The document discusses the normal distribution, often referred to as the 'bell curve' or 'gaussian curve,' highlighting its characteristics such as symmetry, the equality of mean, median, and mode, and the area under the curve representing probabilities. It also explains kurtosis and skewness as measures of distribution shape, classifying kurtosis into mesokurtic, leptokurtic, and platykurtic types, and providing criteria for assessing skewness. Various examples of real-world phenomena that follow normal distribution patterns, along with statistical rules regarding skewness, are mentioned.

1. The Normal Distribution

2. The Normal Distribution
• It is sometimes called the "bell curve,".
•It is also called the "Gaussian curve" after the
mathematician Karl Friedrich Gauss.
•Strictly speaking, it is not correct to talk about "the
normal distribution" since there are many normal
distributions.
•Normal distributions can differ in their means and
in their standard deviations.

3. The Normal Distribution
Figure 1 shows three normal distributions.
1. The green (left-most) distribution has a mean of
-3 and a standard deviation of 0.5,
2. the distribution in red (the middle distribution)
has a mean of 0 and a standard deviation of 1,
3. and the distribution in black (right-most) has a
mean of 2 and a standard deviation of 3.
4. These as well as all other normal distributions
are symmetric with relatively more values at the
center of the distribution and relatively few in
the tails.

4. The Normal Distribution
Figure 1. Normal distributions differing in mean and standard deviation.

5. The Normal Distribution
Features of Normal Distribution
1. Normal distributions are symmetric around their mean.
2. The mean, median, and mode of a normal distribution
are equal.
3. The area under the normal curve is equal to 1.0.
4. Normal distributions are denser in the center and less
dense in the tails.
5. Normal distributions are defined by two parameters,
the mean (μ) and the standard deviation (σ).
6. 68% of the area of a normal distribution is within one
standard deviation of the mean.
7. Approximately 95% of the area of a normal distribution
is within two standard deviations of the mean.

6. The Normal Distribution
Data can be distributed (dispersed) in different
ways
It can be spread out
more on the left
Or more on the right
Or it can be all jumbled up

7. The Normal Distribution
But there are many cases where the data tends to be around a
central value with no bias left or right, and it gets close to a "Normal
Distribution" like this:

8. The Normal Distribution
It is often called a "Bell Curve"because it looks like
a bell.
Many things closely follow a Normal
Distribution:
•heights of people
•size of things produced by machines
•errors in measurements
•blood pressure
•marks on a test

9. The Normal Distribution
We say the data is "normally distributed":
The Normal Distribution has:
•mean = median = mode
•symmetry about the center
•50% of values less than the mean and 50% greater
than the mean

10. The Normal Distribution
What is Kurtosis?

11. The Normal Distribution
1. Distributions of data and probability distributions are
not all the same shape.
2. Some are asymmetric and skewed to the left or to
the right.
3. Other distributions are bimodal and have two peaks.
4. Another feature to consider when talking about a
distribution is the shape of the tails of the
distribution on the far left and the far right.
5. Kurtosis is the measure of the thickness or heaviness
of the tails of a distribution.

12. The Normal Distribution
The kurtosis of a distributions is in one of three
categories of classification:
1. Mesokurtic
2. Leptokurtic
3. Platykurtic

13. The Normal Distribution
1-MESOKURTIC
1. Kurtosis is typically measured with respect to
the normal distribution.
2. A distribution that has tails shaped in roughly the
same way as any normal distribution, not just
the standard normal distribution, is said to be
mesokurtic.
3. The kurtosis of a mesokurtic distribution is neither
high nor low, rather it is considered to be a baseline
for the two other classifications.
4. Besides normal distributions, binomial distributions
for which p is close to 1/2 are considered to be
mesokurtic.

14. The Normal Distribution
2-LEPTOKURTIC
1. A leptokurtic distribution is one that has kurtosis
greater than a mesokurtic distribution.
2. Leptokurtic distributions are sometimes identified by
peaks that are thin and tall.
3. The tails of these distributions, to both the right and
the left,
distributions
are thick and heavy. Leptokurtic
are named by the prefix "lepto"
meaning "skinny."
4. There are many examples of leptokurtic
distributions.
5. One of the most well known leptokurtic distributions
is Student's t distribution.

15. The Normal Distribution
3-PLATYKURTIC
1. The third classification for kurtosis is platykurtic.
Platykurtic distributions are those that have slender
tails.
2. Many times they possess a peak lower than a
mesokurtic distribution.
3. The name of these types of distributions come from
the meaning of the prefix "platy" meaning "broad."
4. All uniform distributions are platykurtic. In addition
to this the discrete probability distribution from a
single flip of a coin is platykurtic.

16. The Normal Distribution
What Is Skewness in Statistics?

17. The Normal Distribution
1. Some distributions of data, such as the bell curve are
symmetric.
2. This means that the right and the left of the
distribution are perfect mirror images of one another.
3. Not every distribution of data is symmetric.
4. Sets of data that are not symmetric are said to be
asymmetric.
5. The measure of how asymmetric a distribution can be
is called skewness.
6. The mean, median and mode are all measures of the
center of a set of data.
7. The skewness of the data can be determined by how
these quantities are related to one another.

18. The Normal Distribution
SKEWED TO THE RIGHT
Data that are skewed to the right have a long tail that
extends to the right. An alternate way of talking about a
data set skewed to the right is to say that it is positively
skewed. In this situation the mean and the median are
both greater than the mode. As a general rule, most of the
time for data skewed to the right, the mean will be greater
than the median. In summary, for a data set skewed to the
right:
1. Always: mean greater than mode
2. Always: median greater than mode
3. Most of the time: mean greater than median

19. The Normal Distribution

20. The Normal Distribution
SKEWED TO THE LEFT
•The situation reverses itself when we deal with data skewed to the
left.
•Data that are skewed to the left have a long tail that extends to the
left.
•An alternate way of talking about a data set skewed to the left is to
say that it is negatively skewed.
•In this situation the mean and the median are both less than the
mode. As a general rule, most of the time for data skewed to the
left, the mean will be less than the median. In summary, for a data
set skewed to the left:
1. Always: mean less than mode
2. Always: median less than mode
3. Most of the time: mean less than median

21. The Normal Distribution

22. The Normal Distribution
So, when is the skewness too much? The rule of thumb seems to
be:
1. If the skewness is between -0.5 and 0.5, the data are
fairly symmetrical
2. If the skewness is between -1 and – 0.5 or between 0.5
and 1, the data are moderately skewed
3. If the skewness is less than -1 or greater than 1, the data
are highly skewed

23. The Normal Distribution
SKEWNESS
•
•
• Skewness is usually described as a measure of a dataset’s
symmetry – or lack of symmetry.
A perfectly symmetrical data set will have a skewness of 0.
The normal distribution has a skewness of 0.
KURTOSIS
1. “Kurtosis is the degree of peakedness of a distribution”
2. “We use kurtosis as a measure of peakedness (or flatness)”