Elements of Statistics

Module 5 Normal Distribution Model

STAT 300 Elements of Statistics I

Objectives

At the end of the module, the student will be able to:

Calculate probabilities of a normal distribution.

Use the normal distribution to solve business situations.

▪ A continuous random variable can assume any value on a

continuum. That is, it can assume decimal values beyond

a counting process.

▪ Examples

o Time to complete a task

o Height, in meters

Continuous Probability Distributions

Normal Distribution

▪ Bell-shaped

▪ Symmetric

▪ The 3 measures of central

tendency (mode, mean and

median) are equal.

▪ The position is determined by the mean, μ

▪ The dispersion is determined by the standard deviation, σ

▪ The random variable has a theoretically infinite range: +  a − 

Mean

= Median

= Mode

X

f(X)

μ

σ

e   

2

− 1  (X−μ) 

f(X) = 2  1

2π

Where,

e = approximate mathematical constant 2.71828

π= approximate mathematical constant 3.14159

μ = population mean

σ =population standard deviation

X = any value of the continuous variable

Density Function of the Normal Distribution

Normal Distributions

By varying the parameters µ and σ, different normal distributions are obtained

Shape of the Normal Distribution

Xμ

σ

f(X)

Shift μ moves the distribution to

the left or right.

Changing σ increases or

decreases dispersion

Developed by Professor Sylvia Y. Cosme Montalvo, MBA

▪ Any normal distribution can be transformed into the

standardized normal distribution (Z).

▪ X units are transformed to Z units.

▪ The standardized normal distribution (Z) has mean 0 and

standard deviation 1.

Standardization of the Normal Distribution

Translation to Standardized Normal Distribution

Z = X − μ

σ

0

The standardized normal distribution (Z) ALWAYS has mean 0 and

standard deviation 1.

Positive Z values exceed the mean.

Negative Z values are below the mean.

Z

▪ Subtract the mean from the X value and divide by the standard deviation:

f(Z)

1

▪ If X is normally distributed with mean equal to $100 and standard

deviation of $50, the Z value for X =$200 is:

Z = X − μ = $200 − $100 = 2.0 σ $50

▪ Implies that X = $200 is 2 standard deviations above the mean of

$100 (2 increments of $50 units).

Example

Contrast of X and Z Units

$100 $200

0 2.0 Z

Note that the shape of the distribution is the same, only the

scale changes.

µ=0, σ=1

$X (µ=$100, σ=$50

Normal Probabilities

a b

f(X)

The probability is measured by the area under the curve.

P (a ≤ X ≤ b )

= P (a < X < b )

(Note that the

probability of any

single value is

zero).

X

μ

Probability as Area Under the Curve

P(−  X  ) = 1.0

The total area under the curve is 1.0, the curve is symmetrical, so 50%

is above the mean and 50% below the mean f(X).

X

P(μ X  ) = 0.5 P(−  X  μ) = 0.5

Standardized Distribution Table

Example:

P(Z < 2.00) = 0.9772

Z0 2.00

The standardized normal distribution table presents the probability

less than a desired value of Z (from negative infinity to Z).

0.9772

Standardized Distribution Table (cont.)

The value in the table gives

the probability Z = −  to the

desired Z value.

.9772

P(Z < 2.00) = 0.9772

The row presents

the value of Z to

the first decimal

place.

2.0

.

.

.

The column gives the value of Z

Z 0.00 0.01 0.02 …

0.0

0.1

▪ To find P(a < X < b) when normally distributed:

1. Draw the normal curve for the exercise in terms of X.

2. Translate the X-values to Z-values.

3. Use the table of standardized values.

Procedure for Finding Normal Probabilities

Finding Normal Probabilities

▪ Let X be the average time in months it takes to solve a case handled by

the Police Department related to petty theft.

