SERII SIMPLE (date negrupate) 0 ULPL�PHGLL 0HGLD�DULWPHWLF 0HGLD� DULWPHWLF HVWH� UH]XOWDWXO� VLQWHWL] ULL� vQWU-R� VLQJXU � H[SUHVLH� QXPHULF � D� WXWXURU�

QLYHOXULORU� LQGLYLGXDOH� REVHUYDWH�� RE LQXW � SULQ� UDSRUWDUHD� YDORULL� WRWDOL]DWH� D� FDUDFteristicii la QXP UXO�WRWDO�DO�XQLW LORU�

n

x

x

n

ii∑

== 1

unde:

ix - UHSUH]LQW �QLYHOXULOH�LQGLYLGXDOH�DOH�YDULDELOHL�

∑=

n

iix

1

- UHSUH]LQW �YROXPXO�FHQWUDOL]DW�DO�YDULDELOHL�

n - UHSUH]LQW �QXP UXO�XQLW LORU observate.

3URSULHW L a) GDF �x1=x2=...=xi=...=xn=xc , atunci cxx =

b) maxmin xxx <<

c) 0)(1

=−∑=

n

ii xx

d) GDF � axx i #=’ , atunci axx #=′ , de unde axx ±= ’

6HULL�GH�GLVWULEX LH�D�IUHFYHQ HORU

e) GDF �h

xx i=" , atunci

h

xx =′′ , de unde hxx ⋅′′= ,

respectiv :

GDF � hxx i ⋅=" , atunci hxx ⋅=′′ , de unde h

xx

′′=

Formule de calcul simplificat al mediei aritmetice:

( )a

n

ax

x

n

ii

±=∑

=1

#

hn

h

x

x

n

i

i

⋅

=∑

=1 sau ( )

hn

hxx

n

ii

:1∑

=

⋅=

ahn

h

ax

x

n

i

i

+⋅

−

=∑

=1

0HGLD�DUPRQLF

0HGLD� DUPRQLF � � hx �� VH� GHILQHúWH� FD� ILLQG� LQYHUVD�PHGLHL�DULWPHWLFH� FDOFXODW �GLQ�YDORULOH�

inverse ale termenilor aceleLDúL�VHULL�

∑=

=n

i i

h

x

nx

1

1

0HGLD�S WUDWLF

0HGLD� S WUDWLF � )( px � HVWH� DFHD� YDORDUH� FDUH� vQORFXLQG� WHUPHQLL� VHULHL� ULGLFD L� OD� S WUDW� QX�

PRGLILF �VXPD�S WUDWHORU�ORU�

n

xx

n

ii

p

∑== 1

2

0HGLD�JHRPHWULF

0HGLD�JHRPHWULF �� gx ��UHSUH]LQW �DFHD�YDORDUH�FX�FDUH��GDF �VH�vQORFXLHVF�WR L�WHUPHQLL�VHULHL�

úL�VH�IDFH�SURGXVXO�ORU��YDORDUHD�OD�FDUH�VH�DMXQJH�HVWH�HJDO �FX�SURGXVXO�WHUPHQLORU�UHDOL�

n

n

iig xx ∏

=

=1

3ULQ�ORJDULWPDUH�VH�RE LQH�

n

xx

n

ii

g

∑== 1

lglg

ÌQWUH�PHGLLOH�SUH]HQWDWH�H[LVW �XUP WRDUHD�UHOD LH�GH�RUGLQH�

pagh xxxx <<<

9DORUL�PHGLL�GH�SR]L LH�VDX�GH�VWUXFWXU Mediana (Me)� UHSUH]LQW � YDORDUHD� FHQWUDO � D� XQHL� VHULL� VWDWLVWLFH� RUGRQDWH� FUHVF WRU� VDX�

GHVFUHVF WRU�FDUH�vPSDUWH�WHUPHQLL�VHULHL�vQ�GRX �S U L�HJDOH�

Locul 2

1+= nMe unde n�UHSUH]LQW �QXP UXO�WHUPHQLORU�VHULHL�

Valoarea medianei: • GDF �QXP UXO�WHUPHQLORU�VHULHL�HVWH�LPSDU��n=2p+1): 1+= pxMe

