StatisticsStatistics

Math 416Math 416

Game PlanGame Plan

IntroductionIntroduction Census / Poll / SurveyCensus / Poll / Survey Population – Sample – BiasPopulation – Sample – Bias Sample Proportion Sample Proportion Mean Median ModeMean Median Mode Box and Whisker PlotBox and Whisker Plot Box and Whisker Interpretation Box and Whisker Interpretation

Stats IntroStats Intro There are lies, there are damn lies and There are lies, there are damn lies and

then there are statistics then there are statistics

- Mark Twain- Mark Twain The goal is by the use of number The goal is by the use of number

describe a characteristic of a describe a characteristic of a population.population.

The idea is to win your argument by The idea is to win your argument by providing facts and too many people providing facts and too many people consider statistics to be absolute facts.consider statistics to be absolute facts.

Stats IntroStats Intro In general, most people do not In general, most people do not

understand statistics.understand statistics. Hypothesis: Student A has a school Hypothesis: Student A has a school

average of 10%average of 10% Conclusion: Student A is a bad Conclusion: Student A is a bad

person. person. The statistic does not measure the The statistic does not measure the

person’s goodness or badness.person’s goodness or badness. What does that statistic mean?What does that statistic mean? If all there marks were the same for If all there marks were the same for

all courses, it would be 10%all courses, it would be 10%

Statistics Statistics Life is a continual battle to get your Life is a continual battle to get your

ideas across and have other people ideas across and have other people trying to get their ideas across to you.trying to get their ideas across to you.

You are constantly being bombarded You are constantly being bombarded by arguments and statistics. by arguments and statistics.

1.1. CommercialsCommercials2.2. TeachersTeachers

To understand the world around you, To understand the world around you, need to be aware of statistics meaning need to be aware of statistics meaning and reliability.and reliability.

Where do statistics come from?Where do statistics come from?

Population Population

First we establish the population.First we establish the population. Population: the complete group that we Population: the complete group that we

are investigating are investigating Characteristic: A particular identifying Characteristic: A particular identifying

object exhibited by the population object exhibited by the population

i.e. hair colouri.e. hair colour

favorite colourfavorite colour

math knowledge, political math knowledge, political opinion etc. opinion etc.

PopulationPopulation

The next problem is interpreting The next problem is interpreting how to measure a characteristic and how to measure a characteristic and obtain the data.obtain the data.

Obtaining the Data: Three methodsObtaining the Data: Three methodsMethod #1: Ask the whole Method #1: Ask the whole population population

- Called a censusCalled a census- Problems – hard to do – Problems – hard to do – depending on population depending on population

CensusCensus

Method #1: Ask the whole Method #1: Ask the whole population population

- Called a censusCalled a census- Problems – hard to do – Problems – hard to do – depending on population depending on population

PollPoll

Method #2: - Ask a Method #2: - Ask a representative “sample” of the representative “sample” of the populationpopulation

- Called a poll- Called a poll

Problems: representative may be Problems: representative may be trickytricky

SurveySurvey

Method #3: Ask only experts of Method #3: Ask only experts of the populationthe population

- called a survey- called a survey

Problems: who is an expertProblems: who is an expert

Representative sample?Representative sample?

BiasBiasBiasBias If data is obtained or presented in an If data is obtained or presented in an

unfair manner than all conclusions are unfair manner than all conclusions are not correct. The results are said to be not correct. The results are said to be biased (or unfair).biased (or unfair).

In collectionIn collection How and who you ask is the main source How and who you ask is the main source

of biasof bias There are 4 types of bias (bad sampling, There are 4 types of bias (bad sampling,

non pertinence, wording of question & non pertinence, wording of question & attitude of pollster).attitude of pollster).

BiasBiasEg asking 5 yr olds their favorite beer Eg asking 5 yr olds their favorite beer Bad sampling Bad sampling

Eg Do you like to play an instrument? (to Eg Do you like to play an instrument? (to find favorite color) find favorite color)

Non-pertinence Non-pertinence

Eg man I hate Bush, are you in favor of Eg man I hate Bush, are you in favor of war?war?

Wording of questions Wording of questions

Eg a policeman asking were you speeding? Eg a policeman asking were you speeding? Attitude of pollster Attitude of pollster

Presentation BiasPresentation Bias

In presentation, imagine you In presentation, imagine you disregard a grade level and disregard a grade level and claim that they do not matter in claim that they do not matter in a school’s decision. a school’s decision.

I need to prove my product is I need to prove my product is the best, how can I get these the best, how can I get these numbers to show that?numbers to show that?

Buy This Stock!Buy This Stock!

$400$300

$0

$100

$200

$500

Jan April

March

Feb Jan

Not!

A statistical presentation is always biased

Stencil #1-3

Representative SampleRepresentative Sample

Creating a representative Creating a representative sample can be an art form in sample can be an art form in itself. The sample should be in itself. The sample should be in all the same proportions, an all the same proportions, an impossibility. impossibility.

You must focus on the You must focus on the characteristics (the poll or characteristics (the poll or survey is focusing on!)survey is focusing on!)

Representative SampleRepresentative Sample Consider a school has 50 boys and Consider a school has 50 boys and

25 girls and a representative 25 girls and a representative sample of 10 needs to be created.sample of 10 needs to be created.

We note the population is We note the population is described in terms of boys and described in terms of boys and girls hence we will need to create girls hence we will need to create our sample on that basisour sample on that basis

Three stepsThree steps

Representative SampleRepresentative Sample1) Relative (by percent)1) Relative (by percent)

50/75 = 67%

25/75 = 33%

n = 75

2) Theory - sample = 10

10x.67=6.7 10x.33 = 3.3

Difficult to get .7 or .3 of a person!

