Introduction to Statistics

Class Overheadsfor

APA 3381 - part 1“Measurement and Data Analysis

in Human Kinetics”

byD. Gordon E. Robertson, PhD, FCSB

School of Human KineticsUniversity of Ottawa

Copyright © D.G.E. Robertson, October 2015

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Introduction to Statistics

Parameter: measurable characteristic of a population.

Population: all members of a definable group.For statistical purposes a population must havedefinable characteristics even if it is not possibleto measure the variable or even count thenumber of members in the population.

Sample: subset or subgroup of a population.Usually obtained by random sampling of a singlepopulation.

Statistic: measurable characteristic of a sample.E.g., height, weight, political affiliation, ethnicity,aerobic capacity, strength, power, ....

Data or Data Set: collection of numerical and/or non-numerical values (plural of datum).

Datum: single measured value (singular of data).

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Statistics

Statistics: 1. plural of statistic, 2. science of conductingstudies to collect, organize, summarize, analyze and drawconclusions from data.

Descriptive statistics: collection, description,organization, presentation and analysis of data.

Inferential statistics: generalizing from samples topopulations, testing of hypotheses, determiningrelationships among variables and making decisions,uses probability theory to make decisions.

Hypothesis: “less than a thesis”, a testableconjecture based on a theory.Thesis: a dissertation or learned argument whichdefends a particular proposition or theory.

Qualitative measurements:typically non-numerical, subjectively measured,

judgmentally determined, categorical.E.g., religious affiliation, teacher/professor

evaluations, emotional states, flavour, gender.

Quantitative measurements:typically numerical, objectively measured, reliability

(repeatability or precision) and validity (accuracy) can beevaluated against a criterion.

E.g., salary, course grade, foot size, IQ, age, girth.

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Types of quantitative measures:Constants: quantities with fixed characteristics.

Physical constants: G, c, h (Planck’s constant)Mathematical constants: p, e, i

Variables: quantities whose characteristics vary.Discrete variables: numerical variables that

have finitely many possibilities (usuallyintegers), countable many possible values

Examples: value of $ bills or coins, card countContinuous variables: numerical variables that

have infinitely many possible values withina range of values (numbers between –1 and+1) or unbounded (Real numbers, numbersgreater than 0).

Examples: height, duration, angle (only a fixednumber of significant figures are reported).

Significant Figures:When reporting numerical information, especially whenobtained by a calculator, usually only 3 or 4 digits arerequired. The general rule that is accurate to 0.5% holdsthat only 4 significant figures are needed if the firstnonzero number is a 1 and 3 when it is not.Examples: 234 000, 1.234, 2.45, 0.003 45, 0.1234,

8910, and 56 100.Exceptions are frequencies and counts when all digits arereported and financial numbers, hich are too nearestdollar or nearest cent depending on the amount.

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Measurement Scales

Nominal: classifies data into mutually exclusive(nonoverlapping), exhaustive categories in which noordering or ranking of the categories is implied.

E.g., colour, flavour, religion, gender, sex,nationality, county of residence, postal code.

Ordinal: classifies data into categories that can beordered or ranked (highest to lowest or vice versa),precise differences between categories does not exist.

E.g., teaching evaluations, letter grade (A+, A, A–, ...F), judges scores (0–10), preferences (polls), skillrankings.

Interval: numerical data with precise differences betweencategories but with no true zero (i.e., zero implies absenceof quantity).

E.g., IQ (0 means could not be measured),temperature (degrees Celsius), z-scores (0 is averagevalue), acidity (pH, 7 is neutral).

Ratio: interval data with a true zero, true ratios existE.g., height, weight, temperature (in Kelvins),

strength, price, age, duration.

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Methods of Sampling

Random: subjects are randomly selected from apopulation, all subjects have equal probability of beingselected, subjects may not be selected twice.

Systematic: subjects are numbered sequentially andevery n subject is selected to obtain a sample of N/nth

subjects (N is number of people in population).

Stratified: population is divided into identifiable groups(strata) by some relevant variable (income, gender, age,education) and each strata is sampled randomly inproportion to the strata’s relative size in the population.

Cluster: subjects are randomly sampled fromrepresentative clusters or regions of the population.Economical method if subjects are widely dispersedgeographically.

Convenience: typically used in student projects and byjournalists, uses subjects that can be conveniently polledor tested. Not suitable for pollsters or medical research.