• GDF �QXP UXO�WHUPHQLORU�HVWH�SDU��n=2p): 2

1++= pp xx

Me

Modul (Mo)�HVWH�YDORDUHD�FDUH�VH�UHSHW �GH�FHOH�PDL�PXOWH�RUL��PRWLY�SHQWUX�FDUH�PDL�HVWH�

FXQRVFXW�vQ�OLWHUDWXUD�GH�VSHFLDOLWDWH�úL�VXE�GHQXPLUHD�GH�dominanta seriei. ,QGLFDWRULL�YDULD LHL

a) ,QGLFDWRULL�VLPSOL�DL�YDULD LHL

Amplitudinea abVROXW (Aa��VH�FDOFXOHD] �FD�GLIHUHQ �vQWUH�QLYHOXO�PD[LP��xmax��úL�QLYHOXO�

minim (xmin) al caracteristicii: Aa = xmax - xmin

$PSOLWXGLQHD�UHODWLY �D�YDULD LHL��A%��VH�FDOFXOHD] �FD�UDSRUW�vQWUH�DPSOLWXGLQHD�DEVROXW �D�

YDULD LHL�úL�QLYHOXO�PHGLX�DO�FDUDFteristicii:

100% ⋅=x

AA a

Abaterile individuale absolute (di)� VH� FDOFXOHD] � FD� GLIHUHQ � vQWUH� ILHFDUH� YDULDQW �

vQUHJLVWUDW �úL�PHGLD�DULWPHWLF �D�DFHVWRUD�

nixxd ii ,1 =−= Abaterile individuale relative (di%)� VH� FDOFXOHD] � UDSRUWkQG� DEaterile absolute la nivelul

mediu al caracteristicii:

nix

xx

x

dd ii

i ,1 100100% =⋅−

=⋅=

ÌQ�DQDOL]D�YDULD LHL��GH�PXOWH�RUL��QH�OLPLW P�OD�D�FDOFXOD�DEDWHULOH�PD[LPH�vQWU-un sens sau altul.

b) ,QGLFDWRULL�VLQWHWLFL�DL�YDULD LHL

$EDWHUHD� PHGLH� OLQLDU )(d � VH� FDOFXOHD] � FD� R� PHGLH� DULWPHWLF � VLPSO � GLQ� DEDWHULOH�WHUPHQLORU�VHULHL�GH�OD�PHGLD�ORU��OXDWH�vQ�YDORDUH�DEVROXW �

n

xx

d

n

ii∑

=−

= 1

$EDWHUHD�PHGLH�S WUDWLF sau abaterea standard )(σ �VH�FDOFXOHD] �FD�R�PHGLH�S WUDWLF �din DEDWHULOH�WXWXURU�YDULDQWHORU�VHULHL�GH�OD�PHGLD�ORU�DULWPHWLF �

n

xxn

ii∑

=

−= 1

2)(σ

&DOFXOD L�SHQWUX�DFHHDúL�VHULH��FHL�GRL�LQGLFDWRUL�YHULILF �UHOD LD�

d>σ ÌQ�OLWHUDWXUD�GH�VSHFLDOLWDWH�VH�DSUHFLD] �F �SHQWUX�R�VHULH�GH�GLVWULEX LH�FX�WHQGLQ �FODU �GH�

QRUPDOLWDWH��DEDWHUHD�PHGLH�OLQLDU �HVWH�HJDO �FX�����GLQ�YDORDUHD�DEDWHULL�PHGLL�S WUDWLFH� Dispersia� �YDULDQ D�� XQHL� FDUDFWHULVWLFL� )( 2σ � VH� FDOFXOHD] � FD� PHGLH� DULWPHWLF � VLPSO � D�

S WUDWHORU�DEDWHULORU�WHUPHQLORU�ID �GH�PHGLD�ORU�� Formulele de calcul sunt:

n

xxn

ii∑

=

−= 1

2

2

)(σ sau

2

11

2

2

−=∑∑

==

n

x

n

xn

ii

n

ii

σ

&RHILFLHQWXO�GH�YDULD LH (v��VH�FDOFXOHD] �FD�UDSRUW�vQWUH�DEDWHUHD�PHGLH�S WUDWLF �úL�QLYHOXO�

mediu al seriei:

100⋅=x

vσ

Coeficientul GH�YDULD LH�VH�PDL�SRDWH�FDOFXOD�úL�GXS �UHOD LD�

100⋅=′x

dv

6(5,,�'(�',675,%8 ,(�81,',0(16,21$/(��GDWH�JUXSDWH� ,QGLFDWRUL�GH�IUHFYHQ H

)UHFYHQ HOH� UHODWLYH ( *in �� VH� RE LQ� UDSRUWkQG� IUHFYHQ D� ILHF UHL� JUXSH� �ni) la totalul

IUHFYHQ HORU� )(1

∑=

k

iin :

∑=

=k

ii

ii

n

nn

1

* sau 100

1

*(%) ⋅=

∑=

k

ii

ii

n

nn

)UHFYHQ HOH�FXPXODWH�VH�QRWHD] �FX� iF sau *iF �vQ�IXQF LH�GH�IHOXO�IUHFYHQ HORU�LQFOXVH�vQ�

calcul (absolute sau relative):

