Normal Distribution

Making only the assumption that any value within some range of values is possible, the normal distribution can be and is used to represent many phenomena. It is used in decision-making to model expectations about the inflation rate, or the future price of oil. This distribution is widely used in metering equipment, in which it can represent the distribution of measurement errors; in reservoir stud­ies it can be used to predict soil permeability, the spaces between the grains and saturation as well as some economic data.

Equation:

Normal Distribution

Подпись: Mean: Подпись: n Подпись: (3.13)

(3.12)

where:

X-arithmetic mean of sample data x = each individual value in sample n = number of values in sample cm = class mark nc = number of values in class

(3.14)

Подпись:where о is the standard deviation, and p is the arithmetic mean.

The Normal distribution is the most commonly used probabil­ity distribution, because it can generate information about a set of measurements without our having to know anything about how the phenomena of interest came to exist in the first place, or whether some values are more likely than others: its sole assumption is that any possible value within some range may be assigned some non­zero probability. As distinct from many of the other distributions discussed below, a normally-distributed variable is always indiffer­ent to the passage of time. Analysis on this basis of measurements of the output of one and the same process are ideal candidates for the application of this tool. Thus, for example, it was found that the Normal distribution is the best probability curve to present con­crete strength from laboratory tests performed on the concrete in most countries of the world. (The moment we have reason to know that the assumption of all outcomes being possible is inapplicable, other distributions should be considered, as will be seen below.)

The most significant characteristics of the Normal distribution for present purposes are:

• distribution is symmetric around the average; more precisely: the arithmetic mean of the curve is divided into two equal halves

• a Normal distribution matches the arithmetic mean and median lines and mode value that is most likely to occur

• area under the curve equals 1 and that the random variable in the outcomes of the cubes to, that this curve represents all the °° to -“break it to take values from potential possibilities of the concrete’s strength.

As a result, each curve depends on the value of the arithmetic mean and standard deviation, and any difference between the two parameters leads to a difference in the shape of the probability distribution. Therefore, the standard normal distribution is used to determine areas under a curve by knowing the standard deviation

and arithmetic mean using another variable, Z, which is obtained from the following equation:

z = ^-± (3.15)

a

Table (3.11) shows the values of the area under the curve by knowing the value of z from the above equation. In the first col­umn, the value of z and first row determine the accuracy to the nearest two decimal digits. From the table one can find that the area under the curve at z is equal to 1.64. The area under the curve for any value less than z is 0.5-0.4495, which is equal to about 0.0505. In other words, the probability of the variables has a value less than or equal to 5% as shown in Figure (3.7).

Normal Distribution

Figure 3.7 Normal distribution curve.

