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Content for DiscussionContent for Discussion

Standard Scores or Z scores

by: Ms. Mary Jane C. Lepiten

Uses of Z scores

.

The Normal Curve

by Ms. Araya I. Mejorada


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What is az-score?

A z score is a raw score expressedin standard deviation units.

Ms. Mary Jane C. Lepiten

z scoresz scores aresometimes calledstandard scoresstandard scores

S

XXz

=Here is the formula for a z score:

=X

zor


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Computational FormulaComputational Formula

Where = any raw score or unit of measurement

s

x=

S

XXz

=

=X

z

or

= mean and standard deviation of thedistribution of scores

= mean and standard deviation of the

distribution of scores

sX ,

Score minus the mean divided by thestandard deviation

,


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You score 80/100 on a statistics test and your friend also

scores 80/100 on their test in another section. Heycongratulations you friend sayswe are both doingequally well in statistics. What do you need to know ifthe two scores are equivalent?

UsingUsing zz scoresscores toto comparecompare twotwo rawrawscoresscores fromfrom differentdifferent distributionsdistributions

the mean?

What if the mean of both tests was 75?

You also need to know the standard deviation

What would you say about the two test scores if the Sin your class was 5 and the S in your friends class is

10?


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What is the z score for your test: raw score = 80; mean= 75, S = 5?

S

XXz

=

15

7580=

=z

Calculating z scoresCalculating z scores

What is the z score of your friends test: raw score = 80;mean = 75, S = 10?

S

XXz

= 5010

7580

.=

=z

Who do you think did better on their test? Why do you

think this?


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Calculating z scoresCalculating z scores

X x z

70 10 1.00S

XXz

= =

=

10

6070

Example: Raw scores are 46, 54, 50, 60, 70. Themean is 60 and a standard deviation of 10.

110

10=

6060 0

60 0 .00

50 - 10 - 1.00

54 - 6 - 0.60

46 - 14 - 1.40

==

10z

10.=

=

=

10

6050z 1

10

10=

=

=

10

6054z 6010

6 .=

=

=

10

6046z 41

10

14.=


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Why zWhy z--scores?scores?

Transforming scores in order to make

comparisons, especially when usingdifferent scales

Gives information about the relativestanding of a score in relation to thecharacteristics of the sample or population

Location relative to meanRelative frequency and percentile


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What does it tell us?What does it tell us?

z-score describes the location of the raw

score in terms of distance from the mean,measured in standard deviations

Gives us information about the location ofthat score relative to the averagedeviation of all scores


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ZZ--score Distributionscore Distribution Mean of zero

Zero distance from the mean

Standard deviation of 1

Z-score distribution always has sameshape as raw score. If distribution waspositively skewed to begin with, zscores made from such a distribution

would be positively skewed.


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Distribution of the various types of standard scores

z Scores -3 -2 -1 0 +1 +2 +3

Navy Scores 20 30 40 50 60 70 80

ACT 0 5 10 15 20 25 30

CEEB 200 300 400 500 600 700 800


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Transformation EquationTransformation Equation Transformations consist of making the

scale larger, so that negative scores areeliminated, and of suing a larger standarddeviation, so that decimals are done away

.

Transformation scores equation:

standard score= z(new standard deviation) + the new mean


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Transformation EquationTransformation Equation

A common form for these transformationsis based upon a mean of 50 and astandard deviation of 10. in equation form

standard score= z(new standard deviation) + the new mean

Or starting with the raw score, we have:

standard score= z(10) + 50

Standard score 5010 +

= )(S

XX


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Fun facts about z scoresFun facts about z scores

Any distribution of raw scores can be converted to a

distribution of z scores

The mean of a distribution has a zzero

Positive z scores represent raw scoresthat are __________ (above or below)the mean?

above

Negative z scores represent raw scoresthat are __________ (above or below)the mean?

below

____


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Comparing scores from differentdistributions

Mr. Johny S. Natad

n erpre ng esr ng n v ua scores

Describing and interpreting sample means


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Student Geography ArithmeticA 60 40

B 72 36

C 46 24

etc.

PART A: RAW SCORES

Spelling140

100

110

Comparing Different VariablesComparing Different Variables Standardizes different scores

Student Geography Spelling Arithmetic Average

A 50 70 80 67

B 62 50 73 62

C 36 55 53 48

PART B: STANDARD SCORES

ean

Standard deviation 10 620

Standard score 5010 +

= )(S

XX

=+= 501010

6060 )(SS 50500 =+)(

Usingtransformation

equation:

ean

Standard deviation


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Student Geography Arithmetic

A 60 40

B 72 36C 46 24

etc.

Mean 60 22

Standard deviation 10 6

PART A: RAW SCORES

Spelling

140

100110

100

20

Interpreting Individual ScoresInterpreting Individual Scores


A 50 70 80 67

B 62 50 73 62

C 36 55 53 48

:

Student As performance is average ingeography, excellent in spelling, and superiorin arithmetic.


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Using standard deviation unitsUsing standard deviation units

to describe individual scoresto describe individual scores

Here is a distribution with a mean of 100 and standard deviation

of 10:

What score is one standard deviation below the mean?

What score is two standard deviation above the mean?