▪ Assume that X is normal with mean 18.0 months and standard deviation

5.0 months. Find P(X < 18.6)

X 18.0

18.6

Z = X − μ = 18.6 −18.0 = 0.12 σ 5.0

.5478

Z .00 .01 .02

0.0 .5000 .5040 .5080

0.1 .5398 .5438 .54.78

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

= 0.5478P(X < 18.6) = P(Z < 0.12)

Finding Normal Probabilities in the Tail

Find P(X > 18.6)

P(X > 18.6) = P(Z > 0.12) = 1.0 - P(Z ≤ 0.12)

= 1.0 - 0.5478 = 0.4522 X

18.0

18.6

Finding Normal Probabilities Between 2 Values

X

Z

18 18.6

0 0.12

P(18 < X < 18.6)

= P(0 < Z < 0.12)

Z = X − μ = 18 − 18 = 0 σ 5

Z = X − μ = 18.6 −18 = 0.12 σ 5

Find P(18 < X < 18.6)

Calculate the Z value

5000

.5478

Z .00 .01 .02

0.0 ..5000 .5040 .5080

0.1 .5398 .5438 .54.78

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

P(18 < X < 18.6)

= P(0 < Z < 0.12)

= P(Z < 0.12) – P(Z ≤ 0)

= 0.5478 - 0.5000 = 0.0478

Steps to follow to find X value of a known probability:

1. Find the Z value of the known probability.

2. Convert the units to the formula:

Normal Probability and X-Value

X = µ + Zσ

Normal Probability and X-Value (cont.)

Example:

▪ Let X be the average time in months it takes to solve a case handled by the Police Department related to petty theft. Assume that X is normal with mean 18.0 and Standard Deviation 5.0.

▪ Find X such that 20% of the case resolution time is less than X.

0.2000

X

Z ? 18.0

? 0

Z … .03 .04 .05

-0.9 … .1762 .1736 .1711

-0.8 … .2033 .2005 .1977

-0.7 … .2327 .2296 .2266

Z = -0.84 X = μ + Zσ

= 18.0 + (−0.84)5.0

= 13.8

Therefore, 20% of

the values are

less than 13.80.

Using Excel to Determine Normal Probability

Given a value X Given a range of values X Given %, Find Z value and X

Normal Probabilities

Mean 7

Standard deviation 2

Probability for X<=

X value 7

Z value 0

P(X<=) 0.5

Probability for X>

X value 9

Z-value 1

P(X<=) 0.1587

Probability for X> or X< 0.6587

Range Probabilities

From value X 5

Up to X-value 9

Z-value for 5 -1

Z-value for 9 1

P(X<=5 0.1587

P(X<=9 0.8413

5<=X<=9) 0.6827

X and Z value given a cumulative %

Cumulative % 10%

Value Z -1.2816

Value X 4.4369

Summary

▪ In this unit you learned:

1. The properties of the normal distribution.

2. To calculate probabilities using formulas and tables.

3. To apply the normal distribution to decision making

exercises.

References

Hesse C., Ofosu J. (2022). Statistical Methods for the Social Sciences. Akrong Publications Ltd.

Ghana. ISBN: 978–9988–2–6060–6

Howell David (2016). Fundamental Statistics for the behavioral sciences. Cengage Learning. ISBN-

10: 1305652975

Oja. (2022). PSYC 2200: Elementary Statistics for Behavioral and Social Sciences. [Vídeo]. Statistics

LibreTexts.

https://stats.libretexts.org/Courses/Taft_College/PSYC_2200:_Elementary_Statistics_for_Behavio

ral_and_Social_Sciences_(Oja)

Pelz, B. (s. f.). Statistics for the Social Sciences | Simple Book Publishing. [Vídeo]. Pressbooks.

https://courses.lumenlearning.com/suny-hccc-wm-concepts-statistics/

S. P. Mukherjee, Bikas K. Sinha Asis, Kumar Chattopadhyay (2018). Statistical Methods in Social

Science Research. Springer Nature Singapore Pte Ltd.

References

Khan Academy. (n.d.). Normal distribution: Area between two points (practice). Khan Academy.

https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/normal-

distributions-calculations/e/z_scores_3

Khan Academy. (n.d.). Deep definition of the normal distribution. [Video]. Khan Academy.

https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/more-on-

normal-distributions/v/introduction-to-the-normal-distribution

Khan Academy. (n.d.). Normal distributions review [Article]. Khan Academy.

https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-

distributions-library/a/normal-distributions-review

Khan Academy. (n.d.). Standard normal distribution and the empirical rule. [Video]. Khan Academy.

https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-

distributions-library/v/ck12-org-exercise-standard-normal-distribution-and-the-empirical-rule

Congratulations you have reviewed the

theoretical summary of this week's topic!

Remember that to successfully build your learning it is important that:

Review as many times as required the information contained in the module folder

(includes this presentation).

Read the reference material to clarify any questions.

Carry out all the activities according to the instructions.

Submit assignments on the indicated date through the educational

platform.

Actively participate in collaborative sessions.