z

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0

0

0.004

0.008

0.012

0.016

0.0199

0.0239

0.0279

0.0319

0.0359

0.1

0.0398

0.0438

0.0478

0.0517

0.0557

0.0596

0.0636

0.0675

0.0714

0.0753

0.2

0.0793

0.0832

0.0871

0.091

0.0948

0.0987

0.1026

0.1064

0.1103

0.1141

0.3

0.1179

0.1217

0.1255

0.1293

0.1331

0.1368

0.1406

0.1443

0.148

0.1517

0.4

0.1554

0.1591

0.1628

0.1664

0.17

0.1736

0.1772

0.1808

0.1844

0.1879

0.5

0.1915

0.195

0.1985

0.2019

0.2054

0.2088

0.2123

0.2157

0.219

0.2224

0.6

0.2257

0.2291

0.2324

0.2357

0.2389

0.2422

0.2454

0.2486

0.2517

0.2549

0.7

0.258

0.2611

0.2642

0.2673

0.2704

0.2734

0.2764

0.2794

0.2823

0.2852

0.8

0.2881

0.291

0.2939

0.2967

0.2995

0.3023

0.3051

0.3078

0.3106

0.3133

0.9

0.3159

0.3186

0.3212

0.3238

0.3264

0.3289

0.3315

0.334

0.3365

0.3389

1

0.3413

0.3438

0.3461

0.3485

0.3508

0.3531

0.3554

0.3577

0.3599

0.3621

1.1

0.3643

0.3665

0.3686

0.3708

0.3729

0.3749

0.377

0.379

0.381

0.383

1.2

0.3849

0.3869

0.3888

0.3907

0.3925

0.3944

0.3962

0.398

0.3997

0.4015

1.3

0.4032

0.4049

0.4066

0.4082

0.4099

0.4115

0.4131

0.4147

0.4162

0.4177

1.4

0.4192

0.4207

0.4222

0.4236

0.4251

0.4265

0.4279

0.4292

0.4306

0.4319

1.5

0.4332

0.4345

0.4357

0.437

0.4382

0.4394

0.4406

0.4418

0.4429

0.4441

1.6

0.4452

0.4463

0.4474

0.4484

0.4495

0.4505

0.4515

0.4525

0.4535

0.4545

1.7

0.4554

0.4564

0.4573

0.4582

0.4591

0.4599

0.4608

0.4616

0.4625

0.4633

Table 3.11 The area under the curve of normal distribution

60 Construction Management for Industrial Projects

1.8

0.4641

0.4649

0.4656

0.4664

0.4671

0.4678

0.4686

0.4693

0.4699

0.4706

1.9

0.4713

0.4719

0.4726

0.4732

0.4738

0.4744

0.475

0.4756

0.4761

0.4767

2

0.4772

0.4778

0.4783

0.4788

0.4793

0.4798

0.4803

0.4808

0.4812

0.4817

2.1

0.4821

0.4826

0.483

0.4834

0.4838

0.4842

0.4846

0.485

0.4854

0.4857

2.2

0.4861

0.4864

0.4868

0.4871

0.4875

0.4878

0.4881

0.4884

0.4887

0.489

2.3

0.4893

0.4896

0.4898

0.4901

0.4904

0.4906

0.4909

0.4911

0.4913

0.4916

2.4

0.4918

0.492

0.4922

0.4925

0.4927

0.4929

0.4931

0.4932

0.4934

0.4936

2.5

0.4938

0.494

0.4941

0.4943

0.4945

0.4946

0.4948

0.4949

0.4951

0.4952

2.6

0.4953

0.4955

0.4956

0.4957

0.4959

0.496

0.4961

0.4962

0.4963

0.4964

2.7

0.4965

0.4966

0.4967

0.4968

0.4969

0.497

0.4971

0.4972

0.4973

0.4974

2.8

0.4974

0.4975

0.4976

0.4977

0.4977

0.4978

0.4979

0.4979

0.498

0.4981

2.9

0.4981

0.4982

0.4982

0.4983

0.4984

0.4984

0.4985

0.4985

0.4986

0.4986

3

0.4987

0.4987

0.4987

0.4988

0.4988

0.4989

0.4989

0.4989

0.499

0.499

3.1

0.499

0.4991

0.4991

0.4991

0.4991

0.4992

0.4992

0.4992

0.4992

0.4993

3.2

0.4993

0.4993

0.4994

0.4994

0.4994

0.4994

0.4994

0.4994

0.4995

0.4995

3.3

0.4995

0.4995

0.4995

0.4996

0.4996

0.4996

0.4996

0.4996

0.4996

0.4997

3.4

0.4997

0.4997

0.4997

0.4997

0.4997

0.4997

0.4997

0.4997

0.4997

0.4998

3.5

0.4998

0.4998

0.4998

0.4998

0.4998

0.4998

0.4998

0.4998

0.4998

0.4998

4.0

0.49997

5.0

0.49999

Economic Risk Analysis 61

Figure (3.8) shows the area under the curve when you add or decrease the value of the standard deviation of the arithmetic mean.

The area under the curve from the arithmetic mean value to one value of the standard deviation is equal to 34.13%, while in the area under the curve in Figure (3.8) for twice the standard deviation values is equal to 47.72%.