Basic stats revision content >> tables

The standard normal table

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This version gives the probability that Z is greater than (or greater than or equal) a value "a" i.e p(Z > a). Graphically this probability is the area under the curve to the right of point "a". This version of the table is in Murdoch and Barnes Statistical tables.

The value of "a", called the percentage point, is given along the borders of the table (in bold) and is to 2 decimal places. The values in the main table are probabilities that Z is GREATER THAN "a".

Notice values running down the table are to 1 decimal place. The numbers along the column change only for the 2nd decimal place.

Z

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0

0.5000

0.4960

0.4920

0.4880

0.4840

0.4801

0.4761

0.4721

0.4681

0.4641

0.1

0.4602

0.4562

0.4522

0.4483

0.4443

0.4404

0.4364

0.4325

0.4286

0.4247

0.2

0.4207

0.4168

0.4129

0.4090

0.4052

0.4013

0.3974

0.3936

0.3897

0.3859

0.3

0.3821

0.3783

0.3745

0.3707

0.3669

0.3632

0.3594

0.3557

0.3520

0.3483

0.4

0.3446

0.3409

0.3372

0.3336

0.3300

0.3264

0.3228

0.3192

0.3156

0.3121

0.5

0.3085

0.3050

0.3015

0.2981

0.2946

0.2912

0.2877

0.2843

0.2810

0.2776

0.6

0.2743

0.2709

0.2676

0.2643

0.2611

0.2578

0.2546

0.2514

0.2483

0.2451

0.7

0.2420

0.2389

0.2358

0.2327

0.2296

0.2266

0.2236

0.2206

0.2177

0.2148

0.8

0.2119

0.2090

0.2061

0.2033

0.2005

0.1977

0.1949

0.1922

0.1894

0.1867

0.9

0.1841

0.1814

0.1788

0.1762

0.1736

0.1711

0.1685

0.1660

0.1635

0.1611

 

 

 

 

 

 

 

 

 

 

 

1

0.1587

0.1562

0.1539

0.1515

0.1492

0.1469

0.1446

0.1423

0.1401

0.1379

1.1

0.1357

0.1335

0.1314

0.1292

0.1271

0.1251

0.1230

0.1210

0.1190

0.1170

1.2

0.1151

0.1131

0.1112

0.1093

0.1075

0.1056

0.1038

0.1020

0.1003

0.0985

1.3

0.0968

0.0951

0.0934

0.0918

0.0901

0.0885

0.0869

0.0853

0.0838

0.0823

1.4

0.0808

0.0793

0.0778

0.0764

0.0749

0.0735

0.0721

0.0708

0.0694

0.0681

1.5

0.0668

0.0655

0.0643

0.0630

0.0618

0.0606

0.0594

0.0582

0.0571

0.0559

1.6

0.0548

0.0537

0.0526

0.0516

0.0505

0.0495

0.0485

0.0475

0.0465

0.0455

1.7

0.0446

0.0436

0.0427

0.0418

0.0409

0.0401

0.0392

0.0384

0.0375

0.0367

1.8

0.0359

0.0351

0.0344

0.0336

0.0329

0.0322

0.0314

0.0307

0.0301

0.0294

1.9

0.0287

0.0281

0.0274

0.0268

0.0262

0.0256

0.0250

0.0244

0.0239

0.0233

 

 

 

 

 

 

 

 

 

 

 

2

0.0228

0.0222

0.0217

0.0212

0.0207

0.0202

0.0197

0.0192

0.0188

0.0183

2.1

0.0179

0.0174

0.0170

0.0166

0.0162

0.0158

0.0154

0.0150

0.0146

0.0143

2.2

0.0139

0.0136

0.0132

0.0129

0.0125

0.0122

0.0119

0.0116

0.0113

0.0110

2.3

0.0107

0.0104

0.0102

0.0099

0.0096

0.0094

0.0091

0.0089

0.0087

0.0084

2.4

0.0082

0.0080

0.0078

0.0075

0.0073

0.0071

0.0069

0.0068

0.0066

0.0064

2.5

0.0062

0.0060

0.0059

0.0057

0.0055

0.0054

0.0052

0.0051

0.0049

0.0048

2.6

0.0047

0.0045

0.0044

0.0043

0.0041

0.0040

0.0039

0.0038

0.0037

0.0036

2.7

0.0035

0.0034

0.0033

0.0032

0.0031

0.0030

0.0029

0.0028

0.0027

0.0026

2.8

0.0026

0.0025

0.0024

0.0023

0.0023

0.0022

0.0021

0.0021

0.0020

0.0019

2.9

0.0019

0.0018

0.0018

0.0017

0.0016

0.0016

0.0015

0.0015

0.0014

0.0014

 

 

 

 

 

 