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Graphing 1

Types of graphs:Pictogram: numeric data are represented by pictures,usually only nominal data are depicted in this way

Example: milk production increases by 200%

Before AfterBiassed way:

• height of cow is doubled but two-dimensionally cow is four timesbigger, three-dimensionally it is eight timesbigger

Unbias se dway:

• increase is correctly depicted as two timesgreater

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Graphing 2Pie chart: used with nominal or frequency data

Example: number of students by province andcountry

• a segment can beemphasized byseparating it

• two-dimensional piecannot create abiassed view

• three-dimensional pies can bias a slice depending onits position

- a slice in front appears- put a slice in the back largerto reduce its size - separating it creates

even more emphasis

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Graphing 3Bar graph: used for nominal data, usually frequencycounts, are depicted by bars proportional to theirmagnitudes

• bars are separated• extreme length bars can be split

Histogram: used forordinal data• bars are

adjacent, nogaps

• one axis isordered, first orlast bars mayincludeextremes

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Graphing 4

Line graph: usedwith interval andratio data

• scaling can create abias

• use large scales tohide changes

• truncated axisreduces “whitespace”

• scaling tominimum andmaximumemphasizeschanges

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Graphing 5

Ogive or cumulative frequency: (pronounced 0-jive) line starts at zero and accumulates to 100%

• useful for determining percentages (by interpolation)

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Rules for Constructing a Frequency Histogram

1. There should be between 5 and 20 classes.• this is strictly for aesthetic purposes

2. The class width should be an odd number.• this ensures that the midpoint has the same number of

decimal places as the original data

3. The classes must be mutually exclusive.• each datum must fall into one class and one class only

4. The classes must be continuous.• there should be no “gaps” in the number line even if a

class has no members

5. The classes must be exhaustive.• all possible data must fit into one of the classes

6. The classes must have equal width.• if not there will be a bias among the classes• you can have open-ended classes at the ends (i.e., for

ages you may use 10 and under or 65 and over, etc.)

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Types of Frequency Distributions

Categorical - for nominal types of data

Ungrouped - for numerical data with few scores

Grouped - for numerical data with many scores

Example: Distribution of the number of hours that boatbatteries lasted.

Class Class Tally Frequency Cumulative CumulativeLimits Bounds frequency percentages

24-30 23.5-30.5 /// 3 3 3/25*100= 12% +)))-

31-37 30.5-37.5 / 1 1+3= 4 4/25*100= 16% +)))-

38-44 37.5-44.5 ///// 5 5+4= 9 9/25*100= 36% +)))-

45-51 44.5-51.5 ///// //// 9 9+9= 18 18/25*100= 72% +)))-

52-57 51.5-57.5 ///// / 6 6+18= 24 24/25*100= 96% +))))-

58-64 57.5-64.5 / 1 1+24= 25 25/25*100= 100%

Total 25 25 100%+))))))))))))))))))))))))))))))))- +)))))))-Use these numbers for frequency polygon. Use these numbers for constructing

cumulative frequency polygon, alsocalled an ogive.

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Frequency Polygon and Ogive

Frequency polygon:

Cumulative frequency or ogive:

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Measures of Central TendencyMode: most frequent score.• best average for nominal data• sometimes none or more than one mode in a sample• bimodal or multimodal distributions indicate several

groups included in sample• easy to determine

Midrange: mean of highest and lowest scores.• easy to compute, rough estimate, rarely used

Median: value that divides distribution in half.• best average for ordinal data• more appropriate average for skewed ratio or interval

data or data on salaries• difficult to compute because data must be sorted• unaffected by extreme data

Arithmetic mean: centre of balance of data.• sum of numbers divided by n• best average for unskewed ratio or interval data• easy to compute

sample mean= population mean =

Other measures: harmonic mean, geometric mean andquadratic mean, also called root mean square (RMS)

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Skewed Data

• direction of skewis the direction ofthe tail

• positive directionof a number lineis to the right, leftis negativedirection

• mean, mode andmedian (MD) arethe same forsymmetricaldistributions

• notice mean isclosest to the tail(i.e., moreinfluenced byextreme values)

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Measures of Variation

Range: highest minus lowest values.• used for ordinal data

R = highest – lowest

Interquartile range: 75 minus 25 percentile.th th

• used for determining “outliers”

3 1IQR = Q – Q

Variance: mean of squared differences between scores and

the mean value (m).• used on ratio or interval data• used for advanced statistical analysis (ANOVAs)

Standard deviation: has same units as raw data.• used on ratio or interval data• most commonly used measure of variation

Coefficient of variation: percentage of standard deviationto mean.