;,...,3,2,1 ;1

kinFi

jji == ∑

=

respectiv ;,...3,2,1 ; 1

** kinFi

jji == ∑

=

0 ULPL�PHGLL 0HGLD�DULWPHWLF &DOFXOXO�FX�IUHFYHQ H�DEVROXWH�

∑

∑

=

==k

ii

k

iii

n

nx

x

1

1

3URSULHW L� a) PHGLD� VH� vQFDGUHD] � vQ� LQWHUYDOXO� FX� IUHFYHQ D� PD[LP � VDX� vQ� XQXO� GLQ� FHOH� GRX �

intervale învecinate.

b) 0)(1

=−∑=

k

iii nxx

c) GDF � axx i #=′ , atunci ax

n

nax

xk

ii

i

k

ii

#

#

==′

∑

∑

=

=

1

1

)(

d) GDFh

xx i=′′ , atunci

1

1

h

x

n

nh

x

xk

ii

n

ii

i

=⋅

=′′

∑

∑

=

=

UHVSHFWLY�GDF � hxx i ⋅=′′ , atunci xhn

nhx

xk

ii

n

iii

⋅=⋅⋅

=′′

∑

∑

=

=

1

1

)(

3URSULHW LOH� GH� OD� SXQFWHOH� F�� úL� G�� VHUYHVF� OD� calculul simplificat al mediei aritmetice. 6H�XWLOL]HD] �vQ�DFHVW�VFRS�IRUPXOHOH�

( )a

n

nax

xk

ii

i

k

ii

±=

∑

∑

=

=

1

1

#

h

n

nh

x

xk

ii

i

k

i

i

⋅

=

∑

∑

=

=

1

1 , respectiv

( )h

n

nhx

xk

ii

i

k

ii

÷⋅

=

∑

∑

=

=

1

1

ah

n

nh

ax

xk

ii

i

k

i

i

+⋅

−

=

∑

∑

=

=

1

1 ,

unde: a - mijlocul unui intHUYDO�GH�RELFHL�FHQWUXO�LQWHUYDOXOXL�FX�IUHFYHQ D�FHD�PDL�PDUH� h -�P ULPHD�LQWHUYDOXOXL�GDF �VHULD�DUH�LQWHUYDOH�GH�YDULD LH�HJDOH� e) GDF �vQWU-R�VHULH�VH�UHGXF�SURSRU LRQDO�WRDWH�IUHFYHQ HOH��PHGLD�FDOFXODW �SH�ED]D�QRLORU�

IUHFYHQ H�U PkQH�QHVFKLPEDW �

x

c

nc

nx

i

ii

=∑

∑

$FHDVW � SURSULHWDWH� VHUYHúWH� OD� FDOFXOXO� PHGLHL� FX� DMXWRUXO� IUHFYHQ HORU� UHODWLYH. În acest caz ∑= inc :

∑= *ii nxx sau

100

*(%)∑= ii nx

x

0HGLD�DUPRQLF

• &DOFXOXO�FX�IUHFYHQ H�DEVROXWH�

∑

∑

=

==k

ii

i

k

ii

h

nx

nx

1

1

1

• &DOFXOXO�FX�IUHFYHQ H�UHODWLYH�

∑=

=k

ii

i

h

nx

x

1

*11

sau

∑=

=k

ii

i

h

nx

x

1

*(%)

1100

)RUP �WUDQVIRUPDW �D�PHGLHL�DULWPHWLFH • &DOFXOXO�FX�IUHFYHQ H�DEVROXWH�

xnx

x

nxx

k

iii

i

k

iii

h ==∑

∑

=

=

1

1

1

• &DOFXOXO�FX�IUHFYHQ H�UHODWLYH�

∑=

=k

iii

i

h

nxx

x

1

*11

sau

∑=

=k

iii

i

h

nxx

x

1

*(%)

1100

0HGLD�S WUDWLF

• &DOFXOXO�FX�IUHFYHQ H�DEVROXWH�

∑

∑

=

==k

ii

i

k

ii

p

n

nxx

1

1

2

• &DOFXOXO�FX�IUHFYHQ H�UHODWLYH�

*

1

2i

k

iip nxx ∑

=

= sau 100

*(%)

1

2i

k

ii

p

nxx

∑==

0HGLD�JHRPHWULF

1

1

∑= = ∏=

k

ii

in k

i

nig xx

Prin lRJDULWPDUH�VH�YD�RE LQH�

∑

∑

=

=⋅

=k

ii

k

iii

g

n

xnx

1

1

lglg

ÌQWUH�PHGLLOH�SUH]HQWDWH�H[LVW �XUP WRDUH�UHOD LH�GH�RUGLQH�

pagh xxxx <<<

9DORUL�PHGLL�GH�SR]L LH�VDX�GH�VWUXFWXU Modul (modulul, dominanta)

