These slides discusss about normal curve and standard scores

1. 1 Thursday, October 23, 2014

2. Posttest day 3
1. Central Tendency ....
Is used to talk about the central point in the distribution of values
in the data.
The measure of central tendency which reports the most
frequently obtained score in the data.
The score which is at the center of the distribution. Half the
scores are above the median and half are below it.
2. Mode is.....
3. Median is...
4. Mean is....
2
The arithmetic average of all scores in a data set.

3. Measures of Central Tendency
• Mean: arithmetic average of
all scores in a distribution
• Median: the point at which
exactly half of the scores in
a distribution are below &
half are above
• Mode: most frequently
occurring score(s)

10. Normal Curve
Objectives
1.Introduce the Normal Distribution
2.Properties of the Standard Normal
Distribution
3.The bell-shaped distribution
4.Z-scores
5.T-scores
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11. Normal Distribution
 For years, scientists have noted that many
variables in the behavioural and physical
sciences are distributed in a bell shape.
These variables are normally distributed in
the population, and their graphic
representation is referred to as the normal
curve or a bell-shaped distribution
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12. Normal Distribution
 If there is no very extreme scores and if
you have 30 or more observations, you
may have a normal distribution.
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13. The beauty of the normal curve:
No matter what Mean and SD are, the area
between Mean - SD and Mean + SD is
about 68%; the area between Mean - 2SD
and Mean+2SD is about 95%; and the area
between Mean - 3SD and Mean + 3SD is
about 99.7%. Almost all values fall within
3 standard deviations.

14. 68-95-99.7 Rule
68% of
the data
95% of the data
99.7% of the data

15. Normal /bell-shaped curve

16. Properties of Normal /bell-shaped curve
• It is a symmetrical distribution
• Most of the scores tend to occur near the center
– while more extreme scores on either side of the
center become increasingly rare.
– As the distance from the center increases, the
frequency of scores decreases.
• The mean, median, and mode are the same.

18. The normal distribution
The normal distribution is actually a group of
distribution, each determined by a mean and
a standard deviation.
 A normal distribution means that most of
the scores cluster around the midpoint of the
distribution, and the number of scores
gradually decrease on either side of the
midpoint.
 The resulting polygon is a bell-shaped
curve
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19. The normal distribution
Why are normal distributions so
important?
Many dependent variables are
commonly assumed to be normally
distributed in the population
If a variable is approximately normally
distributed we can make inferences
about values of that variable
Example: Sampling distribution of the
mean
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20. Key Areas under the Curve
 For normal
distributions
+ 1 SD ~ 68%
+ 2 SD ~ 95%
+ 3 SD ~ 99.9%

21. Normal /bell-shaped curve

23. Skewed Distribution
Negative Skew Positive skew
 Test items were easy.
 Testees performed well.
 The score are far from
zero.
 Test items were difficult.
 Testees performed poorly.
 The scores are near zero.

24. The 68/95/99.7 Rule
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25. The normal distribution
Regardless of the exact shapes of the normal
distributions, all share four characteristics:
1. The curve is symmetrical around the vertical axis
(half the scores are on the right side of the axis,
and half the scores are on its left).
2. The scores tend to cluster around the center (i.e.,
around the mean, or the vertical axis in the center).
3. The mode, median, and mean have the same
values.
4. The curve has no boundaries on either side (the
tails of the distribution are getting very close to the
horizontal axis, but never quite touch it).*
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Keep in mind that this is a theoretical model. In reality, the number of scores in a given distribution is
finite, and certain scores are the highest and the lowest points of that distribution

26. Standard Scores
Two types of scores:
1.Individual scores (raw scores): scores are
obtained by individuals on a certain measure.
2.Group scores (mode, median, mean, range,
variance, and standard deviation): are summary
scores that are obtained for a group of scores.
However, both types of scores are scale specific and cannot be
used to compared scores on two different measures, each with
its own mean and standard deviation
26

