BUKIDNON STATE UNIVERSITYGraduate External StudiesSurigao Study CenterSurigao City

Standard Scores and

the Normal Curve

Report Presentation by:

Mary Jane C. LepitenAraya I. Mejorada

Johny S. Natad

16 January 2010

Content for DiscussionContent for Discussion

� Standard Scores or Z scores

by: Ms. Mary Jane C. Lepiten

� Uses of Z scores

by: Johny S. Natadby: Johny S. Natad

� The Normal Curve

by Ms. Araya I. Mejorada

What is a z-score?

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

Ms. Mary Jane C. Lepiten

z scores z scores are sometimes called standard scoresstandard scores

S

XXz

−=Here is the formula for a z score:

σ

µ−=X

zor

Computational FormulaComputational Formula

Where = any raw score or unit of measurementX

sX ,

s

x=

S

XXz

−=

σ

µ−=X

zor

= mean and standard deviation of the

distribution of scores

= mean and standard deviation of the

distribution of scores

sX ,

Score minus the mean divided by thestandard deviation

σµ ,

You score 80/100 on a statistics test and your friend alsoscores 80/100 on their test in another section. Heycongratulations you friend says—we are both doingequally well in statistics. What do you need to know ifthe two scores are equivalent?

the mean?

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 is10?

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

S

XXz

−= 1

5

7580=

−=z

Calculating z scoresCalculating z scores

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

S

XXz

−= 50

10

7580.=

−=z

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

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

z 00

.=60 0 .00

50 - 10 - 1.00

54 - 6 - 0.60

46 - 14 - 1.40

=−

=10

6060z 0

10

0.=

=−

=10

6050z 1

10

10−=

−

=−

=10

6054z 60

10

6.−=

−

=−

=10

6046z 41

10

14.−=

−

Why zWhy z--scores?scores?

�Transforming scores in order to makecomparisons, especially when usingdifferent scales

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

�Location relative to mean

�Relative frequency and percentile

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

� z-score describes the location of the rawscore in terms of distance from the mean,measured in standard deviations

� Gives us information about the location ofthat score relative to the “average”deviation of all scores

ZZ--score Distributionscore Distribution

� Mean of zero◦ Zero distance from the mean

� Standard deviation of 1

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

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

Transformation EquationTransformation Equation

� Transformations consist of making thescale larger, so that negative scores areeliminated, and of suing a larger standarddeviation, so that decimals are done awaywith.with.

� Transformation scores equation:

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

Transformation EquationTransformation Equation

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

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

this becomes:

� Or starting with the raw score, we have:

standard score= z(10) + 50

Standard score 5010 +−

= )(S

XX

Fun facts about z scoresFun facts about z scores

• Any distribution of raw scores can be converted to adistribution of z scores

The mean of a distribution has a zscore of ____?

zero

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

above

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

below

score of ____?

� Comparing scores from differentdistributions

� Interpreting/desribing individual scores

Mr. Johny S. Natad

� Interpreting/desribing individual scores

� Describing and interpreting sample means

Student Geography Arithmetic

A 60 40

B 72 36

C 46 24

etc.

Mean 60 22

PART A: RAW SCORES

Spelling

140

100

110

100

Comparing Different Variables Comparing Different Variables

� Standardizes different scores

Mean

Student Geography Spelling Arithmetic Average

A 50 70 80 67

B 62 50 73 62

C 36 55 53 48

PART B: STANDARD SCORES

Mean 60 22

Standard deviation 10 6

100

20

Standard score 5010 +−

= )(S

XX

=+−

= 501010

6060)(SS 50500 =+)(

Using transformation equation:

MeanStandard deviation

Student Geography Arithmetic

A 60 40

B 72 36

C 46 24

etc.

Mean 60 22

Standard deviation 10 6

PART A: RAW SCORES

PART B: STANDARD SCORES

Spelling

140

100

110

100

20

Interpreting Individual ScoresInterpreting Individual 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

Student A’s performance is average ingeography, excellent in spelling, and superiorin arithmetic.

