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. T L I \
T S T I C S F O R I N T R O D U G T O R Y C O U R S E S
J STATISTICS - A set of tools for collecting, o r e a n i z i n g , p r e s e n t i n g , a n d a n a l y z i n g numerical facts or observations. I. Descriptive Statistics - procedures used to organize and present data in a convenient, useable. and communicable form.
- Inferential Statistics - procedures^ employed t o a r r i v e a t b r o a d e r^ g e n e r a l i z a t i o n s^ o r inferences from sample data to populations.
-l STATISTIC - A number describing a sample characteristic. Results from the manipulation of sample data according to certain specified procedures. J DATA^ - Characteristics or numbers that a r e c o l l e c t e d b y o b s e r v a t i o n. J POPULATION - A complete set of actual or potential observations. J PARAMETER - A^ number describing a population characteristic; typically, inferred f r o m s a m p l e s t a t i s t i c. f SAMPLE^ - A^ subset of^ the population selectedaccording to some scheme. J RANDOM^ SAMPLE^ - A^ subset selected i n s u c h a w a y t h a t e a c h m e m b e r o f^ t h e population has an equal opportunity^ to be selected. Ex.lottery^ numbers in afair^ lottery J VARIABLE - A phenomenon that may take on different values.
f MEAN -The ooint in a distribution of measurements about which the summeddeviationsare equal^ to zero. Average value of a sample or population. POPULATION MEAN SAMPLE MEAN
p: (^) +!,, o:#2,
Note: The mean ls very sensltlveto extrememeasure- mentsthat are not balancedon both sides. I WEIGHTED MEAN^ - Sum of a setof observations multiplied by their respectiveweights, divided by the sum of the weights: (^) 9, *, *,
WEIGHTED MEAN -L-
,\r*'
w h e r ex r , :^ w e i g h t , ' x ,-^ o b s e r v a t i o n ;G :^ n u m b e ro f o b s e r v a i i o ng r d u p s. ' C a l c u l a t e df r o m a p o p u l a t i o n. sample.or gr6upings in a frequencydistribution. Ex. In the FrequencVDistribution below, the meun is 80.3: culculatbd by- using frequencies for the wis. When grouped, use clossmidpointsJbr xis. J MEDIAN - Observationor potenlialobservationin a set that divides the set so that the same^ number of observationslie on each side of it. For an odd number of values.it is the middle value; for an even^ number it is the averageof the middle two. Ex. In the Frequency Distribution table below, the median is 79.5. f MODE - Observationthat occurs with the greatest tiequency. Ex. In the Frequency Distributioln nble below. the mode is 88.
O SUM OF SOUARES fSSr- Der^ iationstiom
the mean.squaredandsummed:
, (I r,),
P o p u l a t i o n S S : I ( X i- l. r x ) ' o r^ I x i ' - t^ N
_ r , \ , ) 2
S a m p l eS S : I ( x i - x ) 2 o r I x i 2 - - -
O VARIANCE - The averageof squarediffer-
encesbetweenobservationsandtheir mean.
POPULANONVARIANCESAMPLEVARIANCE
VARIANCESFOH GBOUPEDDATA
POPUIATION SAMPLE
^ { G - ' { G
o 2 : * i t , ( r , - p ) t s 2 = ; 1 i t i l m ' - x ; 2
l I ; _ r t = 1
D STANDARD DEVIATION - Squareroot of
the variance: Ex. Pop. S.D. o -
n
Y
I
U
fi
z
D BAR GRAPH - A form of graph that uses bars to indicate the frequency of occurrence of observations. o Histogram - a form of bar graph used rr ith interval or ratio-scaled variables.
- I n t e r v a l S c a l e - a q u a n t i t a t i v e s c a l e t h a t permits the use of arithmetic operations. The zero point in the scale is arbitrary.