27. To illustrate this point, let’s look at
the following example.
Suppose we want to compare the scores obtained by a
student on two achievement tests, one in English and one
in mathematics, Let’s say that the student received a score
of 50 in English and 68 in mathematics. Because the two
tests are different, we cannot conclude that the student
performed better in mathematics than In English. Knowing
the student’s score on each test will not allow you to
determine on which test the student performed better. We
do not know, for example, how many items were on each
test, how difficult the test were, and how well the other
students did on the tests. Simply put, the two tests are not
comparable. 27

28. To illustrate this point, let’s look at
the following example.
To be able to compare scores from different tests,
we can first convert them into standard scores.
A standard scores is a derived scale score that
expresses the distance of the original score from
the mean in standard deviation units.
 Once the scores are measured using the same
units, they can then be compared to each other.
Two types of standard scores are discussed in this
slides : z scores and T scores. (T scores are not
related to the t-test that will be discussed later). 28

29. Z-Scores
 The z score is a type of standard score that
indicates how many standard deviation units
a given score is above or below the mean for
that group.
 The z scores create a scale with a mean of
0 and a standard deviation of 1.
 The shape of the z score distribution is the
same as that of the raw scores used to
calculate the z scores.
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30.  Standard Scores
To compare scores on different
measurement scales
Z-Scores: the commonest score
 Z-score properties
How many scores above/below the mean
The mean being set at zero
The SD being set at one

31. Z-Scores
 To convert a raw score to a z score, the raw
score as well as the group mean and
standard deviation are used. The conversion
formula is:
Where X= Raw score
= Group mean
SD= Group standard deviation 31

32. Table 1 below presents the raw scores of one student on four tests (social
studies, language arts, mathematics, and reading). The table also displays
the means and standard deviations of the student’s classmates on these
test and shows the process for converting raw scores into z scores.
Table 1 Student’s score, Class Means, Class standard Deviations, and z Scores on Four Tests
Subject Raw
Score
Mean SD Z score
Social studies 85 70 14
Language arts 57 63 12
Mathematics 65 72 16
Reading 80 50 15
32

33. T-score
 The T score is another standard score measured
on a scale with a mean of 50 and a SD of 10.
 In order to calculate T scores, z scores have to be
calculate first.
 Using this standard score overcomes problems
associated with z scores. All the scores on the T
score are positive and range from 10 to 90.
 Additionally, they can be reported in whole
numbers instead of decimal points.
 In order to convert scores from z to T, we multiply
each z score by 10 and add a constant of 50 to that
product. This is the formula:

34.  Standard Scores
T-Score: A standard score whose
distribution has a mean of 50 and a
standard deviation of 10.
Advantages of T-score
Enabling us to work with whole numbers
Avoiding describing subjects’
performances with negative numbers

35.  Standard Scores
T-Score: A standard score whose
distribution has a mean of 50 and a
standard deviation of 10.
Advantages of T-score
Enabling us to work with whole numbers
Avoiding describing subjects’
performances with negative numbers

36. Table 2 conversion of z Scores to T Scores
Subject Z score T score
Social studies +1.07 10(+1.07) + 50= 60.7 or 61
Language arts -0.50 10 (-0.5) + 50 = 45.0 or 45
Mathematics -0.44 10 (-0.44) + 50 = 45.6 or 46
Reading +2.00 10 (+2.00) + 50 = 70.0 or 70
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37. References
Main Sources
Coolidge, F. L.2000. Statistics: A gentle introduction. London: Sage.
Kranzler, G & Moursund, J .1999. Statistics for the terrified. (2nd ed.). Upper Saddle
River, NJ: Prentice Hall.
Butler Christopher.1985. Statistics in Linguistics. Oxford: Basil Blackwell.
Hatch Evelyn & Hossein Farhady.1982. Research design and Statistics for Applied
Linguistics. Massachusetts: Newbury House Publishers, Inc.
Ravid Ruth.2011. Practical Statistics for Educators, fourth Ed. New York: Rowman &
Littlefield Publisher, Inc.
Quirk Thomas. 2012. Excel 2010 for Educational and Psychological Statistics: A Guide
to Solving Practical Problem. New York: Springer.
Other relevant sources
Field, A. (2005). Discovering statistics using SPSS (2nd ed.). London: Sage.
Moore, D. S. (2000). The basic practice of statistics (2nd ed.). New York: W. H.
Freeman and Company.
37 Thursday, October 23, 2014