3) Reality

7 3 Total of 10 & has added bias

Some RulesSome Rules

If it starts at zero it stays at zeroIf it starts at zero it stays at zero If it appears to be zero be If it appears to be zero be

careful!careful! Make decisions on a category Make decisions on a category

not overallnot overall

Creating a SampleCreating a Sample1)1) Given the following, create a sample Given the following, create a sample

of 10of 10 Hudson Non-HudsonHudson Non-Hudson

Youngn = 109

Relative

Middle AgedOld

0 120 2431 33

MA 18% 22%

0 28% 30%

Y 0 0 Is it really 0 people?

Creating a SampleCreating a Sample

Y

open here

Reality

MAO

0 01.8

2.22.

83

Y 0 0

Theory

MAO

2 23 3

Stencil 4,5, 6 Do relative, theory and reality for #4; in #5 & 6 put theory & relative together

Statistics Central Tendency Statistics Central Tendency - Mean- Mean

Mean meansMean meansthe averageSymbol x

Found by dividing the sum ∑xi

by the number of elements n. i.e. x = ∑ xi

Means which value would all values be equal to if they were the same i.e. (5,9,3,6)x = ∑ xi

n

n

= (5+9+3+6)/4 = 5.75

ModeMode

Symbol MSymbol M It is the number that appears the It is the number that appears the

mostmost It is possible, not to have any or to It is possible, not to have any or to

have more than one modehave more than one mode Eg (1,2,5) Eg (1,2,5) Eg (1,6,6,8)Eg (1,6,6,8) Eg (1,3,3,4,4,8)Eg (1,3,3,4,4,8)

M = (nothing repeats)

M = 3 & 4M = 6

MedianMedian Symbol Symbol MM Median is found as the middle valueMedian is found as the middle value Note the sample must be in order!Note the sample must be in order! There are two possibilities (odd & There are two possibilities (odd &

even)even) Consider (1,5,7) n = 3Consider (1,5,7) n = 3 Odd only 1 middle; M = 5Odd only 1 middle; M = 5 (1,5,7,8) n = 4(1,5,7,8) n = 4 You must find the mean of both You must find the mean of both

middles (5 + 7)/2 = 6middles (5 + 7)/2 = 6

Do #7

Box & Whisker PlotBox & Whisker Plot (2,5,1,6,9,8,)(2,5,1,6,9,8,) The Construction The Construction 1) Make sure your sample is in order1) Make sure your sample is in order (1, 2,5,6,8,9)(1, 2,5,6,8,9) 2) Find the min, max & median2) Find the min, max & median Min = 1 ; max = 9 median = 5.5 = QMin = 1 ; max = 9 median = 5.5 = Q22

These three points will serve you as These three points will serve you as part of the box and whisker diagram. part of the box and whisker diagram. Draw it on board…Draw it on board…

Box & Whisker PlotBox & Whisker Plot

-1 -0 1 2 3 4 5 6 7 8 9 10

113) Create a number line with vertical line at the3) Create a number line with vertical line at the

three points hingesthree points hinges4) Find median between min and Q4) Find median between min and Q2 2

called Q1called Q15) Find median between Q2 and max called Q3

It is 2 and make another hingeIt is 2 and make another hinge

It is 8 & make another hinge.

(1,2,5,6,8(1,2,5,6,8,9),9)

Complete it!

Words & FactsWords & Facts We have broken the data into We have broken the data into

four parts called quartiles. four parts called quartiles.

Whiskers

MinBox

WhiskersQ1

Q3

Q2Interquar

tile range = Q3-Q1

Max

Words & FactsWords & Facts Each quartile should hold about ¼ Each quartile should hold about ¼

of the data BUT you cannot be sureof the data BUT you cannot be sure You cannot tell the mean or the You cannot tell the mean or the

modemode Do not jump to conclusions! Do not jump to conclusions! A box and whisker gives you an idea A box and whisker gives you an idea

about the spread or concentration about the spread or concentration or dispersion of dataor dispersion of data

Example #1Example #1

1 2 3 4 5 6 7 8 9 10

11 A general ViewA general View

This data is very close together This data is very close together below 4. There is more of a spread below 4. There is more of a spread between 4 and 11 and once again between 4 and 11 and once again between 11 and 12. between 11 and 12. Some Questions…Some Questions…What is the Mean?

12

No idea

QuestionsQuestions What is the mode?What is the mode? What is the median?What is the median? What is the What is the

interquartile range?interquartile range? How many are below How many are below

11?11?

No idea

Q2 = 4Q3-Q1 = 11-3 = 8

75% but no idea of the numberThe lowest concentration of

numbers lie where?

Lowest concentration vs. highest concentration

Between 4 - 11

Example #2Example #2

30

40

50

60

80

70

100

90

n = 20 Class An = 40 Class B

a) Which class did better?

Hard to tell but class Ab) What are the

meansNo idea

¾ x 20 + 2/4 x 40 = 35c) All together approximately how many were over 60%

Example #2 – More Example #2 – More QuestionsQuestions

Which class and which mark was the Which class and which mark was the highest?highest?

Class B at approximately 97%Class B at approximately 97% Which class has lowest range?Which class has lowest range? Class AClass A 87-55 = 3287-55 = 32 Class BClass B 97-40 = 5797-40 = 57 Answer: Class AAnswer: Class A Finish Stencil Finish Stencil