ÌQ�FD]XO�XQHL�VHULL�GH�GLVWULEX LH�SH�YDULDQWH��PRGXO�HVWH�YDULDQWD�FX�IUHFYHQ D�PD[LP � ÌQ�FD]XO�JUXS ULL�SH�LQWHUYDOH��ORFXO�PRGXOXL�HVWH�LQWHUYDOXO�FX�IUHFYHQ D�PD[LP �LDU�YDORDUHD�

VH�FDOFXOHD] �DVWIHO�

21

10 ∆+∆

∆+= hxMo

în care:

x0 - UHSUH]LQW �OLPLWD�LQIHULRDU �D�LQWHUYDOXOXL�PRGDO� h - P Uimea intervalului modal;

1∆ - GLIHUHQ D�GLQWUH�IUHFYHQ D�LQWHUYDOXOXL�PRGDO�úL�D�FHOXL�SUHFHGHQW�

2∆ - GLIHUHQ D�GLQWUH�IUHFYHQ D�LQWHUYDOXOXL�PRGDO�úL�D�FHOXL�XUP WRU�

Mediana (Me) a) În cazul datelor grupate pH� YDULDQWH�� ORFXO� PHGLDQHL� HVWH� YDULDQWD� D� F UHL� IUHFYHQ �

FXPXODW � HVWH� SULPD� PDL� PDUH� GHFkW�2

1+n iar valoarea medianei este chiar varianta

UHVSHFWLY �

b) ÌQ� FD]XO� GDWHORU� JUXSDWH� SH� LQWHUYDOH�� ORFXO�PHGLDQHL� HVWH� LQWHUYDOXO� D� F UXL� IUHFYHQ �

FXPXODW �HVWH�SULPD�PDL�PDUH�GHFkW�2

11

∑=

+k

iin

��LDU�YDORDUHD�PHGLDQHL�VH�FDOFXOHD] �GXS �

formula:

m

m

ii

k

ii

n

nn

hxMe∑

∑ −

=

= −+

⋅+=

1

1

1

0

2

1

,

unde: h - P ULPHD�LQWHUYDOXOXL�PHGLDQ� m - indexul intervalului median;

∑−

=

1

1

m

iin

- VXPD� IUHFYHQ HORU� SUHFHGHQWH� LQWHUYDOXOXL� PHGLDQ� �IUHFYHQ D� FXPXODW �a intervalului precedent celui median);

nm - IUHFYHQ D�DEVROXW �D�LQWHUYDOXOXL�PHGLDQ� Cuartilele VXQW�DFHOH�YDORUL�DOH�FDUDFWHULVWLFLL��FDUH�VHSDU �VHULD�vQ�SDWUX�S U L�HJDOH�

u

u

ii

k

ii

n

nn

hxQ∑∑

−

==

−++=

1

1101

)1(4

1

,

unde u�HVWH�LQGH[XO�LQWHUYDOXOXL�FDUH�FRQ LQH�Q1

MeQ =2

v

v

ii

k

ii

n

nnhxQ

∑∑−

==

−++=

1

1103

)1(4

3

,

unde v�HVWH�LQGH[XO�LQWHUYDOXOXL�FDUH�FRQ LQH�Q3

Decilele GLYLG�VHULD�vQ�]HFH�S U L�HJDOH�IRORVLQG�vQ�DFHVW�VFRS�QRX �GHFLOH

1

11

1101

)1(10

1

d

d

ii

k

ii

n

nnhxD

∑∑−

==−+

+= ,

unde d1�HVWH�LQGH[XO�LQWHUYDOXOXL�FDUH�FRQ LQH�D1

eMD =5

9

19

1109

)1(10

9

d

d

ii

k

ii

n

nnhxD

∑∑−

==

−++= ,

unde d9 HVWH�LQGH[XO�LQWHUYDOXOXL�FDUH�FRQ LQH�D9.

,QGLFDWRULL�YDULD LHL

a) ,QGLFDWRULL�VLPSOL�DL�YDULD LHL

$PSOLWXGLQHD�DEVROXW (Aa) Aa=xL-xl ,

unde: xL – liPLWD�VXSHULRDU �D�XOWLPXOXL�LQWHUYDO� xl –�OLPLWD�LQIHULRDU �D�SULPXOXL�LQWHUYDO� $PSOLWXGLQHD�UHODWLY �D�YDULD LHL��A%)

100% ⋅=x

AA a

b) ,QGLFDWRULL�VLQWHWLFL�DL�YDULD LHL

$EDWHUHD�PHGLH�OLQLDU )(d • &DOFXOXO�FX�IUHFYHQ H�DEVRlute:

