Topic IV

Topic 4
Normal distribution and Probabilities
Bachelor in Global Studies

First Term 2022-2023
First Year
1
Content
1. Density curve
2. Normal distribution
3. Probabilities and 68-95-99.7 Rule
4. Probabilities and standard normal distribution
2
1. Density Curve
3
Density curve main features (vs. Histogram)
Main differences with histogram:

1. Area
• Histogram: groups data and obtains columns
• Density curve: an equation (curve) is obtained by using all the data. The
area under the curve is 1.
2. Notation
• μ and σ: denote the mean and the standard deviation of the sample, and
will be critical to construct the density
The normal distribution will be a particular behavior of a variable, which will

produce a unique density curve (bell shaped).
4
Density curve - Measures of center
5
Density curve - Measures of spread
6
2. Normal Distribution
7
Normal distribution - Properties
1. Symmetrical
• Skewness = 0 (perfectly symmetric distribution). The farther away from 0,
the more non-normal the distribution
• Kurtosis = 0 (in stata = 3)
2. Unimodal (a mountain with a single peak)

3. Bell-shaped
X is labeled as a normal variable with parameters μ and 𝜎 correspond to

the mean value and the standard deviation, or for short:
𝑿~𝑵(𝝁, 𝝈)
8
Different expected value, same standard deviation:
9
Same expected value, different standard deviation:
10
Normal distribuAon – How to detect if it is normal?
How to detect if a dataset is normal distributed?
1. Visual diagnosis: check if the density curve or histogram is symmetric,

fairly regular, etc.
2. Numerical diagnosis: skewness and kurtosis coefficients close to 0

(kurtosis=3 in Stata)
11
Normal distribution - 68-95-99.7 Rule
At a normal distribution it holds that:
• 68% of the observations are located between: 𝝁 − σ and 𝝁 + σ

• 95% of the observations are located between: 𝝁 − 2σ and 𝝁 + 2σ
• 99.7% of the observations are located between: 𝝁 − 3σ and 𝝁 + 3σ
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Example: 𝑿~𝑵(𝟏𝟏𝟎, 𝟐𝟓) • 68% of the observations located

between: 𝝁 − σ and 𝝁 + σ
• 95% of the observations located

between: 𝝁 − 2σ and 𝝁 + 2σ
• 99.7% of the observations

located between: 𝝁 − 3σ and 𝝁
+ 3σ
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Example: An IQ score for 20 to 34 age group is normal distributed with

𝑿~𝑵(𝟏𝟏𝟎, 𝟐𝟓)
1. About what percent of people in this age group have scored above
110?
2. About what percent have scored above 160?
3. In what range do the middle 95% of all scores lie?
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3. Standard normal distribution
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Standard normal distribuAon - ProperAes
A normal distribution is standard when it has a mean value of 0 and a
standard deviation of 1, and we denote it by Z:
𝒁~𝑵(𝟎, 𝟏)
Idea: most of the times, when we have a random variable X, we want to

standardize it because the nice properties of the Z.
Standarize: convert a normal variable into the standard normal.
/01
Property =𝑧
2
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3.1 Standard normal distribution
-
Looking for proportion
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Standard normal distribution – Looking for proportion
When we are looking for a propor\on (area represen\ng a percentage of
the popula\on) for a given range of values, we need to follow this 3 steps:
1. State the problem

2. Standardize x-value/s to z-value/s
3. Find percentage of the corresponding area by using the standard
normal tables (Moore)
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Use the table Standard Normal (Moore) to find the corresponded
percentage and draw a density curve to visualize the area that you are
trying to figure out
𝑋~𝑁(𝜇, 𝜎)
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Example: We have a dataset following 𝑋~𝑁 3,2 . What percentage of the

total amount of observations has a value equal or lower than 6.2?
1. State the problem 2. Standardize variable X to Z

