This document discusses evaluating the normality of data distributions. It covers probability, normal distributions, z-scores, empirical rules, and tests for skewness and kurtosis. Normal distributions are symmetric and bell-shaped. The normality of data can be determined using z-scores and empirical rules. Skewness measures asymmetry in a distribution, while kurtosis measures tail weight. Normality tests like Shapiro-Wilk can determine if a dataset comes from a normal distribution.

1. EVALUTION OF
NORMALITY IN THE
DISTRIBUION OF DATA
Prepared by:
WAQAR AKRAM

2. CONTENTS
Probability
Probability distributions and its types
Normal distribution
Z- table and Z- score calculation
Empirical rule
Skewness
kurtosis

3. Probability
• The set of all possible distinct outcomes to an experiment.
No. of ways event can occur/ No. of outcomes
Probability range
0 To 1
0 percent to 100 percent
Yes or No

4. Example of a dice
Value of probability
is always equal to or
less than 1

5. Probability Distributions
• DISCRETE DISTRIBUTIONS
• Binomial
• Poisson
• Uniform
• CONTINUOUS
DITRIBUTIONS
• Normal
• Distribution

6. NORMAL PROBABILITY DISTRIBUTION OF
DATA
• Normal probability distribution, also called Gaussian
distribution refers to a family of distributions that are bell
shaped.
• These are symmetric in nature and peak at the mean, with
the probability distribution decreasing away before and
after.
• Define by two parameters ‘ its MEAN value and S.D.

7. NORMAL PROBABILITY
DISTRIBUTION OF DATA

8. CHARACTERISTICS OF NORMALLY
DISTRIBUTED DATA
• Symmetry
Mean = Median = Mode
• Normally distributed value can be large in both positive and negative
direction
• Probability of any specific value will be zero
• Unimodal
• Defined by its Mean an S.D completely

9. METHODS TO DETERMINE PROBABILITY
FROM NORMAL DISTRIBUTION OF DATA
• Z- score based
• (Z-Distribution)

12. How to calculate Z – Score???
• U= Mean
• Sigma = S.D
• X = Data at a point

13. • Exercise:
• Question: For a process with a mean of 100, a standard deviation of 10 and
an upper specification of 120, what is the probability that a randomly
selected item is defective (or beyond the upper specification limit)?

14. • Answer:
• The Z-score is equal to = (120 –100) / 10 = 2.
• This means that the upper specification limit is 2standard
deviations above the mean.
• Now that we have the Z-score, we can use the Z-table to
find the probability.
• From the Z-table (the complementary cumulative table),
the area under the curve for a Z-value of 2 = 0.9772
• Now (1 – 0.9772 = 0.02275 or 2.275%).
• •This means that there is a chance of 2.275%for any
randomly selected item to be defective.

15. METHODS TO DETERMINE PROBABILITY
FROM NORMAL DISTRIBUTION OF DATA
• Empirical Rule

16. EXERCISE

17. SKEWED DATA
• Skewness in statistics represents an imbalance and an
asymmetry from the mean of a data distribution. In a
normal data distribution with a symmetrical bell
curve, the mean and median are the same. In a
skewed data distribution, the median and the mean
are different values. Normal value = 0

18. TYPES OF SKEWED DATA

19. TEST OF SKEWNESS
• In order to ascertain whether a distribution is skewed or not the following tests may be
applied. Skewness is present if:
• The values of mean, median and mode do not coincide.
• When the data are plotted on a graph they do not give the normal bell shaped form i.e.
when cut along a vertical line through the centre the two halves are not equal.
• The sum of the positive deviations from the median is not equal to the sum of the
negative deviations.
• Quartiles are not equidistant from the median.
• Frequencies are not equally distributed at points of equal deviation from the mode.

20. MEASURE OF SKEWNESS
The measures of skewness are:
• Karl Pearson's Coefficient of skewness
• Bowley’s Coefficient of skewness
• Kelly’s Coefficient of skewness

21. Karl Pearson's Coefficient of skewness
• The formula for measuring skewness as given by Karl Pearson is as follows
• Where, SKP = Karl Pearson's Coefficient of skewness, σ = standard deviation

22. • In case the mode is indeterminate, the coefficient of skewness is
• The value of coefficient of skewness is zero, when the distribution is
• symmetrical.
• Normally, this coefficient of skewness lies between +1.96 to -1.96.
• If the mean is greater than the mode, then the coefficient of skewness will be
positive, otherwise negative.

23. Excercise
Mean = 70.5.
Median = 80.
Mode = 85.
Standard deviation = 19.33
Measure skweness ?
Answer
Pearson’s Coefficient of Skewness
Step 1: Subtract the mode from the mean: 70.5 – 85 = -
14.5.
Step 2: Divide by the standard deviation: -14.5 / 19.33 = -
0.75.
so data is positively skewed

24. Kurtosis
• Kurtosis is a measure of whether the data are heavy-tailed or light-tailed
relative to a normal distribution. Normal value= 3
• Kurtosis represents the attribute of flatness
• Or peakedness of a distribution

25. TYPES OF KURTOSIS

26. MEASURE OF KURTOSIS

27. When do we do normality test?
A lot of statistical tests (e.g. t-test) require that
our data are normally distributed and therefore
we should always check if this assumption is
violated.

28. Example Scenario on SPSS
Given a set of data, we would like to check if its distribution is normal.
In this example, the null hypothesis is that the data is normally distributed and the
alternative hypothesis is that the data is not normally distributed.

29. Step 1
Select "Analyze -> Descriptive Statistics -> Explore".

30. • A new window pops out.

31. Step 2
From the list on the left,
select the variable "Data"
to the "Dependent List".
• Click "Plots" on the right. A
new window pops out.
Check "None" for boxplot,
uncheck everything for
descriptive and make sure
the box "Normality plots
with tests" is checked.

32. Step 3
The results now pop out in the "Output" window.

33. The test statistics are shown in the third table.
Here two tests for normality are run. For data-
set small than 2000 elements, we use the
Shapiro-Wilk test, otherwise, the Kolmogorov-
Smirnov test is used. In our case, since we have
only 20 elements, the Shapiro-Wilk test is used.
From A, the p-value is 0.316. We can reject the
alternative hypothesis and conclude that the
data comes from a normal distribution.

35. REFERENCES
Biostatistics by Stanley
Z-table.com
statistics.laerd.com
maths-statistics-tutor.com