• used to compare variability among data with differentunits of measure. Not suitable for interval data.

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Biased and Unbiased Estimators

• sample mean is an unbiased estimate of the populationmean (m)

• variances and standard deviations are biased estimatorsbecause mean is used in their computation

Why?• Last score can be determined from mean and all other

scores, therefore, it is not free to vary or add tovariability. To compensate divide sums of squares byn–1 instead of n.

• Instead of using the standard formula a computingformula is used so that running totals of scores andscores squared may be used to compute variability.

Computing Formulae

Variance: s = sample variance2

Standard deviation: S = sample standard deviation

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Measures of Position

Percentile: score which exceeds a specified percentage of thepopulation.• suitable for ordinal, ratio or interval data

2• median (MD or Q ) is 50 percentileth

1 3• first and third quartiles (Q and Q ) are 25 and 75th th

percentiles• easier for non-statisticians to understand than z-scores• scores are all positive numbers

Standard or z-scores: based on mean and standarddeviation and the “normal” distribution.• suitable for ratio and interval numbers• approximately 68% of scores are within 1 standard

deviation of the mean, approximately 95% are within 2standard deviations and approximately 99% are within 3standard deviations

• half the scores are negative numbers• mean score is zero• excellent way of comparing measures or scores which

have different units (i.e., heights vs. weights, metric vs.Imperial units, psychological vs. physiologicalmeasures)

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Measures of Position and Outliers

Other measures of position:Deciles: 1 2 10 10 , 20 , ... 100 percentiles (D , D , ...D )th th th

• often used in education or demographic studies

Quartiles: 1 2 3 25 , 50 and 75 percentiles (Q , Q , Q )th th th

• frequently used for exploratory statistics and to

2determine outliers (Q is same as median)

Outliers: extreme values that adversely affect statisticalmeasures of central tendency and variation.

Method of determining outliers:• compute interquartile range (IRQ)• multiply IRQ by 1.5

1• lower bound is Q minus 1.5 × IRQ

3• upper bound is Q plus 1.5 × IRQ• values outside these bounds are outliers and may be

removed from the data set• it is assumed that outliers are the result of errors in

measurement or recording or were taken from anunrepresentative individual

Alternate method for normally distributed data:• +/– 4 or 5 standard deviations

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Counting Techniques

Fundamental Counting Rule:In a sequence of “n” events with each event having “k”

possibilities, the total number of outcomes is:k = k × k × k × . . . × kn

Examples:• How many 6 digit student ID numbers

10 × 10 × 10 × 10 × 10 × 10 = 1 000 000

• How many ways of throwing 5 dice (Yahtze) 6 × 6 × 6 × 6 × 6 = 7 776

• How many ways of selecting 3 letters 26 × 26 × 26 = 17 576

1In a sequence of “n” events in which there are “k ” possibilities

2for the first event, “k ” possibilities for the second event and

3“k ” for the third, etc., the total number of possible outcomes is:

1 2 3 nk × k × k × . . . × kExamples:• How many 3 digit phone exchanges before 1980

8 × 2 × 10 = 160 (minus 911, 311, 411, 511)

• How many 3 digit phone exchanges after 19808 × 10 × 10 = 800 (minus 911, 311, 411, 511)

• How many 7 character Ontario licence plates (4letters, 3 numbers)26 × 26 × 26 × 26 × 10 × 10 × 10 = 456 976 000

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Factorial Notation

Factorials:Factorial numbers are identified with an exclamation

mark (n!). They are defined:

n! = n × n–1 × n–2 × . . . × 1

0! is defined to be 1

Examples: 1! = 1

2! = 2 × 1 = 2

5! = 5 × 4 × 3 × 2 × 1 = 120

12! = 479 001 600

20! . 2.433 × 1018

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Permutations and Combinations 1

Permutations:Rule 1: How many ways are there of arranging ALL “n” unique

items if replacement is NOT allowed?n! = n × n–1 × n–2 × . . . × 1

Examples:• How many ways of arranging 6 items in a display.

n = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

• How many ways of ordering three experiments.n = 3! = 6

Rule 2: How many ways are there of selecting “r” items from“n” unique possibilities if replacement is NOT allowed?

n rP = n! / (n-r)!

