Warm up

x 74 8, In a class where 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 DataMathematics of Data Management (Nelson)MDM 4U

AGENDA

Comparing Data Standard Normal Distribution Ex. 1: z-scores Ex. 2: Percentage of data below/above Ex. 3: Percentiles Ex. 4: Ranges MSIP / Homework

Comparing Data

Consider the following two students: Student 1

MDM 4U, Mr. Norbraten, Semester 1 Mark = 84%,

Student 2MDM 4U, Mr. Lieff, Semester 2 Mark = 83%,

Can we compare the two students fairly when the mark distributions are different?

x 74 8,

x 70 9 8, .

Mark Distributions for Each Class

Semester 1 Semester 2

74665850 82 90 99.489.679.87060.250.440.698

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

Co

un

t

123456

a1 2 3 4 5 6 7 8

Collection 1 Histogram

Co

un

t

2

4

6

b

4 5 6 7 8 9 10 11

Collection 1 Histogram

Standard Normal Distribution

34% 34%

13.5% 13.5%

2.35% 2.35%

95%

99.7%

0 1 2-1-2 3-3

68%

The Standard Normal Distribution A distribution with mean zero and standard deviation

of one X~N(0,1²) z-score translates from any Normal distribution to the

Standard Normal Distribution z-score is the number of standard deviations below

or above the mean Positive z-score data lies above the mean Negative below

xx

z

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 = 7z = 7 – 10 2 z = -1.5

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)

b. x = 18.5

z = 18.5 – 10

2

z = 4.25

Example continued…

34% 34%

13.5% 13.5%

2.35% 2.35%

95%

99.7%

10 12 1486

7

16

18.5

Standard Deviation

A recent math quiz offered the following data

z-scores offer a way to compare scores among members of the class, find out what % had a mark greater than yours, indicate position (percentile) in the class, etc.

mean = 68.0 standard deviation = 10.9

Co

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t

2

4

6

8

10

marks40 45 50 55 60 65 70 75 80 85 90

Test 1 Histogram

Example 2:

If your mark was 64, what % of the class scored lower?

Compare your mark to the rest of the class 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 What % scored higher? 100 – 35.6 = 64.4%

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 4: Ranges

Now 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

Back to the two students...

Student 1

Student 2

Student 2 has the lower mark, but a higher z-score, so he/she did better compared to the rest of her class.

z

84 74

81 25.

83 701.326

9.8z

MSIP / Homework Read through the examples on pages 180-

185 Complete p. 186 #2-5, 7, 8, 10

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