 

 

 

 

 

3

0.0013

0.0013

0.0013

0.0012

0.0012

0.0011

0.0011

0.0011

0.0010

0.0010

3.1

0.0010

0.0009

0.0009

0.0009

0.0008

0.0008

0.0008

0.0008

0.0007

0.0007

3.2

0.0007

0.0007

0.0006

0.0006

0.0006

0.0006

0.0006

0.0005

0.0005

0.0005

3.3

0.0005

0.0005

0.0005

0.0004

0.0004

0.0004

0.0004

0.0004

0.0004

0.0003

3.4

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

0.0002

3.5

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

3.6

0.0002

0.0002

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

3.7

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

3.8

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

0.0001

3.9

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

4

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000



Examples on how to use the table

1. Find p(Z < 0)

Solution

We want the chance that the variable Z takes a value less than (or less than or equal to) zero.

1st look down the rows to 1 decimal place. In this case 0.0.

2nd run along the columns until the number in the 2nd decimal places matches the one in the question, namely 0.

We use the result P(A) = 1 - P(A) i.e. the chance that an event "A" happens is 1 minus the chance the event doesn't happen.

The answer is p(Z < 0) = 1 - p(Z >0)= 1 - 0.5 = 0.5

This answer should make intuitive sense to you, once you know that: (1) the area under the curve is 1, and (2) the curve is symmetrical about Z=0.



2. Find p(Z < 0.65)

Solution.

We want the chance that variable Z takes a value less than (or less than or equal to) 0.65.Using common sense we know this number must be bigger than 0.5.

1st, read down the column to 1 decimal place. In this case, locate the row 0.6.

2nd, run along the columns until the number in the 2nd decimal place matches the one in the question, namely 0,05.

The answer is p(Z < 0.65) = 1 - p(Z > 0.65) = 1 - 0.2578 = 0.7422

So there's 74.22% chance that Z is less than 0.65.




3. Find p(Z > 1.12)

Solution.

Note this time we want the chance Z is GREATER than a number (in this case, 1.12), which we can get by reading the table directly.

p(Z > 1.12) = 0.1314.




4. Find p(Z < 1.125)

Solution.

Here you note that the number 1.125 is to 3 decimal places. But the table tabulates to only 2 decimal places? You could use what is called interpolation, but it is far quicker and acceptable to use a bit of common sense. If the value in the 3rd decimal place is

(a) less than five, then round down to 0. Eg 1.32 becomes 1.30 and this makes sense cos the probability for z<1.32 is closer to 1.3 rather than 1.4.
(b) above 5 then round up eg 2.48 becomes 2.50
(c) is 5 then round up or round down or take the average probability of number rounded up and down.

Our case is (c). Let's take the probability for 1.12 and 1.13 and average them for the answer:

p(Z < 1.12) = 1 - p(Z >1.12) = 0.8686 and p(Z < 1.13) = 1 - p(Z > 1.13) = 0.8708, so our guess for p(Z < 1.125) is the average of the 2 probabilities: (0.8686+0.8708)/2 = ?




5. Find p(0 < Z < 1.12)

Solution.

Here is a case where we want the chance that Z falls within an INTERVAL. For these questions it may help to sketch the curve and shade in the area, which should give you an idea of the value of the answer.

We can see the answer must be less than 0.5.(Sketch the curve to see why)

p( 0< Z < 1.12) = p(Z < 1.12) - p(Z < 0) = (1 - p(Z>1.12)) - (1 - p(Z>0.5)) =  0.8686 - 0.5 = 0.3686

A mistake that could be made is you find p(Z < 0) - p(Z < 1.12). But if you did this you'd get a negative answer, which is wrong since probabilities are between 0 and 1. It would look very bad if you wrote down a negative probability in an exam!




6. Find p(Z < - 0.52)

Solution.

Now we want the probability that Z is less than -0.52. We see that the table give values only for positive numbers. (You could have a number with negative numbers but this would be a waster of space as I'll now explain to you.) When you get negative values, you use the idea that the curve is symmetrical about zero.

The chance Z < -0.52 is less than 0.5 as can be seen in a sketch.

Using symmetry property, p(Z < -0.52) = 1 - p( Z< 0.52) = 1 - (1 - p(Z>0.52)).

And this is p( Z < -0.52) = 0.3015




7. Find b such that p(Z < b) = 0.975

Solution.

We want the value "b" such that the probability that Z is less than "b" is 97.5%. What we have here is the reverse problem to those encountered so far. We are given the probability and have to find the percentage point, b.

To do this see that the question is equivalent to p(Z > b) = 0.025:

Now, look in the body of the table for 0.025. Hopefully, you find b = 1.96




If you have any difficulties using the table that has not been addressed here, discuss your problem in theforum