CHAPTER 3: THE NORMAL DISTRIBUTION

Chapter 3: The Normal Distribution
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Chaptter 3::
THE NORMAL DIISTRIIBUTIION
Upon completion of this chapter, you should be able to:
explain what is the normal distribution
assess normality using graphical techniques - histogram
assess normality using graphical techniques - box plots
assess normality using graphical techniques - normality plots
assess normality using statistical techniques
CHAPTER OVERVIEW
What is normal distribution
Why is the normal distribution
important
What are the parameters of the
normal distribution
Assessing normality
Assessing normality using
graphical methods
o Histogram
o Boxplots
o Normal probability plots
Assessing normality using
statistical method
o Kolmogorov-Smirnoff
o Sharpiro-Wilks
Chapter 1: Introduction
Chapter 2: Descriptive Statistics
Chapter 3: The Normal Distribution
Chapter 4: Hypothesis Testing
Chapter 5: T-test
Chapter 6: Oneway Analysis of Variance
Chapter 7: Correlation
Chapter 8: Chi-Square
This chapter discusses the issue of normal distribution. Oftentimes researchers tend to
ignore the importance of the normal distribution. The normal distribution is important
several of the statistical you will be using assume that your sample is normally
distributed. Whether a distribution is normal can be determined by both graphical and
statistical method. The graphical methods are the histogram, the boxplot and the normal
probability plot. The statistical methods are Kolmogorov-Smirnoff and the Shapiro-Wilks,

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WHAT IS THE NORMAL DISTRIBUTION
Now that you know what is the mean and standard deviation of a set of scores,
we can examine the concept of a normal distribution. The normal curve was
developed mathematically in 1733 by DeMoivre as an appropximation to the binomial
distribution. Laplace used the normal curve in 1783 to describe the distribution of
errors. However, it was Gauss who popularised the normal curve when he used it to
analyse astronomical data in 1809 and it became known as the Gaussian distribution.
The term normal distribution refers to a particular way in which scores or
observations will tend to pile up or distribute around a particular value rather than be
scattered all over. The normal distribution which is bell-shaped is based on a
mathematical equation (which we will not get into!).
While some argue that in the real world, scores or observations are seldom
normally distributed, others argue that in the general population, many variables such
as height, weight, IQ scores, reading ability, job satisfaction, blood pressure turn out
to have distributions that are bell-shaped or normal.
WHY IS THE NORMAL DISTRIBUTION IMPORTANT
The normal distribution is important for the following reasons:
Many physical, biological and social phenomena or variables are normally
distributed. However, some variables are only approximately normally
distributed.
Many kinds of statistical tests (such as the t-test, ANOVA) are derived from a
normal distribution. In other words, most of these statistical tests argue that
they work best when the sample tested is distributed normally.
FORTUNATELY, these statistical tests work very well even if the distribution is
only approximately normally distributed. Some tests work well even with very wide
deviations from normality. They are described as 'robust' tests that are able to tolerate
the lack of a normal distribution.
WHAT ARE THE PARAMETERS OF THE NORMAL CURVE
A normal distribution (or normal curve) is completely determined by the mean and
standard deviation. i.e. two normally distributed variables having the same mean and
standard deviation must have the same distribution. We often identify a normal curve
by stating the corresponding mean and standard deviation and calling those the
parameters of the normal curve.
A normal distribution is symmetric and centred at the mean of the variable,
and its spread depends on the standard deviation of the variable. The larger the
standard deviation, the flatter and more spread out is the distribution.

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A) ILLUSTRATION OF THE NORMAL DISTRIBUTION OR THE
NORMAL CURVE
55 70 85 100 115 130 145
Figure 3.1 Graph showing a normal distribution of IQ scores among
adolescents
See Figure 3.1 which is a graph showing a normal distribution of IQ scores among a
sample of adolescents.
Mean is 100
Standard Deviation is 15.
As you can see, the distribution is symmetric. If you folded the graph in the centre,
the two sides would match, i.e. they are identical.
B) MEAN. MODE AND MEDIAN and THE NORMAL CURVE
The centre of the distribution is the mean. The mean of a normal distribution is
also the most frequently occurring value (i.e. the mode), and it is also the value that
divides the distribution of scores into two equal parts (i.e. the median). In any normal
distribution, the mean, median and the mode all have the same value (i.e. 100 in the
example above).