Examples:• How many ways of selecting 2 items from group of 6.

• How many ways of selecting committee of four froma staff of 20 if order of selection is significant.

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Permutations and Combinations 2

Rule 3:How many ways are there of selecting “n” items if

1 2 nreplacement is NOT allowed but k , k , ... k items areidentical?

1 2 nn! / (k ! ×k ! × . . . × k !)

Examples:• How many words from the letters in “OTTAWA”.

n = 6! / (2! × 2!) = 180

• How many words from the letters in “TORONTO”.n = 7! / (2! × 3!) = 420

Combinations:How many ways are there of selecting “r” items from

“n” unique possibilities if replacement is NOT allowed andorder is not important?

n r n rC = n! / [(n-r)! × r!] = P /r!

Examples:• How many ways are there of selecting 2 items from a

group of 6.

• How many ways of selecting committee of four froma staff of 20 if order of selection is unimportant.

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Probability

Basic Concepts:Probability experiment: process that leads to well-defined results, called outcomes.

Outcome: result of a single trial of a probabilityexperiment (a datum).

Sample space: all the possible outcomes of aprobability experiment, i.e., the population of outcomes.

Event: consists of one or more outcomes of aprobability experiment, i.e., a sample of outcomes.

I. Classical Probability: ratio of number of ways for anevent to occur over total number of outcomes in thesample space, i.e.,

where E is the event, S is the sample space,P(E) is the probability of the event E occurring,n(E) is the number of possible outcomes of event E, n(S) isthe number of outcomes in the sample space

To obtain a probability using classical methods the eventand sample space must be countable. Often the rules fordetermining permutations and combinations arerequired.

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Classical Probability 1

Examples:• Probability of rolling “snake eyes” (1, 1) with two dice.

Sample space for rolling two dice (one white, one black):ÎØ ÎÙ ÎÚ ÎÛ ÎÜ ÎÝÏØ ÏÙ ÏÚ ÏÛ ÏÜ ÏÝÐØ ÐÙ ÐÚ ÐÛ ÐÜ ÐÝÑØ ÑÙ ÑÚ ÑÛ ÑÜ ÑÝÒØ ÒÙ ÒÚ ÒÛ ÒÜ ÒÝÓØ ÓÙ ÓÚ ÓÛ ÓÜ ÓÝsevens elevensn(S) = 36 possible outcomes

Notice, there are four different ways of reporting a probability(proper fraction, ratio, decimal and percentage).

• Probability of 7 or 11 from rolling two dice.

• Probability of doubles from two dice.

• Probability of drawing a queen from a card deck.

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Classical Probability 2Examples:• Probability of drawing a spade.

• Probability of drawing a red card.

• Probability of flipping “heads” in a coin toss.

• Probability of flipping “heads” after 10 cointosses of heads in a row.

• coin cannot “remember” its history of outcomes

• Probability of “red” on a “double zero” roulettewheel.• wheel has numbers 1 to 36, half are red and half

are black, plus green zero and double zero (n=38)

• Probability of not getting a red on a roulettewheel.

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Rules of Probability

Rule 1: all probabilities range from 0 to 1 inclusively0 # P(E) # 1

Rule 2: probability that an event will never occur is zeroP(E) = 0

Rule 3: probability that an event will always occur is oneP(E) = 1

Rule 4: if P(E) is the probability that an event will occur, theprobability that the event will not occur is (also calledthe complement of an event):

P (not E) = 1 – P(E)

Venn diagrams:

Rule 5: the sum of probabilities of all outcomes in a samplespace is one

P(S) = S P(E) = 1

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Empirical Probability

II. Empirical Probability: obtained empirically bysampling a population and creating a representativefrequency distribution.• for a given frequency distribution, probability is the

ratio of frequency of an event class to the totalnumber of data in the frequency distribution, i.e.,

Examples:• Probability of a girl baby.

Assume that a population has a blood type distribution of:2% AB, 5% B, 23% A, and 70% O

• Probability of a person having type B or ABblood.

P(AB or B) = 2% + 5% = 7.00% = 0.0700

• Probability of strongly left-handed person.

P(strongly left-handed) = 0.050 = 5.00%

• Probability of “natural” blues eyes.