∑

∑

=

=

−=

k

ii

i

k

ii

n

nxx

d

1

1

• &DOFXOXO�FX�IUHFYHQ H�UHODWLYH�

*

1i

k

ii nxxd ∑

=−= sau

100

(%)*

1

i

k

ii nxx

d∑

=

−=

$EDWHUHD�PHGLH�S WUDWLF sau abaterea standard )(σ :

¾�&DOFXOXO�FX�IUHFYHQ H�DEVROXWH�

∑

∑

=

=

−=

k

ii

i

k

ii

n

nxx

1

1

2)(σ

¾�Calculul cu fUHFYHQ H�UHODWLYH�

i

k

ii nxx *

1

2)(∑=

−=σ sau 100

)( (%)1

2i

k

ii nxx∑

=

−=σ

&HL�GRL�LQGLFDWRUL�YHULILF �UHOD LD�

d>σ

&RHILFLHQWXO�GH�YDULD LH (v):

100⋅=x

vσ

, respectiv 100⋅=′x

dv

Dispersia (variaQ D� )( 2σ : • IRUPXOH�GHULYDWH�GLQ�GHILQL LH

¾�FDOFXOXO�FX�IUHFYHQ H�DEVROXWH�

∑

∑

=

=

−=

k

ii

i

k

ii

n

nxx

1

1

2

2

)(σ

¾�FDOFXOXO�FX�IUHFYHQ H�UHODWLYH�

i

k

ii nxx *

1

22 )(∑=

−=σ sau 100

)( (%)*

1

2

2i

k

ii nxx∑

=

−=σ

• ca moment centrat de ordinul doi ¾�calculul cu freFYHQ H�DEVROXWH�

2

1

1

1

1

2

2

−=∑

∑

∑

∑

=

=

=

=k

ii

k

iii

k

ii

i

k

ii

n

nx

n

nxσ

¾�FDOFXOXO�FX�IUHFYHQ H�UHODWLYH�

2

1

**

1

22

−= ∑∑

==

k

iiii

k

i

i nxnxσ sau

2

1

*(%)

*(%)

1

2

2

100100

−=∑∑

==

k

iiii

k

ii nxnx

σ

• prin formula de calcul simplificat ¾�SHQWUX�R�VHULH�GH�IUHFYHQ H�DEVROXWH�

22

1

1

2

2 )( axhn

nh

ax

k

ii

k

ii

i

−−⋅

−

=∑

∑

=

=σ

¾�pentru o serie cu frecven H�UHODWLYH�

22

1

*

2

2 )( axhnh

axk

ii

i −−⋅

−

= ∑=

σ sau

221

*(%)

2

2 )(100

axh

nh

axk

ii

i

−−⋅

−

=∑

=σ

1RW � a�úL�h�DX�DFHOHDúL�VHPQLILFD LL�FD�OD�FDOFXOXO�PHGLHL�DULWPHWLFH� &RUHF LD�OXL�6KHSSDUG�

12)(

222 h−=′ σσ ,

unde h �HVWH�P ULPHD�LQWHUYDOXOXL�GH�JUXSDUH� $FHDVW �FRUHF LH�VH�DSOLF �QXPDL�SHQWUX�VHULLOH�FDUH�SUH]LQW �XUP WRDUHOH�SURSULHW L� • UHSDUWL LD�HVWH�QRUPDO �VDX�XúRU�DVLPHWULF � • UHSDUWL LD�DUH�LQWHUYDOH�GH�JUXSDUH�HJDOH�

9DULD LD�LQWHUFXDUWLOLF �úL�LQWHUGHFLOLF $EDWHUHD�LQWHUFXDWLOLF ��Qd):

22

)()( 1331 QQMeQQMeQd

−=

−+−=

CoeILFLHQWXO�GH�YDULD LH�LQWHUFXDUWLOLF ��9T�:

Me

QQ

Me

QQ

Me

QV d

q 22 13

13

−=

−

==

$EDWHUHD�LQWHUGHFLOLF :

22

)()( 1991 DDMeDDMeDd

−=

−+−=

&RHILFLHQWXO�GH�YDULD LH�LQWHUGHFLOLF :

Me

DD

Me

DD

Me

DV d

d 22 19

19

−=

−

==

0HGLD�úL�GLVSHUVLD�FDUDFWHULVWLFLL�DOWHUQDWLYH 'LVWULEX LD�GH�IUHFYHQ H�D�FDUDFWHULVWLFLL�DOWHUQDWLYH�VH�SUH]LQW �vQWU-un tabel de forma:

Tabel 3.1.