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C) THE THREE-STANDARD-DEVIATIONS RULE
The normal distribution shows the area under the curve. The Three-standarddeviations
rule, when applied to a variable states that almost all the possible
observations or scores of the variable lie within three standard deviations to either
side of the mean. The normal curve is close to (but does not touch) the horizontal axis
outside the range of the three standard deviations to either side of the mean. Based on
the graph above, you will notice that with a mean of 100 and a standard deviation of
15;
68% of all IQ scores fall between 85 (i.e. one standard deviation less then the
mean which is 100 - 15 = 85) and 115. (i.e. one standard deviation more than
the mean which is 100 + 15 = 115).
95% of all IQ scores fall between 70 (i.e. two standard deviations less then the
mean which is 100 - 30 = 70) and 130. (i.e. two standard deviations more than
the mean which is 100 + 30 = 130.
99% of all IQ scores fall between 55 (i.e. three standard deviations less then
the mean which is 100 - 45 = 55) and 145. (i.e. three standard deviations more
than the mean which is 100 + 45 = 145.
A normal distribution can have any mean and standard deviation. But the percentage
of cases or individuals falling within one, two or three standard deviations from the
mean is always the same. The shape of a normal distribution does not change. Means
and standard deviations will differ from variable to variable. But the percentage of
cases or individuals falling within specific intervals is always the same in a true
normal distribution.
LEARNING ACTIVITY
1. Precisely what is meant by the statement that a
population is normally distributed
2. Two normally distributed variables have the same
means and the same standard deviations. What can
you say about their distributions Explain your
answer.
3. Which normal distribution has a wider spread: the
one with mean 1 and standard deviation 2 or the
one with mean 2 and standard deviation 1 Explain
you answer.
4. The mean of a normal distribution has no effect on
its shape. Explain.
5. What are the parameters for a normal curve

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D) INFERENTIAL STATISTICS AND NORMALITY
Often in statistics one would like to assume that the sample under
investigation has a normal distribution or an approximate normal distribution.
However, such an assumption should be supported in some way some technique. As
mentioned earlier, the use of several inferential statistics such as the t-test and
ANOVA require that the distribution of the variables analysed are normally
distributed or at least approximately normally distributed. However, as discussed in
Chapter 1, if a simple random sample is taken from a population, the distribution of
the observed values of a variable in the sample will approximate the distribution of
the population. Generally, the larger the sample, the better the approximation tends to
be. In other words, if the population is normally distributed, the sample of observed
values would also be normally distributed if the sample is randomly selected and it is
large enough.
ASSESSING NORMALITY
Assessing normality means determining whether the sample of students,
teachers, parents or principals you are studying are normally distributed. When you
draw a sample from a population that is normally distributed, it does not mean that
your sample will necessarily have a distribution that is exactly normal. Samples vary,
so the distribution of each sample may also vary. However, if a sample is reasonably
large and it comes froma normal population, its distribution should look more or less
normal.
For example, when you administer a questionnaire to a group of school
principals, you want to be sure that your sample of 250 principals is normally
distributed. WHY The assumption of normality is a prerequisite for many inferential
statistical techniques and there are two main ways of determining the normality of
distribution.
a) Using graphical methods (such as histograms, stem-and-lead plots and
boxplots)
b) Using statistical procedures.(such as the Kolmogorov-Smirnov
statistic and the Shapiro-Wilks statistics)