P(blues eyes) = 0.065 = 6.50%

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Addition Rules 1

Addition Rule 1: if two events are mutually exclusive,(i.e., no outcomes in common) then the probability ofA or B occurring is:

Venn diagramsof events thatare mutuallyexclusive:

If three events are mutually exclusive:

Examples:• Probability of selecting a spade or a red card.

• Probability of drawing a face card (king or queen orjack).

• Probability of 7, 11 or doubles with 2 dice.

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Addition Rules 2

Addition Rule 2: if two events are not mutuallyexclusive, the probability of A or B occurring is:

Venn diagrams ofevents that are NOTmutually exclusive:

When three events are not mutually exclusive there is aregion that is common to all three events. This area getsadded and subtracted three times and therefore must beadded back once. That is,

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Addition Rules 2, cont’d

Examples:• Probability of selecting a spade or a face card.

• 13 spades• 12 face cards (3 per suit)

• Probability of selecting a female student or athird-year student from 103 students.

• 53 are female• 70 students in third year• 45 females in third year

• Probability of selecting a male or person withtype O blood from 100 people.

• half are males • 70% are O-type blood

• Probability of selecting a left-handed person or a Liberal

from 1000 people.

• 32% Liberals, 5% left-handed

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Independence

Definition: two events are independent if the occurrenceof one event has no effect on the occurrence of theother event. In probability experiments where there isno “memory” from one event to the next, the eventsare called independent.

Examples of Independent Events:Coin tosses. Even when 10 heads are flipped in a row the next

coin toss still as a 50:50 chance of being a head.

Roulette wheel spins. Each spin of the wheel is theoreticallyindependent. Each number on the wheel has equalprobability of occurring at each spin.

Rolling dice repeatedly. The dice cannot “remember” whatthey rolled from one toss to another.

Drawing cards with replacement. “With replacement”means after a card is drawn it is put back in the deck; thusall cards are equally likely to be drawn each time.

Examples of Dependent Events:Drawing cards without replacement. Once a card is drawn

it cannot be drawn a second time. This changes thecharacteristics of the remaining deck of cards.

Bingo numbers. Once a ball is drawn it is not replaced.

Lottery 6/49. All numbers (1– 49) are equally likely to bechosen but can only be chosen once.

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Multiplication Rules 1

Multiplication Rule 1: if two events are independent (i.e.,have NO on influence of each other’s probability)then the probability of A and B occurring is:

Venn diagram:

Examples:Coin and dice tossing, lotteries, slot machines,

roulette wheels etc., any game or experiment whereknowledge of an outcome is not “remembered” by the nextgame or experiment.

• Probability of tossing heads twice.

• Probability of rolling seven twice with two dice.

• Probability of having nine daughters in a row.

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Multiplication Rules 2

Multiplication Rule 2: if two events are dependent thenthe probability of A and B occurring is:

where P(B|A) means the probability of B occurringgiven that A occurs or occurred, also called theconditional probability

Examples:Card games where the cards are not replaced or

selections where replacement is not allowed. Results fromone experiment affect outcome of next.

• Probability of a drawing a two then a three.

• Probability of a drawing an ace then a face card.

• Probability of a drawing a pair.

• Probability of a drawing a pair of aces.

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Probability Distributions

Definition: distribution of the values of a random variable andtheir probability of occurrence.

Random variable: discrete or continuous variable whosevalues are determined by chance.

Examples:1. Probability distribution of acoin toss (approximately 1 half)

2. Probability distribution of a “fair” die toss (each 1/6 )th

3. Probability distribution ofpolls (correct 19 times out of 20)

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Mean, Variance and Expectation

Mean: of a probability distribution (weighted average)

i iwhere X is the i outcome and P(X ) is its probability.th

Examples:1. Mean number of heads for tossing two coins

2. Mean number of “spots” for tossing a single die

Notice that the answer does not have to be possible.

Variance and Standard Deviation:

Expectation: the expectation or expected value of aprobability distribution is equal to the mean.• for predicting the cost of playing games and lotteries

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Expectation, cont’d

Examples:

1. Compute the expectation of playing a lottery where 100tickets are sold for $1 and the winning prize is worth $100.

This is considered a “fair” game. If the prize was $50 theexpectation would be –$0.50. Any negative value is a loser for theplayer; any positive value is a good game for the player.