Valoarea

caracteristicii 5 VSXQVXO�

înregistrat )UHFYHQ H�DEVROXWH )UHFYHQ H�UHODWLYH

0 1 2 3

x1 = 1 DA

M �QXP UXO�XQLW LORU�

FDUH�SRVHG �caracteristica)

N

Mp =

x2 = 0 NU

(N-M) �QXP UXO�GH�XQLW L�FDUH�QX�SRVHG �caracteristica)

p1N

MNq −=−=

Total N = M + (N - M) p + q = 1

Media:

N

Mp =

Dispersia:

)1(sau 2p

2p ppqp −⋅=⋅= σσ

$EDWHUHD�PHGLH�S WUDWLF :

qpp ⋅=σ

Asimetria

$VLPHWULD�DEVROXW ( As ) MoxAs −=

&RHILFLHQ LL�GH�DVLPHWULH�SURSXúL�GH�.DUO�3HDUVRQ��SHQWUX�VHULL�GH�GLVWULEX LH�XúRU�DVLPHWULFH��

⇒<

⇒>

⇒=−=

0

0

0

as

as

as

as

C

C

C

MoxC

σ

Acest coeficient poate lua valori cuprinse între -�� úL� +1; cu cât este mai mic în valoare

DEVROXW �FX�DWkW�DVLPHWULD�HVWH�PDL�PLF � ÌQ�FD]XO� FkQG�VH�FXQRDúWH�PHGLDQD�VHULHL�� FRHILFLHQWXO�GH�DVLPHWULH� )( asC ′ se poate calcula

XWLOL]kQG�UHOD LD�

σ)(3 Mex

Cas

−=′

3HDUVRQ�PDL�SURSXQH�úL�XQ�Dlt coeficient pentru calculul gradului de asimetrie al unei serii IRUPDW �GLQWU-XQ�QXP U�IRDUWH�PDUH�GH�REVHUYD LL�

=−

=

−=

=′′

∑∑

∑∑

22

2

3

3

32

23

)(

)(

unde ,

σµ

µ

µ

µ

i

ii

i

ii

as

n

nxx

n

nxx

C

Coeficientul propus de Yule:

)()(

)()(

13

13

QMeMeQ

QMeMeQCasY −+−

−−−= ,

unde [ ]1,1−∈asYC

Coeficientul propus de Bowley:

)()(

)()(

19

19

DMeMeD

DMeMeDCasB −+−

−−−= ,

unde [ ]1,1−∈asBC

,QGLFDWRULL�FRQFHQWU ULL Coeficientul de concentrare propus de statisticianul italian Corado Gini:

kigC iG ,1 2 == ∑ ,

unde :

1

∑=

=k

i

ii

iii

nx

nxg

VHULH�VLPHWULF

DVLPHWULH�SR]LWLY

DVLPHWULH�QHJDWLY

Acest coeficient ia valori în intervalul

1,

1

n.

Coeficientul de concentrare propus de R. Struck:

1

12

−−

= ∑n

gnC i

S

Acest coeficient ia valori în intervalul [ ]1,0 .

6(5,,�'(�',675,%8 ,(�%,',0(16,1$/(

$��&DOFXOXO�FX�IUHFYHQ H�DEVROXWH

O serie de distrLEX LH�ELGLPHQVLRQDO �VH�SUH]LQW �vQWU-un tabel de forma:

Tabel 3.2.

Valorile caracteristicii de grupare X

Variantele sau valorile caracteristicii dependente Y

Volumul grupei

(ni.)

Medii pe grupe

)y( i

y1 y2 … yj … ym x1 n11 n12 ... n1j … n1m n1. )y( 1

x2 n21 n22 … n2j … n2m n2. )y( 2

... ... ... … ... … ... ... ... xi ni1 ni2 … nij … nim ni. )y( i

... ... ... … ... … ... ... ... xr nr1 nr2 … nrj … nrm nr. )y( r

Total n.1 n.2 … n.j … n.m ∑=∑==

m

1jj

r

1ii nn ..

y

9ROXPXO��IUHFYHQ D��JUXSHL�i:

.1

i

m

jij nn =∑

=

0 ULPL�PHGLL

• 0HGLLOH�GH�JUXS ( iy ):

∑

∑

=

==m

jij

m

jijj

i

n

ny

y

1

1

• Media pe total:

∑

∑

∑

∑

=

=

=

===

r

ii

r

iii

m

jj

m

jjj

n

ny

y

n

ny

y

1.

1.

1.

1.

sau

,QGLFDWRULL�YDULD LHL

• 'LVSHUVLD�GH�JUXS sau GLVSHUVLD�SDU LDO )( 2iσ :

∑

∑

=

=

−=

m

jij

m

iijij

i

n

nyy

1

1

2

2

)(σ ,

unde: yj �UHSUH]LQW �YDULDQWD�VDX�PLMORFXO�LQWHUYDOXOXL�j al caracteristicii dependente;

iy media grupei i; nij IUHFYHQ HOH�FRUHVSXQ] WRDUH�ILHF UHL�YDULDQWH��LQWHUYDO�GH�YDORUL��GLQ�FDGUXO�JUXSei.