100 110 1209080

-1 s 1 s 2 s-2 s

90

120


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Using standard deviation units to

describe individual scoresHere is a distribution with a mean of 100 andstandard deviation of 10:

How many standard deviations below the mean is ascore of 90?

How many standard deviations above the mean is a

score of 120?

2

1

100 110 1209080

-1 s 1 s 2 s-2 s


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A 50 70 80 67

B 62 50 73 62

C 36 55 53 48


Describing Individual ScoresDescribing Individual Scores

--

1. What is the standard deviation of50? ___

20 30 40 50 60 70 80

67 8070Student A 50

scores



3. What is the standard deviation of 80? ___

0

2

3

1.7

-

=

Xz

10

5067 =

711017 .==


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A 50 70 80 67

B 62 50 73 62

C 36 55 53 48



--

20 30 40 50 60 70 80

67 8070Student A 50

scores

0 2 31.7

-

Student A is at mean in Geography, 2 standard deviationabove the mean in Spelling, 3 standard deviation abovethe mean in Arithmetic and has an average of 67 which is

1.7 standard deviation above the mean.


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A 50 70 80 67

B 62 50 73 62

C 36 55 53 48



--

20 30 40 50 60 70 80

67 8070Student A 50

scores

0 2 31.7

-

Student B

Student C

736250

5348 5536

0 1.2 2.3

0.3 0.5-0.2-1.4

62


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Using the zUsing the z--TableTable

Important when dealing with decimal z-

scores

Gives information about the area between

t e mean an t e z an t e area eyon zin the tail

Use z-scores to define psychologicalattributes


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Using zUsing z--scores to Describe Samplescores to Describe SampleMeansMeans

Useful for evaluating the sample and for inferential

statistical procedures Evaluate the sample means relative standing

by plotting all possible means with that samplesize and is always approximately a normaldistribution

Sometimes the mean will be higher, sometimes

lower The mean of the sampling distribution always

equals the mean of the underlying raw scores ofthe population


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Random variation conforms to a

particular probability distribution knownas the normal distribution, which isthe most commonl observed

Ms. Araya I. Mejorada

probability distribution.

de Moivre

Mathematicians de Moivre

and Laplace used thisdistribution in the 1700's


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German mathematician and

physicist Karl FriedrichGauss used it to analyzeastronomical data in 1800's,

The Standard Normal Curve

known as the Gaussiandistribution among thescientific community.

Karl Friedrich Gauss

The shape of the normal distributionresembles that of a bell, so it sometimes isreferred to as the "bell curve".


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Symmetric - the mean coincides with aline that divides the normal curve into parts.It is symmetrical about the mean becausethe left half of the curve is just equal to theright half.

Bell Curve Characteristic

-said to be normal if the mean, median andmode coincide at a single point

Extends to +/- infinity - left and right tailsare asymptotic with respect to the horizontallines

Area under the curve = 1


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CompletelyCompletely DescribedDescribed byby TwoTwoParametersParameters

The normal distribution can be completelyspecified by two parameters:

1.mean

.s an ar ev a on

If the mean and standard deviation areknown, then one essentially knows asmuch as if one had access to every pointin the data set.


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Drawing of a Normal curveDrawing of a Normal curve

Normal Curve

Standardized

Normal Curve


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Areas Under the Normal CurveAreas Under the Normal Curve

.3413 of the curve falls between the mean and onestandard deviation above the mean, which meansthat about 34 percent of all the values of a normallydistributed variable are between the mean and one

standard deviation above it


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The normal curve and the area under the curve between

units

about 95 percent of the values lie within twostandard deviations of the mean, and 99.7 percent of

the values lie within three standard deviations


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68.26%

PercentagePercentage underunder thethe NormalNormal CurveCurve atatvariousvarious standardstandard deviationdeviation unitsunits fromfromthethe meanmean

Approximately 68.26% of scores will fallwithin one standard deviation of the mean

In a normal distribution:

-3s -2s -1s +1s +2s +3s

13.59% 13.59%2.15% 2.15%


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PointsPoints inin thethe NormalNormal CurveCurve

Points in the normal curve above or belowwhich different percentage of the curve lie

90%

10%

c10 c90

28.1=z 28.1=z

16

80

000,1

=

=

=

N


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S

XX

z

=

Areas Cut Off Between different

Points

80110 =

16

30=

8751.=80=X

16 cases

110=X

8751.=z


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=X

z

Equation of z for a different

unknown

80X

Similarly, the raw-score equivalentof the point below which 10 percentof the case fall is:

16

=.

).( 2811680 =X

482080 .+=X

5100.=X

16

80281

=.

482080 .=X

482080 .=X

559.=X


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Application of Normal Curve ModelApplication of Normal Curve Model

Can determine the proportion of scores

between the mean and a particular score

Can determine the number of peoplew r u r r r ymultiplying the proportion by N

Can determine percentile rank

Can determine raw score given thepercentile


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Acknowledgement of References:Acknowledgement of References:

N.M Downie and R.W Heath. BasicStatistical Methods, 5th Edition. Harper &

Row Publisher, 1983

Robert Nileshttp://www.robertniles.com/stats/stdev.shtml

Rosita G. Santos, Phd, et. al. Statistics.Escolar University, 1995.

Leslie MacGregor. z Scores & the NormalCurve Model (presentation)


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