://www.coolschool.ca/content/display.php?file=conte
nt/pmath12/standnorm
Back to the two students...

Student 1

Student 2
84  74
z
 125
.
8
83  70
z
 1.326
9.8

Student 2 has the lower mark, but a higher zscore!
Exercises

Read through the examples on pp. 180-185
Complete p. 186 #2-5, 7, 8, 10

Tomorrow:



Group Design Challenge: Build a Normal
Distribution using Cube-A-Links
Chapter 3 Problem
Group Design Challenge: Build a Normal
Distribution using Cube-A-Links




Split into 4 groups
Each group will be given a bucket of Cube-ALinks
Your task is to build a Normal Distribution (20
mins)
We will perform a peer review of each
distribution
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 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 = 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

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 = TB where TB = 1B + 2B×2 + 3B×3 + HR×4
and 1B = singles, 2B = doubles,
AB
3B = triples, HR = homeruns
Slugging Percentage Example

e.g. 1B Miguel Cabrera, Detroit Tigers
http://sports.yahoo.com/mlb/players/7163



2008 Statistics: 616 AB, 180 H, 36 2B, 2 3B, 37 HR
NOTE: H (hits) includes 1B as well as 2B, 3B and HR
So



1B = H – (2B + 3B + HR)
= 180 – (36 + 2 + 37)
= 105
Slugging Percentage Example cont’d
SLG = (H + 2×2B + 2×3B + 3×HR) / AB
= (105 + 2×36 + 3×2 + 4×37) / 616
= 331 / 616
= 0.537 (3 decimal places)
 This means Miggy attained 0.537 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  Technical chart
n-Day Moving Average
Useful for showing long term trends
Other Examples
1) 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/cgibin/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’d
2) 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’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


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 2009), 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