• 0HGLD�GLVSHUVLLORU�GH�JUXS )( 2

/

2

ryσσ =

∑

∑

=

==r

ii

i

r

ii

n

n

1.

.1

2

2

σσ ,

unde: 2iσ - dispersia grupei i;

.in - volumul grupei i.

• Dispersia dintre grupe )( 2

/

2

xyσδ = :

∑

∑

=

=−

==r

ii

r

iii

xy

n

nyy

1.

1.

2

22/

)(

δσ

• DisSHUVLD�WRWDO )( 22

yσσ = :

∑

∑

=

=−

==m

jj

m

jjj

y

n

nyy

1.

1.

2

22

)(

σσ

5HJXOD�DGXQ ULL�GLVSHUVLLORU:

2/

2/

2ryxy σσσ +=

Pe baza regulii de adunare a dispersiilor se pot calcula indicatori statistici cu caracter de P ULPL�UHODWLYH�GH�VWUXFWXU � • Gradul de dHWHUPLQD LH )( 2

/ xyR :

1002

2/2

/ ⋅=σ

σ xyxyR

'DF � )R( x/y !�����DGPLWHP�F �IDFWRUXO�GH�JUXSDUH�HVWH�KRW UkWRU��VHPQLILFDWLY��GHWHUPLQDQW��

SHQWUX�YDULD LD�IDFWRUXOXL�GHWHUPLQDW��Y). • *UDGXO�GH�QHGHWHUPLQD LH:

1002

2/2

/ ⋅=σ

σ ryxyK

$EDWHUHD�PHGLH�S WUDWLF �OD�QLYHOXO�JUXSHL:

2ii σσ =

$EDWHUHD�PHGLH�S WUDWLF �SH�WRWDO:

2σσ =

&RHILFLHQWXO�GH�YDULD LH�OD�QLYHOXO�JUXSHL�

100⋅=i

ii y

vσ

,

unde:

iσ -�DEDWHUHD�PHGLH�S WUDWLF �D�JUXSHL�

iy - media grupei. &RHILFLHQWXO�GH�YDULD LH�SH�WRWDO�

100⋅=y

vσ

,

unde: σ -�DEDWHUHD�PHGLH�S WUDWLF �SH�WRWDO� y - media pe total.

%��&DOFXOXO�FX�IUHFYHQ H�UHODWLve 2�VHULH�GH�GLVWULEX LH�ELGLPHQVLRQDO �VH�SUH]LQW �vQWU-un tabel de forma:

Tabel 3.3.

Valorile caracteristicii

de grupare X )UHFYHQ H�UHODWLYH���� Total

(%) Ponderea grupei

(ni(%)) y1 y2 … yj … ym

x1 n*11 n*12 ... n*1j … n*1m 100 n1(%) x2 n*21 n*22 ... n*2j … n*2m 100 n2(%) ... ... ... … ... … ... ... ... xi n*i1 n*i2 ... n*ij … n*im 100 ni(%) ... ... ... … ... … ... ... ... xr n*r1 n*r2 ... n*rj … n*rm 100 nr(%)

Total 100

• 0HGLLOH�GH�JUXS ( iy ):

1001

*∑==

m

j

ijj

i

ny

y

• Media pe total:

100

1(%)∑

==

r

iii ny

y

• 'LVSHUVLD�GH�JUXS �VDX�GLVSHUVLD�SDU LDO )( 2

iσ :

100

)(1

*2

2∑=

−=

m

i

ijij

i

nyy

σ

• 0HGLD�GLVSHUVLLORU�GH�JUXS )( 2

/

2

ryσσ =

100

(%)1

2

2

i

r

ii n∑

==

σ

σ

• Dispersia dintre grupe )( 2

/

2

xyσδ = :

100

)(1

(%)2

22/

∑=

−

==

r

iii

xy

nyy

δσ

'LVSHUVLD�WRWDO )( 22yσσ = :

2222 δσσσ +== y

,QGLFDWRULL�PHGLL�úL�DL�YDULD LHL�SHQWUX�FDUDFWHULVWLFL�DOWHUQDWLYH 'LVSHUVLD�GH�JUXS ( 2

ipσ ):

)1(sau 22iipiip ppqp

ii−== σσ ,

în care: pi -�UHSUH]LQW �PHGLL�GH�JUXS � qi -�IUHFYHQ HOH�UHODWLYH�DOH�XQLW LORU�FDUH�QX�SRVHG �FDUDFWHULVWLFD�vQ�ILHFDUH�JUXS �

0HGLD�GLVSHUVLLORU�SDU LDOH )( 2

pσ :

∑

∑

=

==r

ii

r

iip

p

N

Ni

1

1

2

2

σσ ,

în care Ni�UHSUH]LQW �QXP UXO�WRWDO�DO�XQLW LORU�REVHUYDWH�vQ�ILHFDUH�JUXS � Dispersia dintre grupe )( 2

pδ :