2. Compute the profit or loss of playing a lottery where thecost of a ticket is $10, there are 1000 tickets sold and theprizes are:

1 place wins $1000,st

2 place wins $500 andnd

five 3 places win $100rd

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Binomial DistributionDefinition: probability distribution in which there are only two

outcomes, or can be reduced to only two by some rule(“an event occurs” and “the event does not occur”).

Examples: heads and tails, true and false, success and failure,boy or girl, equal to a value and not equal, roll a “1” andnot roll an “1” with a die.

Rules: - only two outcomes per trial- fixed number of trials- independence from trial to trial- probability same from trial to trial

Notation:p = probability of successq = probability of failuren = number of trialsx = number of successes where 0 # x # n

n xP(x) = C × p × q x n–x

Note, since p + q = 1 therefore q = 1 – p

Examples:1. Probability of 4 sixes in 4 tosses of a die.

2. Probability of tossing five heads in seven tosses.

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Binomial Distributions

Examples:1. Tossing of a “fair” coin (1 trial and 4 trials)

2. Rolling a “six” with a fair die (rolling a die ismultinomial)

3. Answering a four-choicemultiple choice questioncorrectly

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Normal Distribution

Many biological and physical processes exhibit a distributioncalled the normal or Gaussian distribution. Values tend tocluster around a mean and extreme values are relatively rare. Eachnormal distribution has different units of measure but they can benormalized using the z-score transform (z = (X - m)/s). Thisdefines the standard normal distribution or z-distribution.

Definition:

e is called the exponential function. e , called Euler’s number, isx 1

a transcendental number equal to:S 1/n! @ 2.718 281 828 459 045 1 ...

Properties of Standard Normal Distribution: • “bell” shaped, unimodal distribution• mean, median and mode are the same and equal to 0• standard deviation is equal to 1• area under curve equals 1• symmetrical about the mean• continuous function (no gaps &

for every x there is a single y)• asymptotic to the x-axis at both ends (y values

approach but never become zero)

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Applications of Normal Distribution 1

Uses: • computing areas and percentiles of scores that are

“normally distributed”• testing hypotheses concerning means of different

populations (are they the same or different?)

Examples:1. Find the percentage of scores between +/–1 standard

deviations.area between 0 and +1s = 0.3413area between 0 and –1s = 0.3413

area between –1s and +1s = 2 × 0.3413 = 0.6826 = 68.3%

2. Find the percentage of scores between +/– 2 and +/–3standard deviations.

area between 0 and +2s = 0.4772area between –2s and +2s = 2 × 0.4772 = 0.954 = 95.4%area between –3s and +3s = 2 × 0.4987 = 0.997 = 99.7%

3. Find the z-score that defines 95% of scores around themean.

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Applications of Normal Distribution 24. Find z-score that defines the lower 90 percent.th

5. Find 25 and 75 percentile z-scores.th th

6. College wants top 15% of students who take a testwhich has a mean of 125 and standard deviation of 25.

7. Determine the 5 and 95 percentile heights of ath th

population that has a mean of 150.0 ±20.0 cm.

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Central Limit Theorem 1

Sampling Distribution of Sample Means: distributionbased on the means of random samples of a specifiedsize (n=constant) taken from a population.

Example: Test scores from a class of four students. Scores were2,4,6,8

(uniform distribution)

List all possible samples of size n=2, allowing replacement. Sample Mean Sample Mean

2,2 2 6,2 4

2,4 3 6,4 5

2,6 4 6,6 6

2,8 5 6,8 7

4,2 3 8,2 5

4,4 4 8,4 6

4,6 5 8,6 7

4,8 6 8,8 8

Sampling distribution of means:Mean frequency

2 13 24 35 46 37 28 1

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Central Limit Theorem 2

As sample size (n) increases the shape of the samplingdistribution of sample means taken from a population with mean,m, and standard deviation, s, will approach a normal

distribution, with mean, m, and standard deviation, .

• the standard deviation of the sampling distribution iscalled the standard error of the mean (notice, bydefinition, it is always less than sample’s standarddeviation when n > 1)

Note, whenever the sample size (n) exceeds 5% of the populationsize (N) the standard error must be adjusted by the FinitePopulation Correction Factor:

That is,

Example:What is the standard error of the mean for a sampling

distribution given a sample of size of 100 and s.d. of 5.00 takenfrom a population of size, 1000.

