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Quick Statistics Formula Cheat Sheet, Cheat Sheet of Statistics

In this cheat sheet you find basic principles of Statistics for introductory courses.

Typology: Cheat Sheet

2019/2020

Uploaded on 10/23/2020

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.TLI\
TSTICS FOR INTRODUGTORY COURSES
J STATISTICS - A set
of tools for collecting,
oreanizing, presenting, and analyzing
numerical facts
or observations.
I . Descriptive Statistics - procedures used to
organize and present data in a convenient,
useable.
and communicable form.
2. Inferential Statistics - procedures
employed
to arrive at broader generalizations or
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
are collected
by
observation.
J POPULATION - A complete set of actual
or potential observations.
J PARAMETER - A number describing a
population characteristic; typically, inferred
from sample
statistic.
f SAMPLE - A subset of the population
selected
according
to some scheme.
J RANDOM SAMPLE - A subset selected
in such a way that each member of the
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 summed
deviations
are
equal
to zero.
Average value of a sample or population.
POPULATION MEAN SAMPLE MEAN
p:
+!,*, o:#2*,
Note: The mean
ls very sensltlve
to extreme
measure-
ments that are
not balanced
on both sides.
I WEIGHTED MEAN - Sum of a
set of observations
multiplied by their respective
weights,
divided by the
sum of the weights: 9, *, *,
WEIGHTED MEAN -L-
,\r*'
where xr,
: weight,'x,
- observation;
G : number of
observaiion
grdups.'Calculated
from a
population.
sample.
or gr6upings in a frequency distribution.
Ex. In the FrequencVDistribution below, the meun is
80.3: culculatbd by-
using frequencies for the wis.
When
grouped,
use closs
midpoints
Jbr xis.
J MEDIAN - Observation
or potenlial
observation
in a
set
that divides
the set so that the same
number of
observations
lie on each
side of it. For an odd number
of values. it is the
middle value;
for an
even
number
it
is the average
of the
middle two.
Ex. In the Frequency Distribution table below, the
median is 79.5.
f MODE - Observation
that occurs
with the greatest
tiequency. Ex. In the Frequency Distributioln nble
below. the mode is 88.
O SUM OF SOUARES fSSr- Der iations
tiom
the mean. squared
and
summed: , (I r,),
PopulationSS:I(Xi
-l.rx)'or
Ixi'- t N
_ r, \,)2
Sample
SS:I(xi -x)2or
Ixi2---
O VARIANCE - The average
of square
differ-
ences
between
observations
and
their
mean.
POPULANONVARIANCE
SAMPLEVARIANCE
VARIANCES
FOH
GBOUPED
DATA
POPUIATION SAMPLE
^{G-'{G
o2:*i t,(r,-p
)t s2=;1i tilm'-x;2
lI ;_r t=1
D STANDARD DEVIATION - Square
root of
the variance:
Ex. Pop. S.D. o -
n
Y
IU
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.
- Interval Scale- a quantitative scale
that
permits the use of arithmetic operations.
The
zero point in the scale is arbitrary.
- R.atio Scale- same
as
interval scale excepl
that there is a true zero point.
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 GROTJPED
FREOUENCY
EilSTRIBUTION
- A frequency distribution in which the values
ofthe variable have been grouped into classes.
J il {il, I a rr I.)'A .l b]|, K I 3artl LQ
xfxtxfxt
100 183 11 74 11f 65 o
99 1ut 11111 75 1111 66 1
98 085 176 11 67 11
gl 086 o77 111 68 1
96 11 87 17A I69 111
95 088 1111111 79 11 70 1111
94 089 111 80 171 0
93 I11 81 11 72 11
92 091 182 I73 111
tr CUMULATUE
FREOUENCY
BISTRI.
BUTION -A distribution
which
shows
the to-
tal frequency
through
the upper
real limit of
each class.
tr CUMUIATIVE PERCENTAGE DISTRI.
BUTION-A distribution
which
shows
the to-
tal percentage
through
the upper
real limit of
each class.
!I! llrfGl:
I il {.ll lNl.l'tlz
CLASS f
ICum
f"
65-67 334.84
6&70 811 17.74
71-73 516 25.81
7+76 925 40.32
Tt-79 631 50.00
80-82 435 56.45
83-85 843 69.35
86-88 8 51 82.26
89-91 657 91.94
92-g 158 93.55
95-97 260 96.77
9&100 262 100.00
15
10
0
NORMAL CURVE
^/T\
./ \
-t
-att? \
CLASS f CLASS t
98-100
15
10
0
SKEWED
CURVE
-- \
/\
-/ LEFT \
J- \
pf3
pf4

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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.

  1. 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.

  1. 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 ,

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