NHL Playoff Support by Province/State

Warm up

3.5 Applying the Normal Distribution: Z-Scores Chapter 3 – Tools for Analyzing Data Learning goal: Calculate the % of data above/below and between values in a Normal Distribution Questions? p. 176 #1, 3b, 6, 8-10 MSIP / Home Learning: Read examples on p p. 186 #2-5, 7, 8, Quiz Wed.

Learning Goals Describe the shape of a distribution given a frequency table, histogram or description 3.2 Calculate the mean, median and mode for grouped data  Select the most representative measure 3.3 Calculate the Range, IQR and Std.Dev.  Compare 2 datasets using spread 3.4 Draw and label a Normal Distribution given the mean and std.dev. Calculate the 68%, 95% and 99.7% confidence intervals

AGENDA Comparing Data in different Normal Distributions The Standard Normal Distribution Ex. 1: z-scores Ex. 2: Percentage of data below/above Ex. 3: Percentiles Ex. 4: Ranges

Comparing Data Consider the following two students: Student 1  MDM 4U Semester 1  Mark = 84%, Student 2  MDM4U Semester 2  2  Mark = 83%, How can we compare the two students when the class mark distributions are different?

Mark Distributions for Each Class Semester 1 Semester

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

The Standard Normal Distribution A Normal distribution with mean 0 and std.dev. 1  X~N(0, 1²) A z-score maps a data point from any Normal distribution to the Standard Normal Distribution A z-score is the number of standard deviations a data point lies below or above the mean  Positive z-score  data lies above the mean  Negative z-score  data lies below the mean

Standard Normal Distribution 34% 13.5% 2.35% 95% 99.7% %

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 = 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% 13.5% 2.35% 95% 99.7%

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 in the class, etc. mean = 68.0 standard deviation = 10.9

Example 2 If your mark was 64, what % of the class scored lower?  Calculate your z-score  z = (64 – 68.0)÷10.9 = Using the z-score table on page 398 we get 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 k th 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 at? z = ( ) ÷ 10.9 = 0.64  From the table on page 398 we get or 73.9%, so the student is at the 74 th 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 = -0.73gives 23.3% For 80:  z = (80-68)÷10.9 = 1.10gives 86.4% 86.4% % = 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.

MSIP / Homework Read through the examples on pages Complete p. 186 #2-5, 7, 8, 10

Minds on! Tell Them From Me Survey What type of study was it? Was it well-written? Was there any possible bias? Were any steps taken to prevent bias?

Minds on! Is it more expensive to live today than it was a year ago? How could you measure it?

3.6 Mathematical Indices Chapter 3 – Tools for Analyzing Data Learning goal: Calculate mathematical indices and draw conclusions Questions? p. 186 #2-5, 7, 8, 10 MSIP/Home Learning: pp #1a (odd), 2-3 ac, 4 (look up recent stats if desired), 8, 9, 11

What is a Mathematical Index? An arbitrarily defined number Most are based on a formula Can be used to make cross-sectional or longitudinal comparisons Does not always represent an actual measurement or quantity

Price Indices Measure the price of a basket of goods at different points in time Indicators of how the cost of living has changed  Consumer Price Index  Fan Cost Index  Christmas Price Index  Big Mac Index

Consumer Price Index (CPI) Managed by Statistics Canada An indicator of changes in Canadian consumer prices Compares the cost of a fixed basket of commodities (600 items) over time Expressed as a % of the base year (2002).

Consumer Price Index A CPI value of 124 in 2014 means the CPI is 24% higher than it was in the base year Source:

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).

Fan Cost Index Which cities do you think are in the top 3? Bottom 3? Compares the prices of: 4 average-price tickets 2 small draft beers 4 small soft drinks 4 regular-size hot dogs 1 parking pass 2 game programs 2 least-expensive, adult-size adjustable caps

Fan Cost Index – the fine print Average ticket price represents a weighted average of season ticket prices fpr general seating categories. 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=$ USD.

Christmas Price Index Totals the cost of the items in “Twelve Days of Christmas” A measure of inflation from year-to-year Created by PNC Bank

Big Mac Index Uses the cost of a Big Mac to compare currencies A Big Mac in Canada costs $5.85 = $4.54 USD  This is 1.22 times the cost in the US However, the current exchange rate is 1 USD = 1.29 CAD  So, the CAD is undervalued according to the BMI because it takes fewer CAD to buy a Big Mac

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  UnderweightBelow 18.5  Normal  Overweight  Obese30.0 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  AVG = (hits) (at bats) Slugging percentage (SLG) takes into account the number of bases that a player earns SLG = where TB = 1B + (2B × 2) + (3B × 3) + (HR × 4) 1B = singles, 2B = doubles, 3B = triples, HR = homeruns Hitting for the cycle

Slugging Percentage Example e.g. 1B Adam Lind, Toronto Blue Jays 2013 Statistics: 465 AB, 134 H, 26 2B, 1 3B, 23 HR NOTE: H (Hits) includes 1B, 2B, 3B and HR So  1B = H – (2B + 3B + HR)  = 134 – ( )  = 84 SLG = (1B + 2×2B + 3×3B+ 4×HR) ÷ AB  = (84 + 2×26 + 3×1 + 4×23) ÷ 465  = 231 ÷ 465  = This means Adam attained bases per AB

Example 3: Moving Average Used when time-series data show a great deal of fluctuation (e.g. stocks, currency exchange, gas) Average of the previous n values e.g. 5-Day Moving Average  cannot calculate until the 5 th day  value for Day 5 is the average of Days 1-5  value for Day 6 is the average of Days 2-6  etc. e.g. Look up a stock symbol at Click CHARTS  Interactive TECHNICAL INDICATORS  Simple Moving Average (SMA) Useful for showing long term trends

Humidex "humidity index“ An index number used by Canadian Meterologists to describe how hot the weather feels to the average person Combines the effect of heat and humidity Unit-less number based on the dew point Equivalent to dry temperature in degrees Celsius  30+ causes "some discomfort“  40+ causes "great discomfort“  45+ is "dangerous“  54+ heat stroke is imminent

MSIP / Home Learning Read pp Complete pp #1a (odd), 2-3 ac, 4 (alt: calculate SLG for 3 players on your favourite team for 2015), 8, 9, 11 Ch3 Review: p. 199 #1a, 3a, 4-6 You will be provided with:  Formulas in Back Of Book  z-score table on p

References Halls, S. (2004). Body Mass Index Calculator. Retrieved October 12, 2004 from Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from