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SPSS Procedures for Assessing Normality:
There are several procedures to obtain the different graphs and statistics to assess
normality, But the EXPLORE procedure is the most convenient when both graphs
and statistics are required.
Select the Analyse menu
Click on Descriptive Statistics and then Explore ....to open the Explore dialogue
box
Select the variable you require and click on the button to move this variable into
the Dependent List: box
Click on the Plots...command pushbutton to obtain the Explore: Plots sub dialogue
box
Click on the Histogram check box and the Normality plots with tests check box,
and ensure that the Factor levels together radio button is selected in the Boxplots
display
Click on Continue
In the Display box, ensure that Both is activated
Click on the Options...command pushbutton to open the Explore: Options subdialogue
box
In the Missing Values box, click on the Exclude cases pairwise (if not selected by
default)
Click on Continue and then OK.
ASSESSING NORMALITY USING GRAPHICAL METHOD
A) The HISTOGRAM
See Figure the Graph which is a histogram showing the distribution of
scores obtained on a Scientific Literacy Test administered to a sample of
students.
The values on the vertical axis indicate the frequency or number of cases.
The values on the horizontal axis are midpoints of value ranges. For
example, the first bar is 20 and the second bar is 30, indicating that each
bar covers a range of 10.
Superimposed on the histogram is the normal curve. Simply looking at the
bars indicates that the distribution has the rough shape of a normal
distribution. The superimposed curve, however shows that there are some
deviations. The question is whether this deviation is small enough to say
that the distribution is approximately normal.

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50
Frequency
60
50
40
30
20 30 40 50 60 70 80 90 100
Figure 3.2 Graph showing the distribution of scores on Scientific Literacy among
a group of students
a) Some Key Characteristics of Distribution Using the Histogram
(i) Skewness
Skewness is the degree of departure from symmetry of a distribution. A normal
distribution is symmetrical. A non-symmetrical distribution is described as being
either negatively or positively skewed. A distribution is skewed if one of its tail is
longer than the other or the tail pulled to either the left or right.
Skewness 1.5
Figure 3.3 Graph
showing a positive skew

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Refer to Figure 3.3 which shows the distribution of the scores obtained by students on
a test. There is a positive skew because it has a longer tail in the positive direction or
the long tail is on the right side (towards the high values on the horizontal axis).
What does it mean It means that more students were getting low scores in the
test which indicates that the test was too difficult. Alternatively, it could mean that the
questions were not clear or the teaching methods and materials did not bring about the
desired learning outcomes.
Skewness - 1.5
Figure 3.4 Graph
showing a negative skew
Refer to Figure 3.4 which shows the distribution of the scores obtained by students on
a test. There is a negative skew because it has a longer tail in the negative direction
or to the left (towards the lower values on the horizontal axis).
What does it mean It means that more students were getting high scores on
the test which may indicate that either the test was too easy or the teaching methods
and materials were successful in bringing about the desired learning outcomes.
(ii) Interpreting the Statistics for Skewness
Besides graphical methods, you can also determine skewness by examining
the statistics reported. A normal distribution has a skewness of 0. See the Table 3.1
which reports the skewness statistics for three independent groups. A positive value
indicated a positive skew while a negative value reflects a negative skew.
Among the three groups, Group 3 is not as normally distributed compared to
the other two groups. Its skewness value of -1.200 which is greater than 1
which indicates that the distribution is non-symmetrical [Rule of thumb - > 1
indicates a non-symmetrical distribution].
The distribution of Group 2 with a skewness value of .235 is closer to being
normal (i.e. 0) followed by Group 1 with a skewness value of .973.

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SPSS Output:
GROUP 1 Skewness .973
GROUP 2 Skewness +.235
GROUP 3 Skewness - 1.200
Table 3.1 Skewness reported for three groups of students
(iii) Kurtosis:
Kurtosis indicates the degree of "flatness" or "peakedness" in a distribution relative to
the shape of normal distribution.
High kurtosis
Low kurtosis
Figure 3.5 Graphs showing high and low kurtosis
Refer to Figure 3.5 which shows:
Low Kurtosis: Data with low kurtosis tend to have a flat top near the mean rather
than a sharp peak.
High Kurtosis: Data with high kurtosis tend to have a distinct peak near the mean
and a sharp decline rather rapidly with a heavy tail.