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Confidence Intervals: Estimators

Point Estimate:• a specific value that estimates a parameter• e.g.,“best estimator” of the population mean (m) is a

sample mean • problem is that there is no way to determine how close a

point estimate is to the parameter

Properties of a Good Estimator:1. must be an unbiased estimator -expected value of

estimator or mean obtained from samples of a given sizemust be equal to the parameter

2. must be consistent -as sample size increases estimatorapproaches value of the parameter

3. must be relatively efficient -estimator must havesmallest variance of all other estimators

Interval Estimate:• range of values that estimate a parameter• precise probabilities can be assigned as to the validity of

the interval• e.g., mean ± standard deviation, 25 ± 10 kg,

5 - to 95 -percentiles, sample mean ± standard errorth th

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Confidence Intervals when s is Known

Confidence Interval:• interval estimate based on sample data and a given

confidence level

Confidence Level:• probability that a parameter will fall within an interval

estimate• related to probability of an error called the alpha (a)

level, that is,Confidence Level (CL) = 1 – a

E.g., CL = 95% means a = 0.05CL = 99% means a = 0.01

Formula for Computing Confidence Intervals

a/2where z is the z-scorethat places the area ½ain each tail of thenormal distribution.For example, if a is 5%

a/2then z is the z-scorethat places 2.5% in theright tail and 2.5% in theleft tail.

a/2That is z = +/–1.960.

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Margin of Error(Maximum Error of Estimate)

Margin of Error alsocalled Maximum Error ofEstimate (E):

Example:Compute the 95 -percentileth

confidence interval from asample of size 30 that has astandard deviation of 5.00 anda mean of 25.0. Since s isunknown use s.

From z-table:

a/2z = +/!1.960

Therefore,

Confidence interval = CI = 25.0 ±1.789 = 23.2 to 26.8

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Confidence Intervals when s is Known

• Data must be randomly sampled• Each datum must be independent of other data• Data must be “approximately” normally distributed• If not normally distributed, sample size should be

greater than 29• When population standard deviation is known

calculate margin of error (E) from:

• Add and subtract E from the sample mean

ExampleHeights of a random sample of 100 people were collected andafter determining they were normally distributed, the mean wascomputed. Given that the population’s standard deviation was8.25 cm and the sample mean was 150.5 cm, compute the 95 -th

percentile confidence interval.

First, determine the z-values that define the 95 -percentileth

a/2confidence interval. I.e., z = +/–1.960

a/2Next, compute margin of error: E = (z x s) / /n

= (1.960 x 8.25)//(100) = 16.17/10 = 1.617 cm

Finally, add and subtract E from sample mean to define theconfidence interval:

CI: 148.9 cm < m < 152.1 cm

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Confidence Intervals when s is Unknown

• Data must be randomly sampled and independent.• If data are normally distributed and s is not known,

which is often the case, use sample standarddeviation, s, and use the t-distribution withn–1degrees of freedom.

• If not normally distributed use t-distribution as longas n > 29.

Formula for Computing Confidence Intervals forSmall Sample Sizes

Decision Tree for Selecting Statistical Method

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t-distribution

• family of curves similar to z-distribution, which becomemore platykurtic (flatter) as sample size decreases

• select distribution using degrees of freedom (df ) that isusually n–1

• mean is 0, area under curve is 1, is asymptotic to x-axis,symmetrical about mean, SD is greater than 1, approachesshape of z-distribution as sample size increases (verysimilar when df =29 or greater (i.e., n >29).

Example:Compute the 95 -percentile confidence interval from a sample ofth

size of 10 which has a standard deviation of 5.00 and a mean of25.0. (Similar to previous example.)From t-table with degrees of freedom (df ) = n–1 = 9:

a/2t = +/–2.262(Use columns labelled “area in two tails” or “two tails”.)

Therefore,

Confidence interval = CI = 25.0 ± 3.58 = 21.4 to 28.6

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Sample Size Estimation

Minimum Sample Size for Interval Estimate ofPopulation Mean:

where n is sample size, s is the population standard deviation andE is the maximum error of estimate. Note, always “round up” tothe next highest integer when there is a fraction.

When s is unknown it may be estimated from the sample standarddeviation, s.

Example:Calculate the sample size needed to estimate muscle strength froma population that has a standard deviation of 140.0newtons if you want to be 95% confident and within50.0 newtons.

n = (1.960 × 140.0 / 50.0) = 30.12

You will therefore need a sample size of 31. (Always round up tobe “conservative”).