- R. a t i o S c a l e - s a m e a s i n t e r v a l s c a l e e x c e p l t h a t t h e r e i s a t r u e z e r o p o i n t. D FREOUENCY CURVE - A form^ of graph representing a frequency distribution in the form of a continuous line that traces a histogram. o Cumulative Frequency Curve - a continuous line that traces a histogram where bars in all the lower classes are stacked up in the adjacent higher class. It cannot have a negative slop€. o Normal curve - bell-shaped curve. o Skewed curve - departs from symmetry and tails-off at one end.
GROUpITG
OF DATA Shows the number of times each observation occurs when the values ofa variable are arranged in order according to their magnitudes.
II GROTJPEDFREOUENCYEilSTRIBUTION
- A frequency distribution in which the values ofthe variable have been grouped into classes.
J il {il, I a rrI.)'A .l b]|, K I 3artl^ LQ
x f x t^ x^ f^ x^ t
100 1 83 1 1^74 11f^65 o
99 1 ut 11111 75 1111 66 1
gl 0 86 o 77 111 68 1
96 11 87 1 7A^ I^69 95 0 88 1111111 79 1 1 70 1111 94 0 89 111 80 1 71 0 93 I 1 1 81 1 1^72 1 1 92 0 91 1 82 I 73 111
tr CUMULATUEFREOUENCYBISTRI.
BUTION -A distributionwhich^ showstheto-
tal frequencythrough the upper real limit of
eachclass.
tr CUMUIATIVE PERCENTAGE DISTRI.
BUTION-A distributionwhich showstheto-
tal percentagethrough the upper real limit of
eachclass.
I il {.ll lNl.l'tlz^ !I!^ llrfGl:
CLASS fI Cum f "
65-67 3 3 4. 6&70 8 1 1^ 17. 71-73 5 1 6 25. 7+76 9 25 40. Tt-79 6 3 1 50. 80-82 4 35 56. 83-85 8 43 69. 86-88 8 5 1^ 82. 89-91 6 57 9 1. 9 4 92-g 1 58 9 3. 5 5 95-97 2 60 96. 9&100 2 62 100.
1 5
1 0
0
N O R M A LC U R V E
^/T\
./ \
-t
C L A S S f C L A S S t^ -att?^ \
1 5
1 0
0
SKEWEDCURVE
-- \
/ \
-/ LEFT \
J- \
Probability of occurrence^t at -Number of outcomafamring^ EwntA oif'ent'l Ant=@
D SAMPLE SPACE - All possibleoutcomesof an
experiment.
N TYPE OF EVENTS
o Exhaustive - two or more events are said to be exhaustive if all possible outcomes are considered. Symbolically, P (A or B or...) -^ l. rNon-Exhausdve -two^ or more events are said to be non- exhaustive if they do not exhaust all possible outcomes. rMutually Exclusive^ -^ Events^ that^ cannot^ occur simultaneously:p (A and B) = 0; and^ p (A or B) = p (A) + p (B). Ex. males, females oNon-Mutually Exclusive - Event-s that can occur s i m u l t a n e o u s l y : p ( A o r B )^ = P ( A ) + p ( B )^ - p ( A^ a n d B ) ' &x. males, brown^ eyes. Slndependent - Events whose probability^ is unaffected by occurrence or nonoccurrence of each other: p(A^ lB) = p(A); ptB In)= p(e); and p(A and B) = p(A) p(B). Ex. gender and eye color SDependent - Events whose probability^ changes deoendlns upon the occurrence or non-occurrence ofeach other: p{.I I bl^ dilfers lrom AA): p(B lA)^ differs from p ( B ) ; a n d p ( A a n d B ) : p ( A ) p ( B l A ) :^ p ( B ) A A I B ) Ex. rsce and eye colon
C JOINT PROBABILITIES - Probabilitythat2 ot
more eventsoccur simultaneously.
tr MARGINAL^ PROBABILITIES^ or Uncondi-
tional Probabilities= summationof probabilities'
D CONDITIONAL PROBABILITIES^ - Probability of I given the existence of ,S,^ written,^ p (Al$. f l E X A M P L E - G i v e n^ t h e^ n u m b e r s^ I^ t o^9 a s o b s e r v a t i o n s i n a s a m p l e s p a c e : .Events mutually exclusive and exhaustive' Example: p (all odd numb ers); p ( all eu-en nurnbers) .Evenls mutualty exclusive but not exhaustive- Example: p (an eien number); p (the numbers 7 and 5) .Events ni:ither mutually exclusive or exhaustive- Example: p (an even number or a 2)
fl SAMPLING DISTRIBUTION^ - A theoretical
probability distribution of a statistic that would
iesult from drawing all possible samples^ of a
given size from some population.