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Figure 3.6 Names assigned to different types of kurtosis
See Figure 3.6 which show the names assigned to different levels of kurtosis:
A normal distribution has a kurtosis of 0 and is called mesokurtic (Graph A).
[Strictly speaking a mesokurtic distribution has a value of 3 but in line with
the practice used in SPSS packages, the adjusted version is 0].
If a distribution is peaked (tall and skinny), its kurtosis value is greater than 0
and it is said to be leptokurtic (Graph B) and has a positive kurtosis.
If, on the other hand, the kurtosis is flat, its value is less than 0, or platykurtic
(Graph C) and has a negative kurtosis.
(iv) Interpreting the Statistics for Kurtosis
Besides graphical methods, you can also determine skewness by examining the
statistics reported. A normal distribution has a kurtosis of 0. See the Table 3.2 which
reports the kurtosis statistics for three independent groups.

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SPSS Output:
GROUP 1 Kurtosis .500
GROUP 2 Kurtosis -1.58
GROUP 3 Kurtosis 1.65
Table 3.2 Kurtosis reported for three groups of students
Group 1 which a kurtosis value of 0.500 (positive value) is more normally
distributed than the other two groups because it is closer to 0.
Group 2 with a kurtosis value of -1.58 has a distribution that is more flattened
and not as normally distributed compare to Group 1.
Group 3 with a kurtosis value + 1.65 has a distribution that is more peaked and
not as normally distributed compared to Group 1.
B) The BOXPLOT
The boxplot also provides information about the distribution of scores. Unlike
the histogram which plots actual values, the boxplot summarises the distribution using
the median, the 25th and 75th percentiles, and extreme scores in the distribution. See
Figure 3.7 which shows a boxplot for the same set of data on scientific literacy
discussed earlier. Note that the lower boundary of the box is the 25th percentile and
the upper boundary is the 75th percentile.
The BOX
The box has hinges that form the outer boundaries of the box. The hinges are
the scores that cut of the top and bottom 25% of the data. Thus, 50% of the scores fall
within the hinges. The thick horizontal line through the box represents the median in
the case of a normal distribution the line runs through the centre of the box.
If the median is closer to the top of the box, then the distribution is negatively
skewed. If it is closer to the bottom of the box, then it is positively skewed.
Whiskers
The smallest and largest observed values within the distribution are represented by the
horizontal lines at either end of the box, commonly referred as whiskers.
The two whiskers indicate the spread of the scores.

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Hinges
The largest observed
value within the
distribution is
represented by the
horizontal line at the end
of the box, referred to as
whisker.
75 percentile
‘wisker’
MEDIAN
25 percentile
‘wisker’
The median is presented
by a horizontal line
through the centre of the
box
Hinges
The smallest observed value
within the distribution is
represented by the horizontal
line at the end of the box,
referred to as wisker.
Figure 3.7 Boxplot showing the distribution of scores on Scientific Literacy
among a group of students
Scores that fall outside the upper and lower whiskers are classed as extreme
scores or outliers. If the distribution has any extreme scores, i.e. 3 or more box lengths
from the upper or lower hinge; these will be represented by a circle (o).
Outliers tell us that we should see why it is so extreme. Could it be that you
may have made an error in data entry.
Why is it important to identify outliers This is because many of the
statistical techniques used involve calculation of means. The mean is sensitive to
extreme scores and it is important to be aware whether you data contain such extreme
scores if you are to draw conclusions from the statistical analysis conducted.

