Important Characteristics of a Normal Curve

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Standard Scores
and Normal Curves
Discussant:
JENEWEL M. AZUELO
MA.Ed Social Studies

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THE NORMAL DISTRIBUTION
In normal distribution,
majority of the population will
yield average scores; a small
part of the population will yield
above average scores; and very
few will yield scores that are
extremely high and low.

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Important characteristics of a
normal distribution
Its equation can help us
determine the area between
any two points under the
normal curve if the mean () and
the standard deviation are
known.

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Normal Curve
• It is a graphical presentation of normal
distribution
•It is a bell-shaped curve that extends
indefinitely on both sides.
•The curve is symmetrical, with the highest
point at the center.
•It is asymptotic to the base line or the x-axis.
•The total area under the normal is 1.0 or
100%.
•The mean, median and mode coincide at
one point at the center of the curve.
•The mean and the standard deviation are
the parameters used to describe the normal
curve.

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Normal curves have different heights and widths, but in all
cases , the following characteristics apply:
1.Approximately 34% will lie between the mean and 1
standard deviation above the mean and 34% will lie
between the mean and 1 standard deviation below the
mean.
2.Approximately 47.5% will lie between the mean and 2
standard deviation above the mean and 47.5% will lie
between the mean and 2 standard deviation below the
mean.
3.Approximately 49.5% will lie between the mean and 3
standard deviation above the mean and 49.5% will lie
between the mean and 3 standard deviation below the
mean.

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Standard Score (z-score)
The standard score is the
distance of the score from the
mean in terms of the standard
deviation. It tells how many
standard deviations the
observed value lies above or
below the mean of the
distribution.

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The standard score is useful
in comparing observed values
with different distributions. To
be able to find areas under
the normal curve, observed
values must first be converted
into standard scores, to solve
statistical problems.

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To change an observed value (x) into
standard score use the equation:
z=x-x̄ where :
s x= raw score/observed value
x = mean̄
s= standard deviation
Note: A positive (+) z-score means that the
score/observed
value is above the mean.
A negative(-) z-score means that the
score/observed
value is below the mean.

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Example: In a given distribution, the
mean (x) is 65 and the standard̄
deviation (s) is 6. Find the standard
score of
a.68
b.59
c.77
d.47

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Solution:
a. The standard score of 68 is
z=x-x This means that 68 is 0.5 standard̄
s deviation above the mean.
z= 68-65
68 6
= 0.5

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Solution:
a. The standard score of 59 is
z=x- x̄
This means that 59 is -1.0 standard
s deviation below the mean.
z= 59-65
59 6
= -1.0

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Solution:
a. The standard score of 77 is
z=x- x̄ This means that 77 is 2.0 standard
s deviation above the mean.
z= 77-65
77 6
= 2.0

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Solution:
a. The standard score of 47 is
z=x- x̄ This means that 47 is -3.0 standard
s deviation below the mean.
z= 47-65
47 6
= 3.0

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Example 2. In the final examination in Mathematics, the
mean grade was 82 and the standard deviation was 8. In
English, the mean grade was 86 and the standard deviation
was 10. Joseph scored 88 in Mathematics and 92 in
English. In which subject was his standing higher?
Solution:
The first thing that has to be done is to change the
scores into standard scores.

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His standing in mathematics, therefore, was
higher than his standing in English. He was only 0.6
standard deviation above the mean in English while he
was 0.75 standard deviation above the mean in
Mathematics.

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