THE STAIUDARDEBROR OF THE MEAN
A theoretical standard deviation^ of sample mean of a given sample si4e, drawn from some speciJied popu- lation. DWhen based on a very large, known population, the s t a n d a r de r r o r i s : (^) 6 _ _ o " r _ ^ ln
EWhen estimated from a sample drawn from very large population, the standard error is:
lThe dispersion of sample means decreasesas sample size is increased.
O = =^ t - S ' f n
RANDOM VARIABLES A mapping or function that assignsone and'onlv one-numerical value to each outcome in an exPeriment.
tl DISCRETE RANDOM VARIABLES - In-
volvesrulesor probabilitymodelsfor assign-
ing or generatingonly distinctvalues(not^ frac-
tionalmeasurements).
C BINOMIAL^ DISTRIBUTION - A model
for the sum of a seriesof n independenttrials
wheretrial resultsin a 0 (failure)^ or I (suc-
cess).Ex. Coin to
" t p ( r ) = (! ) n ' l - t r l " - '
wherep(s) is the probabilityof s successin n
trials with a constantn probability per trials,
a n dw h e r e( , 1 \ = , - " - " ' - ' - t s / s! ( n - s ) !n!
B i n o m i a l m e a n : (^)! : n x Binomial variance: o': n, (l -^ tr)
A s n i n c r e a s e s ,t h e B i n o m i a l^ a p p r o a c h e st h e Normal distribution. D HYPERGEOMETRIC^ DISTRIBUTION^ - A model for the sum of a series of n trials where each trial results in a 0 or I and is drawn from a small population with N elements split between N1 successesand N2 failures. Then the probabil- ity of splitting the n trials between xl successes and x2 failures is:^ Nl! (^) {_z!
p ( x l a n d t r r : W 't 4tlv-r;lr
Hypergeometric mean: pt :E(xi^ - + andvariance: o2: ffit+][p]
D POISSON DISTRIBUTION - A model for the number of^ occurrences of an event x^ : 0 , 1 , 2 ,... ,w h e n t h e p r o b a b i l i t y o f o c c u r r e n c e i s s m a l l , b u t t h e n u m b e r o f o p p o r t u n i t i e s f o r t h e o c c u r r e n c ei s l a r g e , f o r x :^ 0 , 1 , 2 , 3... .a n d )v > 0. otherwise P(x) =. 0. e $ t = f f P o i s s o nm e a n a n d r a r i a n c e : , t.
Fo r c ontinuo u s t'a ri u b I es. .fi'eq u en t'^ i es u re e.tp re^ ssed in terms o.f areus under u t'ttt.re. D CONTINUOUS RANDOM VARIABLES
- Variable that may take on any value along an uninterrupted interval of a numberline. D NORMAL DISTRIBUTION - bell cun'e; a distribution whose values cluster symmetri- cally around the mean (also^ median and mode).
f ( x ) = - 1 ,
( x - P ) 2 1 2 o 2 o"t'2x
wheref (x): frequency.at.a givenrzalue o :^ s t a n d a r dd e v i a t l o no f t h e d i s t r i b u t i o n
lt : approximatelyI 111q
approximately2.
p : the meanof the distribution
x : any scorein the distribution
D STANDARD NORMAL DISTRIBUTION
- A normalrandomvariableZ. thathasa mean
of0. andstandarddeviationof l.
