Download standard scores and the normal curve normal curve area under the

9/11/2013
NORMAL CURVE
STANDARD SCORES AND THE
NORMAL CURVE
Prepared by:
Jess Roel Q. Pesole
CHARACTERISTICS OF THE NORMAL
CURVE
• Theoretical distribution of
population scores
represented by a bell-shaped
curve obtained by a
mathematical equation
• Used for:
(1) Describing distributions of
scores
(2) Interpreting the standard
deviation
(3) Making statements of
probability
SIGNIFICANCE OF THE NORMAL CURVE
1. Symmetrical – can be
divided into the equal
halves
2. Unimodal – has only one
peak of maximum
frequency; it is also where
the mean, median, and
mode is found
3. Asymptotic – theoretically,
the tails never touch the
base line but extend to
infinity in either direction
• Some variables are assumed to be normally
distributed; as such, the sampling
distributions of various statistics are known or
assumed to be normal
• This assumption does not differ radically from
the real world (ex. Height, IQ), but some
variables may not conform to this assumption
(ex. Distribution of wealth)
AREA UNDER THE NORMAL CURVE
AREA UNDER THE NORMAL CURVE
• PRINCIPLE # 1: The area under the normal
curve is the area that lies between the curve
and the base line containing 100% or all of the
cases in any given normal distribution
• PRINCIPLE # 2: A constant proportion of the
total area under the normal curve will lie
between the mean and any given distance
from the mean, as measured in sigma units (σ)
1
9/11/2013
AREA UNDER THE NORMAL CURVE
AREA UNDER THE NORMAL CURVE
• PRINCIPLE # 3: Any given sigma distance
above the mean contains the identical
proportion of distance as the same sigma
distance below the mean
Important Proportions:
• μ to 1σ = 34.13% of cases or p = 0.3413
• μ to 2σ = 47.72% of cases or p = 0.4772
• μ to 3σ = 49.87% of cases or p = 0.4987
AREA UNDER THE NORMAL CURVE
Important Proportions:
• -1σ to +1σ = 68.26% of cases or p = 0.6826
• -2σ to +2σ = 95.44% of cases or p = 0.9544
• -3σ to +3σ = 99.74% of cases or p = 0.9974
STANDARD SCORES
(Z SCORES)
STANDARD SCORES
(Z SCORES)
• Transformed score that designates how many
standard deviation units the corresponding raw
score is above or below the mean
• Allows us to compare scores that are otherwise
not directly comparable
= (for population data)
= (for sample data)
STANDARD SCORES
(Z SCORES)
• Characteristics:
1. The z-scores have the same shape as the set
of raw scores.
2. The mean of the z-scores always equals zero.
3. Standard deviations of z-scores always equals
1.
2
9/11/2013
STANDARD SCORES
(Z SCORES)
• Example # 1 (Finding the z-score):
You got a score of 80 in your midterm
examination for history, with a mean of 60 and a
standard deviation of 11. For economics, you
got a score of 70, with a mean of 55 and a
standard deviation of 13. Compare the relative
position of the two scores.
STANDARD SCORES
(Z SCORES)
INTERPRETATION:
Since the z-score
of tiyr exam score in
History is higher, this
means that its relative
position is higher than
that of the z-score for
the Economics exam.
STANDARD SCORES
(Z SCORES)
= =
− = − =
=
=
= 1.82
= 1.15
STANDARD SCORES
(Z SCORES)
• Example # 2 (Finding the z-score):
A person scores 81 on a test of verbal
ability and 6.4 on a test of quantitative ability.
For verbal ability test, the mean for the people
in general is 50 and the Sd is 20. For the
quantitative ability test, the mean for people in
general is 0 and the Sd is 5. Which is this
person’s stronger ability, verbal or quantitative?
Explain and interpret your answer.
3
9/11/2013
STANDARD SCORES
(Z SCORES)
• Example # 3 (Finding the area given the raw
score):
A child scored 109 on the Wechsler IQ test.
This test is scaled in the population to have a
mean of 100 and a standard deviation of 15.
What is its percentile rank?
STANDARD SCORES
(Z SCORES)
Example # 3 (Finding the
area given the raw score):
STEP 1: Convert into zscores.
Z =
=
= 0.60
STEP 2: Sketch your curve.
STANDARD SCORES
(Z SCORES)
Example # 3 (Finding the area
given the raw score):
STEP 3: Determine the percent of
area at or below the z-score.
NOTE: Use the table of the areas
under the curve for this step.
Based on the table, 22.57% of the
distribution lie between the
mean and z. Add 50% to cover
the other end of the curve. This
gives you 72.57%.
STANDARD SCORES
(Z SCORES)
Example # 3 (Finding the
area given the raw score):
ANSWER:
The percentile rank of
the child is 72.57 %. This
means that 72.57 percent of
those in the population who
took the exam got a score
lower than the child.
STANDARD SCORES
(Z SCORES)
Example # 4 (Finding the area given the raw
score):
The Scholastic Achievement Test (SAT) is
standardized to be normally distributed with a
mean of 500 and a standard deviation of 100.
What percentage of the SAT scores falls
above 600?
STANDARD SCORES
(Z SCORES)
Example # 4 (Finding the
area given the raw score):
STEP 1: Convert into zscores.
Z =
=
=1
STEP 2: Sketch your curve.
4
9/11/2013
STANDARD SCORES
(Z SCORES)
Example # 4 (Finding the area
given the raw score):
STEP 3: Determine the percent of
area at or below the z-score.
Based on the table, 15.87% of the
distribution lie beyond the z.
ANSWER:
15.87% of the scores are found
above 600.
STANDARD SCORES
(Z SCORES)
Example # 5 (Finding the
area given the raw score):
STEP 1: Convert into zscores.
Z =
=
=
Z =
=
=1
0.5
STEP 2: Sketch your curve.
EXERCISE
• IQ scores are normally distributed with a
mean μ = 100 and a standard deviation σ = 15.
Based on this distribution, determine:
a. The percentage of scores between 88 and
120.
b. The percentage of scores that are 110 or
above
c. The percentile rank corresponding to an IQ
score of 125
STANDARD SCORES
(Z SCORES)
Example # 5 (Finding the area given the raw
score):
The Scholastic Achievement Test (SAT) is
standardized to be normally distributed with a
mean of 500 and a standard deviation of 100.
What percentage of the SAT scores falls
between 450 and 600?
STANDARD SCORES
(Z SCORES)
Example # 5 (Finding the area given the
raw score):
STEP 3: Determine the percent of area at
or below the z-score.
Based on the table, 34.13% of the
distribution lie between the mean and the
z-score for 600. On the other hand,
19.15% of the distribution lie between the
mean and the z-score for 450.
ANSWER:
53.28% of the scores are found between
450 and 600.
• Pediatric data reveal that the average child is
toilet trained at 26 months, but that there is a 2month standard deviation from this norm.
A. What percentage of children are toilet trained
by 23 months?
B. A mother is concerned that her son was trained
at 30 months. What is the percentile rank of her
child? How common is it for toilet training to
occur at or beyond 30 months?
C. A mother is pleased that her son is trained at 18
months. What is the percentile rank of this
prodigy? How likely is it that a child would be
toilet trained by this age?
5