Module 5 Normal Distribution Model

STAT 300 Elements of Statistics I

Objectives

At the end of the module, the student will be able to:

Calculate probabilities of a normal distribution.

Use the normal distribution to solve business situations.

▪ A continuous random variable can assume any value on a

continuum. That is, it can assume decimal values beyond

a counting process.

▪ Examples

o Time to complete a task

o Height, in meters

Continuous Probability Distributions

Normal Distribution

▪ Bell-shaped

▪ Symmetric

▪ The 3 measures of central

tendency (mode, mean and

median) are equal.

▪ The position is determined by the mean, μ

▪ The dispersion is determined by the standard deviation, σ

▪ The random variable has a theoretically infinite range: +  a − 

Mean

= Median

= Mode

X

f(X)

μ

σ

e   

2

− 1  (X−μ) 

f(X) = 2  1

2π

Where,

e = approximate mathematical constant 2.71828

π= approximate mathematical constant 3.14159

μ = population mean

σ =population standard deviation

X = any value of the continuous variable

Density Function of the Normal Distribution

Normal Distributions

By varying the parameters µ and σ, different normal distributions are obtained

Shape of the Normal Distribution

Xμ

σ

f(X)

Shift μ moves the distribution to

the left or right.

Changing σ increases or

decreases dispersion

Developed by Professor Sylvia Y. Cosme Montalvo, MBA

▪ Any normal distribution can be transformed into the

standardized normal distribution (Z).

▪ X units are transformed to Z units.

▪ The standardized normal distribution (Z) has mean 0 and

standard deviation 1.

Standardization of the Normal Distribution

Translation to Standardized Normal Distribution

Z = X − μ

σ

0

The standardized normal distribution (Z) ALWAYS has mean 0 and

standard deviation 1.

Positive Z values exceed the mean.

Negative Z values are below the mean.

Z

▪ Subtract the mean from the X value and divide by the standard deviation:

f(Z)

1

▪ If X is normally distributed with mean equal to $100 and standard

deviation of $50, the Z value for X =$200 is:

Z = X − μ = $200 − $100 = 2.0 σ $50

▪ Implies that X = $200 is 2 standard deviations above the mean of

$100 (2 increments of $50 units).

Example

Contrast of X and Z Units

$100 $200

0 2.0 Z

Note that the shape of the distribution is the same, only the

scale changes.

µ=0, σ=1

$X (µ=$100, σ=$50

Normal Probabilities

a b

f(X)

The probability is measured by the area under the curve.

P (a ≤ X ≤ b )

= P (a < X < b )

(Note that the

probability of any

single value is

zero).

X

μ

Probability as Area Under the Curve

P(−  X  ) = 1.0

The total area under the curve is 1.0, the curve is symmetrical, so 50%

is above the mean and 50% below the mean f(X).

X

P(μ X  ) = 0.5 P(−  X  μ) = 0.5

Standardized Distribution Table

Example:

P(Z < 2.00) = 0.9772

Z0 2.00

The standardized normal distribution table presents the probability

less than a desired value of Z (from negative infinity to Z).

0.9772

Standardized Distribution Table (cont.)

The value in the table gives

the probability Z = −  to the

desired Z value.

.9772

P(Z < 2.00) = 0.9772

The row presents

the value of Z to

the first decimal

place.

2.0

.

.

.

The column gives the value of Z

Z 0.00 0.01 0.02 …

0.0

0.1

▪ To find P(a < X < b) when normally distributed:

1. Draw the normal curve for the exercise in terms of X.

2. Translate the X-values to Z-values.

3. Use the table of standardized values.

Procedure for Finding Normal Probabilities

Finding Normal Probabilities

▪ Let X be the average time in months it takes to solve a case handled by

the Police Department related to petty theft.