Q Z-VALUES - The numberof standarddevia-
tionsa specificobservationliesfrom^ themean:
' : x - 1 1
tr LEVEL OF SIGNIFICANCE-Aprobabilin
valueconsideredrarein thesamplingdistribution.
specifiedunderthenull hypothesiswhereoneis
willing to acknowledgetheoperationof chance
factors. Common significance levels are 170, 5 0 , l 0 o. A l p h a ( a ) l e v e l :^ t h e l o w e s tl e v e for which the null hypothesis can be rejected. The significanceleveldeterminesthecritical region. [| NULL HYPOTHESIS (flr)^ - A^ statement that specifies hypothesized value(s) for one or more of the population parameter. lBx. Hs=^ a coin is unbiased.That isp :^ 0.5.] tr ALTERNATM HYPOTHESIS (.r/1) - A statement that^ specifies^ that^ the^ population parameter is some value other than the one specified underthe null trypothesis.[Ex. I1r:^ a coin is biased That isp * 0.5. I. NONDIRECTIONAL HYPOTHESIS^ - an alternative hypothesis (H1) that states onll that the population parameter is different from the one ipicified under H 6. Ex. [1^ f lt + !t Two-Tailed Probability Value is employed^ when the alternative hypothesis is non-directional.
- DIRECTIONAL HYPOTHESIS -^ an alternative hypothesis that statesthe direction rn which the population parameter differs fiom the one specified under 11* Ex. Ilt: Ir > pn^ r-trHf lr ' t One-TailedProbability Value is employedu'hen the alternative hypothesis is directional. D NOTION OF INDIRECT^ PROOF^ - Stnct interpretation ofhypothesis testing reveals that thc' null hypothesis can never be proved. [Ex. Ifwe toi. a coin 200 times and^ tails comes^ up 100 times. it i^ s no guarantee that heads will come up exactly hali the time in the long run; small discrepancies migfrt exist. A bias can exist even at a small magnitude. We can make the assertion however that NO B A S I S E X I S T S F O R R E J E C T I N G T H E H Y P O T H E S I S T H A T^ T H E^ C O I N^ I S UNBIASED. (The null hypothesisis not reieued. When employing the 0.05 level of significa reject the null hypothesis when a given res occurs by chance 5% of the time or less.] ] TWO TYPES OF ERRORS
- (^) Type 1 Error (Typea Error) = the rejectionof
11,whenit is actuallytrue.The^ probabilityof
a type 1 error is givenby a.
-TypeII Error(TypeBError)=The acceptance
offl, whenit is actuallyfalse.Theprobabilin
of a type II error is given by B.
(for sample mean X) rlf x 1, X2, X3,... xn , is a simple random sample of n elements from a large (infinite) population, with mean mu(p) and standard deviation o, then the distribution of T takes on the bell shaped distribution^ of^ a normal random variable as n increases andthe distribution ofthe ratio: 7-! 6l^J n approaches the standard normal distribution^ as n goes t o ' i n f i n i t y. I n p r a c t i c e. a n o r m a l a p p r o x i m a t i o n i s acceptable for samples of 30 or larger.
Percentage Cumulative Distribution for selected (^) Z values under a normal curye
Z - v a l u e - 3^ - 2^ - l^0 + 1 + 2 + 3 PercentifeScore o-13 2.2a 15.87 50.00 a4.13 97.72 99.a
G O R R E L A T I O N
Q *PEARSON r'METHOD (Product-Moment
CorrelationCoefficient) - Cordlationcoefficient
employedwith interval-or ratio-scaledvariables.
Ex.: Givenobservationsto twovariablesXandIl
we cancomputetheir correspondingu values:
Z, :(x-R)/s" andZ, :{y-D/tv.
'The formulas for the Pearsoncorrelation(r):
: {* -;Xy -y)
JSSI SSJ
- Use the above formula for large samples.
