source: D.C. Montgomery - Applied Statistics and Probability for Engineers

pb.4

562 869 708 775 775 704 809

856 655 806 878 909 918 558

768 870 918 940 946 661 820

898 935 952 957 693 835 905

939 955 960 498 653 730 753

(a) Compute the sample mean, variance, and standard deviation.

(b) Find the sample upper and lower quartiles.

(c) Find the sample median.

(d) Construct a box plot of the data.

(e) Find the 5th and 95th percentiles.

c) calculati mediana; d) trasati graficul tip box-plot; e) gasiti percentilele 5 si 95

a) calculati media, variatia si deviatia standard; b) gasiti cuartila superioara si cea inferioara

Construct a cumulative frequency plot and histogram for the solar intensity data. Use

6 bins

Calculate the sample mean and sample standard deviation. Prepare a dot diagram of

these data. Indicate where the sample mean falls on this diagram. Provide a practical

interpretation of the sample mean.

The following data are direct solar intensity measurements on different

days at a location in southern Spain:

Date reprezinta masuratori ale intensitatii razelor solare in diferite zile intr-o zona din

sudul Spaniei.

Calculati media si deviatia standard. Trasati diagrama prin puncte ale acestor valori.

Indicati unde este apare media pe aceste grafic. Oferiti o interpretare practica a mediei

Trasati histograma valorilor intensitatii solare si diagrama fercventelor cumulate. Folosind

6, respectiv 12 bin-uri

min 498

max 960

amplitude 462

1st quartile 719

3rd quartile 918

sample mean 810,51

standard deviation 128,32

variance 15995,2

median 835

mode 775

490 590 690 790 890 990

dot diagram of direct solar intensity measurements

Cumulative frequency Bin Frequency This table use Data Analysis from Data menu

490 0490 0 570 3 490-570570 3 650 0 570-650650 3 730 7 650-730730 10 810 6 730-810810 16 890 6 810-890890 22 970 13 890-970970 35 More 0

Input range: C5:I9

Frequencies

Activate first Data Analysis option from Excel options (File menu ) -> Add-Ins -> select Analysis Toolpak -> the press Go

After that should be appear in Data Analysis in Data menu

Use now Data -> Data Analysis -> Histogram

Bin Range: O5:O11Output range: select area for display results

before finish your command press keyCtrl+Shift+EnterThis will become an array function

0

0,0005

0,001

0,0015

0,002

0,0025

0,003

0,0035

505

525

545

565

585

605

625

645

665

685

705

725

745

765

785

805

825

845

865

885

905

925

945

965

985

Probability mass function

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

150

552

554

556

558

560

562

564

566

568

570

572

574

576

578

580

582

584

586

588

590

592

594

596

598

5

Cumulative distribution function

0

2

4

6

8

10

12

14

490-570 570-650 650-730 730-810 810-890 890-970

frequency

490 0,006248251 0,000137351

495 0,006969353 0,000151277

500 0,007762953 0,000166362

505 0,008635013 0,000182673

510 0,009591838 0,000200279

515 0,010640074 0,000219249

520 0,011786712 0,000239651

525 0,013039086 0,000261555

530 0,014404873 0,000285027

535 0,015892083 0,000310135

540 0,017509054 0,000336942

545 0,019264441 0,000365511

550 0,021167203 0,000395901

555 0,023226583 0,000428168

560 0,025452091 0,000462361

565 0,027853481 0,000498527

570 0,030440722 0,000536707

575 0,033223968 0,000576935

580 0,036213528 0,000619236

585 0,039419822 0,000663631

590 0,042853347 0,00071013

595 0,04652463 0,000758734

600 0,05044418 0,000809434

605 0,054622436 0,000862213

610 0,059069719 0,000917039

615 0,063796169 0,000973872

620 0,06881169 0,001032657

625 0,074125888 0,00109333

630 0,079748009 0,001155812

635 0,085686871 0,001220011

640 0,091950803 0,001285821

645 0,098547575 0,001353126

650 0,105484334 0,001421793

655 0,112767537 0,001491678

660 0,120402887 0,001562625

665 0,128395268 0,001634461

670 0,136748682 0,001707007

675 0,145466195 0,001780068

680 0,154549876 0,001853439

685 0,164000747 0,001926907

690 0,173818734 0,002000248

x

Cumulative

distribution function

Probability mass

function

=NORMDIST(P15;$M

$12;$M$13;TRUE)

=NORMDIST(P15;$

M$12;$M$13;

FALSE)

695 0,184002626 0,00207323

700 0,194550035 0,002145614

705 0,205457366 0,002217157

710 0,21671979 0,00228761

715 0,228331226 0,00235672

720 0,24028433 0,002424235

725 0,25257049 0,0024899

730 0,265179828 0,002553464

735 0,278101214 0,002614678

740 0,291322284 0,002673298

745 0,304829468 0,002729084

750 0,318608023 0,002781809

755 0,332642081 0,002831249

760 0,346914695 0,002877197

765 0,361407901 0,002919454

770 0,376102782 0,002957838

775 0,390979544 0,00299218

780 0,40601759 0,003022328

785 0,42119561 0,003048148

790 0,436491666 0,003069525

795 0,451883294 0,003086362

800 0,467347595 0,003098584

805 0,482861344 0,003106134

810 0,498401088 0,003108978

815 0,51394326 0,003107104

820 0,529464277 0,00310052

825 0,544940657 0,003089256

830 0,560349118 0,003073363

835 0,575666686 0,003052912

840 0,590870798 0,003027997

845 0,605939402 0,002998729

850 0,62085105 0,002965238

855 0,635584991 0,002927673

860 0,65012126 0,002886198

865 0,664440753 0,002840994

870 0,678525304 0,002792255

875 0,692357752 0,002740188

880 0,705922003 0,002685013

885 0,719203079 0,002626957

890 0,732187164 0,002566257

895 0,744861645 0,002503156

900 0,757215136 0,002437902

905 0,769237499 0,002370747

910 0,780919863 0,002301945

915 0,792254622 0,002231748

920 0,803235436 0,002160409

925 0,81385722 0,002088177

930 0,824116127 0,002015298

935 0,834009523 0,001942012

940 0,843535958 0,001868552

945 0,852695126 0,001795143

950 0,86148783 0,001722001

955 0,869915926 0,001649333

960 0,87798228 0,001577335

965 0,885690705 0,001506192

970 0,89304591 0,001436075

975 0,900053433 0,001367145

980 0,906719583 0,001299549

985 0,91305137 0,001233421

990 0,91905644 0,001168882

995 0,924743014 0,001106039

1000 0,930119814 0,001044987

skewness -0,75223 Skewness: indicator used in distribution analysis as a sign of asymmetry and deviation from a normal distribution.

kurtosis -0,29608 Interpretation:

Skewness = 0 - mean = median, the distribution is symmetrical around the mean.

Kurtosis - indicator used in distribution analysis as a sign of flattening or "peakedness" of a distribution.

Interpretation:

Kurtosis = 3 - Mesokurtic distribution - normal distribution for example.

Kurtosis < 3 - Platykurtic distribution, flatter than a normal distribution with a wider peak. The probability for extreme values is

less than for a normal distribution, and the values are wider spread around the mean.

Skewness > 0 - Right skewed distribution - most values are concentrated on left of the mean, with extreme values to the right.

Skewness < 0 - Left skewed distribution - most values are concentrated on the right of the mean, with extreme values to the

left.

Kurtosis > 3 - Leptokurtic distribution, sharper than a normal distribution, with values concentrated around the mean and

thicker tails. This means high probability for extreme values.