What percentage has a value 6.2 − 3
𝑧= = 1.6
equal or lower than 6.2? 2
3. Draw and look for the z-

value (next slide)
20
3. Find percentage of the corresponding area by using the standard normal

tables (Moore)
21
Example: We have a dataset following 𝑋~𝑁 3,2 . What percentage has a

value equal or lower than 6.2?
94.52% of the values are equal or lower than 6.2
Remark: the tables only give us the value for equal or lower!
In case the question would be: what percentage has a value value equal or higher
than 6.2? you have to calculate 100% - 94.52% = 5.48%
22
Take-home example (solved during the seminar session): We have a

representative data of the birth weight of babies for a given country. The
dataset is normally distributed with a mean weight of 3500g and standard
deviation 500g. What is the probability of newborn to weight between
3100g and 3900g?
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3.2 Standard normal distribution
-
Looking for a value of the x variable
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Standard normal distribuAon – Looking for x-values
Until now we were looking for the corresponding proportion (area
representing a percentage of the population) for a given value or range of
values
We can find an x-value or range of x-values corresponding to a given

proportion, by following similar steps:
1. State the problem

2. Find the z-value/s corresponding to the stated percentages
3. Transform z-values to x-values
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Standard normal distribution – Looking for x-values
Example: The average annual income in an imaginary country is 24,000

euros, with a standard deviation of 5,000 euros. Assume that annual
income follows a normal distribution over the population of the country
• What income do you need to have to belong to the richest 1%?
Step 1: state the problem

We are looking for a value x such that we are in the richest 1% of the
distribution (the area to the right is 0.01)
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Step 2: Find the z-value/s corresponding to the stated percentages

Remember that the tables only give us the area to the leb (the propor\on
of values smaller than z). In this case we must look for the z such that the
area to the leb is 0.99.
The corresponding z=2.33
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Step 3: Transform z-values to x-values
2.33=(x-24.000)/5000=35.650
A BUNCH OF NEW EXERCISES TO PRACTICE WILL BE SOLVED DURING THE

SEMINAR SESSION!!!!
28

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Topic 4

Normal distribution and Probabilities

Bachelor in Global Studies

3. Probabilities and 68-95-99.7 Rule

4. Probabilities and standard normal distribution

Main differences with histogram:

The normal distribution will be a particular behavior of a variable, which will

2. Unimodal (a mountain with a single peak)

X is labeled as a normal variable with parameters μ and 𝜎 correspond to

Different expected value, same standard deviation:

Same expected value, different standard deviation:

How to detect if a dataset is normal distributed?

1. Visual diagnosis: check if the density curve or histogram is symmetric,

2. Numerical diagnosis: skewness and kurtosis coefficients close to 0

At a normal distribution it holds that:

• 68% of the observations are located between: 𝝁 − σ and 𝝁 + σ

Example: 𝑿~𝑵(𝟏𝟏𝟎, 𝟐𝟓) • 68% of the observations located

• 95% of the observations located

• 99.7% of the observations

Example: An IQ score for 20 to 34 age group is normal distributed with

2. About what percent have scored above 160?

3. In what range do the middle 95% of all scores lie?

Idea: most of the times, when we have a random variable X, we want to

Standarize: convert a normal variable into the standard normal.

1. State the problem

Example: We have a dataset following 𝑋~𝑁 3,2 . What percentage of the

1. State the problem 2. Standardize variable X to Z

3. Draw and look for the z-

3. Find percentage of the corresponding area by using the standard normal

Example: We have a dataset following 𝑋~𝑁 3,2 . What percentage has a

94.52% of the values are equal or lower than 6.2

Take-home example (solved during the seminar session): We have a

We can find an x-value or range of x-values corresponding to a given

1. State the problem

Example: The average annual income in an imaginary country is 24,000

• What income do you need to have to belong to the richest 1%?

Step 1: state the problem

Step 2: Find the z-value/s corresponding to the stated percentages

Step 3: Transform z-values to x-values

A BUNCH OF NEW EXERCISES TO PRACTICE WILL BE SOLVED DURING THE

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