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C) The NORMAL PROBABILITY PLOT
Besides the histogram and the box plot, another frequently used graphical
technique of determining normality is the "Normal Probability Plot" or "Normal
Q-Q Plot". The idea behind a normal probability plot is simple. It compares the
observed values of the variable to the observations expected for a normally distributed
variable. More precisely, a normal probability plot is a plot of the observed values of
the variable versus the normal scores (the observations expected for a variable having
the standard normal distribution).
In a normal probability plot, each observed or value (score) obtained is paired
with its theoretical normal distribution forming a linear pattern. If the sample is from
a normal distribution, then the observed values or scores fall more or less in a straight
line. The normal probability plot is formed by:
Vertical axis: Expected normal values
Horizontal axis: Observed values
SPSS Procedures
1. Select the Analyze menu.
2. Click on Descriptive Statistics and then Explore .....to pen the Explore
dialogue box
3. Select the variable you require (i.e. mathematics score) and click on button to
move this variable to the Dependent List: box
4. Click on the Plots....command pushbutton to obtain the Explore: Plots
subdialogue box
5. Click on the Histogram check box and the Normality plots with tests check
box and ensure that the Factor levels together radio button is selected in the
Boxplots display
6. Click on Continue
7. In the Display box, ensure that Both is activated
8. Click on the Options....command pushbutton to open the Explore: Options
sub-dialogue box.
9. In the Missing Values box, click on the Exclude cases pairwise radio button.
If this option is not selected then, by default, any variable with missing data will
be excluded from the analysis. That is, plots and statistics will generated only for
cases with complete data.
10.Click on Continue and then OK
Note that these commands will give you the 'Histogram', 'Stem-and-leaf plots',
'Boxplots' and Normality Plots.

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Outlier
Outlier
Figure 3.8 Normal Probability Plot showing the distribution of scores on
Scientific Literacy among a group of students
Figure 3.8 Normal Probability Plot showing the distribution of scores on
Scientific Literacy among a group of students
When you use a normal probability plot to assess the normality of a variable, you
must remember that the decision of whether the distribution is roughly linear and is
normal is a subjective one. Figure 3.8 is an example of a normal probability plot.
Though none of the value fall exactly on the line, most of the points are very close to
the line.
Value that are above the line represent units for which the observation is larger
than its normal score.
Value that are below the line represent units for which the observation is
smaller than its normal score.

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Note that there is one value that falls well outside the overall pattern of the plot. It is
called an outlier and you will have to remove the outlier from the sample data and
redraw the normal probability plot.
Even with the outlier, the values are close to the line and you can conclude
that the distribution will look like a bell-shaped curve. If the normal scores plot
departs only slightly from having all of its dots on the line, then the distribution of the
data departs only slightly from a bell-shaped curve. If one or more of the dots departs
substantially from the line, then the distribution of the data is substantially different
from a bell-shape.
Outliers:
Refer to the normal probability plot above. Note that there are possible outliers
which are values lying off the hypothetical straight line.
Outliers are anomalous values in the data which may be due to recording
errors, which may be correctable, or they may be due to the sample not being entirely
from the same population.

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Skewness to the left:
Refer to the normal probability plot above. Both ends of the normality plot fall below
the straight line passing through the main body of the values of the probability plot,
then the population distribution from which the data were sampled may be skewed to
the left.
Skewness to the right:
If both ends of the normality plot bend above the straight line passing through the
values of the probability plot, then the population distribution from which the data
were sampled may be skewed to the right.

Chapter 3: The Normal Distribution
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LEARNING ACTIVITY
Refer to the output of a Normal Probability Plot above for
the distribution of mathematics scores by eight students:
a) Comment on the distribution of scores
b) Would you consider the distribution normal
c) Are there outliers
ASSESSING NORMALITY USING STATISTICAL TECHNIQUES
The graphical methods discussed present qualitative information about the
distribution of data that may not be apparent from statistical tests. Histograms, box