- Use this formula (also knovsn asthe Mean-Devistion Makod of camputngthe Pearsonr) for small samples. 2( z-2,,) r:__iL Q RAW SCORE METHOD is quicker^ and can be used in place of the first^ formula above when the sample values are available. (IxXI/)
Definirton - Carrelation refers to the relatianship baween two variables,The Correlstion CoefJicientis a measurethst exFrcssesthe extent ta which two vsriables we related
z
tU
I
0
D STANDARD ERROR OF THE DIFFER-
ENCE betweenMeansfor CorrelatedGroups.
The seneralformula is:
where r is Pearsoncorrelation o By matching samples on a variable correlated with the criterion variable, the magnitude of the standard error ofthe^ difference can be reduced. o The higher the correlation, the greater the reduction in the standard error ofthe difference.
^ 2 ^ 2 n
""r* " 7r- zrsrrsr,
N S A M P L I N G^ D I S T R I B U T I O N^ O F^ T H E
DIFFERENCE BETWEEN MEANS- If a num- ber of pairs of samples were taken from the same population or from two different populations, then: r (^) The distribution of differences between pairs of samplemeanstendsto be normal (z-distrjbution). r The mean of these differences between means F 1 , 1 " i s e q u a l t o t h e d i f f e r e n c e b e t w e e n t h e population means. that is ltfl-tz. I Z-DISTRIBUTION: or and ozure known o The standard error ofthe difference between means o", - ",
={toi) | \ + @',)I n 2 o Where (u, - u,) reDresentsthe hvpothesizeddif- ferencein rirdan!.'ihefollowins statisticcan be used for hypothesis tests: _ ( 4 - t r ) - ( u t -^ u z )
z=
"r.r,
o When n1 and n2 qre >30, substifuesy and s2 for ol and 02. respecnvely.
(To-qbtain sum of squaygs(SS) (^) see Measures of Cen- tral Tendencyon'page l) D POOLED^ '-TEST o Distribution is normal o n < 3 0 r o1 and 02 are zal known but assumed equal
- The hvoothesistest mav be 2 tailed (: (^) vs. *) or I talled:.1i.is1t, andrhe.alternativeis 1rl > lt2 @r 1t, 2 p 2 and the alternatrves y f p2.)
- degreesof freedom(dfl : (n (^) r-I)+(n y 1)=n (^) fn 2 -2.
;U.r11!9 to determrnest,-x-,.giyen formula below for estimating 6riz
- Determine the critical region for reiection by as- sieningan acceptablelevel-ofsisnificdnceand[ook- in! at ihe r-tablb with df, nt + i2-2. o_Use the following formula for the estimated stan- dard error: n I + n 2 n 1 * n 2 - 2 n tn
(j;.#)
(n1- l)sf +(n2- l)s
D HETEROGENEITY OF VARIANCES mav
be determinedby using the F-test:
o s2lllarger variance' stgmaller variance)
D NULL HYPOTHESIS- Variancesare equal
and their ratio is one.
TIALTERNATM HYPOTHESIS-Variances
differ andtheir ratio is not one.
f, Look at "Table C" below to determine if the
variancesare significantly different from each
other. Use degrees of freedom from the 2
samples:(n1-1,nyI).
Top row=.05,Bottom row=. points for distribution of F
of freedom for numerator
D PURPOSE- Indicatespossibility^ of overall
meaneffectof the experimentaltreatmentsbefore
investigatinga specific hypothesis.
D ANOVA- Consistsof obtaining independent
estimatesfrom populationsubgroups.It allowsfor
the partition of the sum of squaresinto known
componentsof variation.
D TYPES OF VARIANCES
' Between4roupVariance(BGV| reflectsthemag-
nihrdeof thedifference(s)amongthegroupmeans.
' Within-Group Variance (WGV)- reflectsthe
dispersionwithin eachtreatmentgroup. It is also
referredto asthe error term.
tI CALCULATING VARIANCES
' Following the F-ratio, when the BGV is large
relativeto theWGV. the F-ratiowill alsobe larse.
66y=
o2(8i'r,,')'
k- : (^) mean of ith treatment group and xr., of all n values across all k treatmeiii
.ss,+ss, ...+^ss
WGV:
"_O
wherethe SS'sare the sumsof squares(seeMea-
sures of Central Tendencyon page 1) of each
subgroup'svaluesaroundthe subgroupmean.