Using standard deviation units Using 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

Using standard deviation units todescribe individual scores

Here is a distribution with a mean of 100 andstandard deviation of 10:

How many standard deviations below the mean is a score 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

Student Geography Spelling Arithmetic Average

A 50 70 80 67

B 62 50 73 62

C 36 55 53 48

PART B: STANDARD SCORES

Describing Individual ScoresDescribing Individual Scores

-1 0-3 -2 21 3

1. What is the standard deviation of 50? ___

20 30 40 50 60 70 80

67 8070Student A 50

scores

4. What is the standard deviation of 67? ___

2. What is the standard deviation of 70? ___

3. What is the standard deviation of 80? ___

0

2

31.7

-1 0-3 -2 21 3

σ

µ−=X

z

10

5067 −=

7110

17.==

Student Geography Spelling Arithmetic Average

A 50 70 80 67

B 62 50 73 62

C 36 55 53 48

PART B: STANDARD SCORES

Describing Individual ScoresDescribing Individual Scores

-1 0-3 -2 21 3

20 30 40 50 60 70 80

67 8070Student A 50

scores

0 2 31.7

-1 0-3 -2 21 3

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 is1.7 standard deviation above the mean.

Student Geography Spelling Arithmetic Average

A 50 70 80 67

B 62 50 73 62

C 36 55 53 48

PART B: STANDARD SCORES

Describing Individual ScoresDescribing Individual Scores

-1 0-3 -2 21 3

20 30 40 50 60 70 80

67 8070Student A 50

scores

0 2 31.7

-1 0-3 -2 21 3

Student B

Student C

736250

5348 5536

0 1.2 2.3

0.3 0.5-0.2-1.4

62

Using the zUsing the z--TableTable

� Important when dealing with decimal z-scores

� Gives information about the area betweenthe mean and the z and the area beyond zthe mean and the z and the area beyond zin the tail

� Use z-scores to define psychologicalattributes

Using zUsing z--scores to Describe Sample scores to Describe Sample MeansMeans

� Useful for evaluating the sample and for inferentialstatistical procedures

� Evaluate the sample mean’s relative standing

� Sampling distribution of means could be createdby plotting all possible means with that sample

� Sampling distribution of means could be createdby plotting all possible means with that samplesize and is always approximately a normaldistribution

� Sometimes the mean will be higher, sometimeslower

� The mean of the sampling distribution alwaysequals the mean of the underlying raw scores ofthe population

� Random variation conforms to aparticular probability distribution knownas the normal distribution, which isthe most commonly observed

Ms. Araya I. Mejorada

the most commonly observedprobability distribution.

de Moivre

� Mathematicians de Moivreand Laplace used thisdistribution in the 1700's

� German mathematician andphysicist Karl FriedrichGauss used it to analyzeastronomical data in 1800's,and it consequently became

The Standard Normal Curve

and it consequently becameknown as the Gaussian

distribution 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".

� 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.

� Unimodal - a probability distribution is

Bell Curve Characteristic

� Unimodal - a probability distribution issaid 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

CompletelyCompletely DescribedDescribed byby TwoTwoParametersParameters

� The normal distribution can be completelyspecified by two parameters:

1.mean

2.standard deviation2.standard deviation

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

Drawing of a Normal curveDrawing of a Normal curve

Normal Curve

Standardized

Normal Curve

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 onestandard deviation above it

The normal curve and the area under the curve between

σ units

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

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 +1sX +2s +3s

13.59% 13.59%2.15% 2.15%

PointsPoints inin thethe NormalNormal CurveCurve

Points in the normal curve above or below which different percentage of the curve lie

90%

10%

c10 c90

28.1−=z 28.1−=z

16

80

000,1

=

=

=

σ

µ

N

S

XXz

−=

Areas Cut Off Between differentPoints

16

80110 −=

16=

16

30=

8751.=80=X

16 cases

110=X

8751.=z

σ

µ−=X

z

Equation of z for a differentunknown

80281

−=X

.

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

80−X16

80281

−=X

.

).( 2811680 =−X

482080 .+=X

5100.=X

16

80281

−=X

.

482080 .−=−X

482080 .−=X

559.=X

Application of Normal Curve ModelApplication of Normal Curve Model

� Can determine the proportion of scoresbetween the mean and a particular score

� Can determine the number of peoplewithin a particular range of scores bywithin a particular range of scores bymultiplying the proportion by N

� Can determine percentile rank

� Can determine raw score given thepercentile

Acknowledgement of References: Acknowledgement of References:

� N.M Downie and R.W Heath. Basic

Statistical Methods, 5th Edition. Harper &Row Publisher, 1983

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

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

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