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plots and normal probability plots are graphical methods are useful for determining
whether data follow a normal curve. Extreme deviations from normality are often
readily identified from graphical methods. However, in many instances the decision is
not straightforward. Using graphical methods to decide whether a data set is normally
distributed involves making a subjective decision; formal test procedures are usually
necessary to test the assumption of normality.
In general, both statistical tests and graphical plots should be used to
determine normality. However, the assumption of normality should not be rejected on
the basis of a statistical test alone. In particular, when the sample is large, available,
statistical tests for normality can be sensitive to very small (i.e., negligible) deviations
in normality. Therefore, if the sample is very large, a statistical test may reject the
assumption of normality when the data set, as shown using graphical methods, is
essentially normal and the deviation from normality too small to be of practical
significance.
a) KOLMOGOROV-SMIRNOV TEST
You could use the Kolmogorov-Smirnov statistic Z test evaluates statistically
whether the difference between the observed distribution and a theoretical normal
distribution is small enough to be just due to chance. If it could be due to chance you
would treat the distribution as being normal. If the distribution between the actual
distribution and the theoretical normal distribution is larger than is likely to be due to
chance (sampling error) then you would treat the actual distribution as not being
normal.
In terms of hypothesis testing, the Kolmogorov-Smirnov test is based on Ho: that the
data are normally distributed. The test is used for samples which have more than 50
subjects.
Ho: µ1 = µ2 OR Ha: µ1 ≠ µ2
If the Kolmogorov-Smirnov Z test yields a significance level of less () than
0.05, it means that the distribution is normal.
Kolmogrorov-Smirnov (a)
Statistic df Sig.
SCORE .21 1598 .000*
* This is lower bound of the true signifcance
(a) Lilliefors Significance Correction

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B) SHAPIRO - WILKS TEST
Another powerful and most commonly employed tests for normality is the W
test by Shapiro and Wilks, also called the Shapiro-Wilks test. It is an effective method
for testing whether a data set has been drawn from a normal distribution.
If the normal probability plot is approximately linear (the data follow a normal
curve), the test statistic will be relatively high.
If the normal probability plot has curvature that is evidence of non-normality
in the tails of a distribution, the test statistic will be relatively low.
In terms of hypothesis testing, the Shapiro-Wilks test is based on the hypothesis (Ho:)
that the data are normally distributed. The test is used for samples which have less
than 50 subjects.
Ho: µ1 = µ2 OR Ha: µ1 ≠ µ2
Reject the assumption of normality if the test of significance reports a p-value
of less () than 0.05.
SPSS Output
Tests of Normality
Shapiro-Wilks
Independent variable group Statistic df Sig.
Group 1 .912 22 .055
Group 2 . .166 14 .442
Group 3 .900 16 .084
Table 3.3 showing the Shapiro-Wilks statistic for assessing normality
See Table 3.3. The Shapiro-Wilks normality tests indicate that the scores are normally
distributed in each of the three groups. All the p-values reported are more than 0.05
and hence you DO NOT REJECT the null hypothesis.

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NOTE:
It should be noted that with large samples even a very small deviation from normality
can yield low significance levels, so a judgement still has to made as to whether the
departure from normality is large enough to matter.
WHAT TO DO IF THE DISTRIBUTION IS NOT NORMAL
You have TWO choices if the distribution is not normal and they are:
Use a Nonparametric Statistic Instead
Transform the Variable to Make to Normal
a) Use a Nonparametric Statistic
In many cases, if the distribution is not normal an alternative statistic will be
available, especially for bivariate analyses such as correlation or comparisons of
means. These alternatives which do not require normal distributions are called
nonparametric or distribution-free statistics. Some of these alternatives are shown
below:
Purpose of Parametric Non-Parametric
Analysis Statistics Statistics
---------------------------------------------------------------------------------
Differences between
- Mann-Whitney U test
two independent t-test - Kolmogorov-Smirnov
two means
sample Z test
Differences between t-test - Wilcoxon's matched
two dependent means
pairs test
Differences between more Oneway - Kurskal-Wallis analysis
than two means ANOVA of ranks
Differences between more Repeated - Friedman's two-way
than two means that is measures analysis of variance
repeated ANOVA - Cochran Q test
Relationship between Pearson r - Spearman Rho
variables
- Kendall's tau
- Chi-square

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b) Transform the Variable to Make it Normal
The shape of a distribution can be changed by expressing it in different way
statistically. This is referred to as transforming the distribution, Different types of
transformations can be applied to "normalise" the distribution. The type of
transformation selected depends on the manner to which the distribution departs from
normality. [We will not discuss transformation in this course]
Kolmogorov-Smirnov (a)
Statistic df Sig.
SCORE 0.57 999 .200*
* This is lower bound of the true significance
(a) Lilliefors Significance Correction
LEARNING ACTIVITY
Examine the SPSS output above and determine if the sample
is normally distributed.
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