Upload
johny-natad
View
28.826
Download
2
Tags:
Embed Size (px)
DESCRIPTION
Z score or Standard Score and the Normal Curve in statistics
Citation preview
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)