D USING F.RATIO- F : BGV/WGV
1 Degreesof freedomare^ k-l for the^ numerator
and n-k for the denominator.
' If BGV > WGV, the experimentaltreatments
are responsiblefor the large^ differencesamong
group means.Null hypothesis:the group means
are estimatesof a commonpopulationmean.
In random samplesof size n, the samplepropor-
tionp fluctuatesaroundthe proportionmean: rc
with a proportionvarianceof I#9^ proportion
standarderror of ,[Wdf;
As the sampling distribution of p increases,it
concentratesmore aroundits targetmean. It also
sets closerto the normal distribution.In which
i o o o .w q o v ' P - f t z : ^ T n ( | - t t y n
z
oMost widely-used non-parametric test. .The y2 mean :^ its degreesof freedom. oThe X2 variance :^ twice its degreesof fieedorl o Can be usedto test one or two independentsamples. o The square of a standardnormal variable is a c h i - s q d a r ev a r i a b l e. o Like the t-distibution" it has different distribu- tions depending on the degrees of freedom. D D E G R E E S O F F R E E D O | M ( d. f. )^. / COMPUTATION v o lf chi-pquare tests for the goodness-of'-fitto a hr - p o t h e s i z ' e dd i s t r i b u t i o n. d.f.: (^) S - I - m, where
. g:.number of groups,or classes.in the^ fiequener dlstrlbutlon. m -^ number of population parametersthat lnu\r b e e s t i m a t e d f r o m
- s a m n l e s t a t i s t i c s t o t e s t t h c h y p o t h e s is. o lf chi-squara testsfor homogeneityor contingener d.f :^ (rows-1) (columns-I) f, GOODNESS-OF-FIT TEST- To anolr tht-- c h i - s q u a r e d i s t r i b u t i o n i n t h i s m a n h ' e r - .t h. ' c r i t i c d l c h i - s q u a r ev a l u e i s e x p r e s s e da s : , (f-_i) ' ,nh".. /p :^ observedfreqd6ncy ofthe variable /" (^) - .lp..,tSd fiequency^ (basedon hypothesrzcd p o p u l a t l o no l s t n D u t r o n). t r T E S T S O F C O N T I N G E N C Y - A p p l i c a t i o nt ' l Chi-squarete.ststo two separatepopulaiionsto te.r s t a t r s t l c a lr n d e o e n d e n c eo f a t t r r b u t e s. D T E S T S O F H O M O G E N E I T Y - (^) A p p l i c a t i o no t Chi-square-teststo tWpqq.mplqsto_test'iTtheycanrc f r o m f o p u l a t i o n s w i t h l i k e 'd i s t r i b u t i o n s. D R U N S T E S T - T e s t s w h e t h e r a s e q u e n c e( t t r c o m p r i s ea s a m p l e ). i sr a n d o m .T h e ' f o l l o ui n g e q u a t r o n sa r e a p p l r e d : r R r = 2 " ' r - r ,.. s l 2 n t n r ( 2 n t n ,
n t ' n 1 1
W h e r e
, n , = , , r , n 1 ' I t ' lt " r r J 1 " , * " r 1 21 n r + n , l 1
7 = mean number of runs n , :^ n u m b e r o f o u t c o m e s o f o n e t y p e n2 = number of outcomes of the other type S4 = standard deviation of the distribution of the n u m D e r o I r u n s.
C u s t o m e rH o t l i n e# 1. 8 0 0. 2 3 0. 9 5 2 2
\ l r n u l l i e t u r e c lb r I l n r i n a t r n gS e r rr c e s .I n c. l. a r g o .f l. i r n d s o l d u n d e r l i c e n s et f o r f I. r. 1 f l o l d r n g :. L L (.. P r l o \ l t o. (. \ l S. { u n d e r o n e o r n r o r e o l t h e t b l l o $ i n g P a t e n t s :L S P r r r r
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