▪ Assume that X is normal with mean 18.0 months and standard deviation

5.0 months. Find P(X < 18.6)

X 18.0

18.6

Z = X − μ = 18.6 −18.0 = 0.12 σ 5.0

.5478

Z .00 .01 .02

0.0 .5000 .5040 .5080

0.1 .5398 .5438 .54.78

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

= 0.5478P(X < 18.6) = P(Z < 0.12)

Finding Normal Probabilities in the Tail

Find P(X > 18.6)

P(X > 18.6) = P(Z > 0.12) = 1.0 - P(Z ≤ 0.12)

= 1.0 - 0.5478 = 0.4522 X

18.0

18.6

Finding Normal Probabilities Between 2 Values

X

Z

18 18.6

0 0.12

P(18 < X < 18.6)

= P(0 < Z < 0.12)

Z = X − μ = 18 − 18 = 0 σ 5

Z = X − μ = 18.6 −18 = 0.12 σ 5

Find P(18 < X < 18.6)

Calculate the Z value

5000

.5478

Z .00 .01 .02

0.0 ..5000 .5040 .5080

0.1 .5398 .5438 .54.78

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

P(18 < X < 18.6)

= P(0 < Z < 0.12)

= P(Z < 0.12) – P(Z ≤ 0)

= 0.5478 - 0.5000 = 0.0478

Steps to follow to find X value of a known probability:

1. Find the Z value of the known probability.

2. Convert the units to the formula:

Normal Probability and X-Value

X = µ + Zσ

Normal Probability and X-Value (cont.)

Example:

▪ Let X be the average time in months it takes to solve a case handled by the Police Department related to petty theft. Assume that X is normal with mean 18.0 and Standard Deviation 5.0.

▪ Find X such that 20% of the case resolution time is less than X.

0.2000

X

Z ? 18.0

? 0

Z … .03 .04 .05

-0.9 … .1762 .1736 .1711

-0.8 … .2033 .2005 .1977

-0.7 … .2327 .2296 .2266

Z = -0.84 X = μ + Zσ

= 18.0 + (−0.84)5.0

= 13.8

Therefore, 20% of

the values are

less than 13.80.

Using Excel to Determine Normal Probability

Given a value X Given a range of values X Given %, Find Z value and X

Normal Probabilities

Mean 7

Standard deviation 2

Probability for X<=

X value 7

Z value 0

P(X<=) 0.5

Probability for X>

X value 9

Z-value 1

P(X<=) 0.1587

Probability for X> or X< 0.6587

Range Probabilities

From value X 5

Up to X-value 9

Z-value for 5 -1

Z-value for 9 1

P(X<=5 0.1587

P(X<=9 0.8413

5<=X<=9) 0.6827

X and Z value given a cumulative %

Cumulative % 10%

Value Z -1.2816

Value X 4.4369

Summary

▪ In this unit you learned:

1. The properties of the normal distribution.

2. To calculate probabilities using formulas and tables.

3. To apply the normal distribution to decision making

exercises.

References

Hesse C., Ofosu J. (2022). Statistical Methods for the Social Sciences. Akrong Publications Ltd.

Ghana. ISBN: 978–9988–2–6060–6

Howell David (2016). Fundamental Statistics for the behavioral sciences. Cengage Learning. ISBN-

10: 1305652975

Oja. (2022). PSYC 2200: Elementary Statistics for Behavioral and Social Sciences. [Vídeo]. Statistics

LibreTexts.

https://stats.libretexts.org/Courses/Taft_College/PSYC_2200:_Elementary_Statistics_for_Behavio

ral_and_Social_Sciences_(Oja)

Pelz, B. (s. f.). Statistics for the Social Sciences | Simple Book Publishing. [Vídeo]. Pressbooks.

https://courses.lumenlearning.com/suny-hccc-wm-concepts-statistics/

S. P. Mukherjee, Bikas K. Sinha Asis, Kumar Chattopadhyay (2018). Statistical Methods in Social

Science Research. Springer Nature Singapore Pte Ltd.

References

Khan Academy. (n.d.). Normal distribution: Area between two points (practice). Khan Academy.

https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/normal-

distributions-calculations/e/z_scores_3

Khan Academy. (n.d.). Deep definition of the normal distribution. [Video]. Khan Academy.

https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/more-on-

normal-distributions/v/introduction-to-the-normal-distribution

Khan Academy. (n.d.). Normal distributions review [Article]. Khan Academy.

https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-

distributions-library/a/normal-distributions-review

Khan Academy. (n.d.). Standard normal distribution and the empirical rule. [Video]. Khan Academy.

https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-

distributions-library/v/ck12-org-exercise-standard-normal-distribution-and-the-empirical-rule

Congratulations you have reviewed the

theoretical summary of this week's topic!

Remember that to successfully build your learning it is important that:

Review as many times as required the information contained in the module folder

(includes this presentation).

Read the reference material to clarify any questions.

Carry out all the activities according to the instructions.

Submit assignments on the indicated date through the educational

platform.

Actively participate in collaborative sessions.