∑

∑

=

=

−=

r

ii

r

iii

p

N

Npp

1

1

2

2

)(

δ ,

în care p este media caracteristicii alternative pe întreaga colectivitate. 'LVSHUVLD�WRWDO )( 2

pσ :

qpp ⋅=2σ

5HJXOD�DGXQ ULL dispersiilor:

222ppp δσσ +=

9HULILFDUHD�VHPQLILFD LHL�IDFWRUXOXL�GH�JUXSDUH�IRORVLQG�WHVWXO�³)´

2/

2/

ry

xycalculat S

SF = ,

unde:

( )1

1.

2

2/ −

⋅−=

∑=

r

nyyS

r

iii

xy (r -�QXP UXO�GH�JUXSH�

( )rn

nyy

S

r

i

m

jijij

ry −

⋅−=

∑∑= =1 1

2

2/

'DF �)calculat> Ftabelar factorul de grupare este semnificativ 'DF �)calculat< Ftabelar factorul de grupare nu este semnificativ Ftabelar� VH� GHWHUPLQ � vQ� IXQF ie de un anumit nivel de semnifica LH� �GH� H[HPSOX�� ������ úL� GH�

gradele de libertate f1=r-1�úL�f2=n-r.

PROBLEME REZOLVATE

1. Într-un magazLQ�OXFUHD] ����YkQ] WRUL��FDUH�vQ�OXQD�RFWRPEULH������DX�UHDOL]DW�XQ�YROXP�DO�

desfacerilor (mil. lei) astfel: 138,8; 146,0; 150,0; 152,3; 158,2; 163,1; 165,0; 170,4; 176,2; 180,0. Se cere: 1. V �VH�IRUPH]H�VHULD�SULYLQG�YROXPXO�GHVIDFHULORU�SULQ�FHQWUDOL]DUea datelor individuale; 2. V � VH� FDOFXOH]H� YROXPXO� PHGLX� DO� GHVIDFHULORU� GLQ� DFHVW� PDJD]LQ� IRORVLQG� PHGLD�

DULWPHWLF �úL�V �VH�YHULILFH�SULQFLSDOHOH�SURSULHW L�DOH�DFHVWHLD� 3. V �VH�IRORVHDVF �SHQWUX�DFHHDúL�VHULH�úL�DOWH�WLSXUL�GH�PHGLL�úL�V �VH�DUDWH��vQ�FH�Uaport de

P ULPH�VH�DIO �HOH�ID �GH�PHGLD�DULWPHWLF � 4. V �VH�FDOFXOH]H�LQGLFDWRULL�PHGLL�GH�SR]L LH��PHGLL�GH�VWUXFWXU �� 5. V � VH� FDUDFWHUL]H]H� JUDGXO� GH� YDULD LH� DO� GLVWULEX LHL� FDOFXOkQG� vQ� DFHVW� VFRS� LQGLFDWRULL�

VLPSOL�úL�VLQWHWLFL�DL�YDULD LHL��SUHFXP�úL�gradul de asimetrie; 6. V �VH�UHSUH]LQWH�JUDILF�VHULD� Rezolvare

1. 6HULD�VH�IRUPHD] �SULQ�RUGRQDUHD�YDORULORU��YH]L�WDEHOXO������FRORDQHOH���úL����

Notând cu xi termenii individuali, volumul centralizat este ∑=

n

iix

1

.

∑=

10

1iix = 138,8 +146,0 +150,0 +....+180,0 = 1600 mil. lei

2. &DOFXOXO�PHGLHL�DULWPHWLFH�úL�YHULILFDUHD�SULQFLSDOHORU�SURSULHW L�DOH�DFHVWHLD se poate

face cu ajutorul unui tabel de calcul (Vezi tabelul 3.4.).

&DOFXOXO�PHGLHL�DULWPHWLFH�VLPSOH�úL�YHULILFDUHD�SULQFLSDOHORU�SURSULHW L Tabelul 3.4.

Nr.crt. Volumul

desfacerilor (mil. lei)

xxi − axi −

a = 138,8 h

xi ; h = 20

A 1 2 3 4 1 138,8 -21,2 0 6,94 2 146,0 -14,0 7,2 7,3 3 150,0 -10,0 11,2 7,5 4 152,3 -7,7 13,5 7,615 5 158,2 -1,8 19,4 7,91 6 163,1 +3,1 24,3 8,155 7 165,0 +5,0 26,2 8,25 8 170,4 +10,4 31,6 8,52 9 176,2 +16,2 37,4 8,01 10 180,0 +20,0 41,2 9,0 Total 1600,0

∑=

10

1iix

0

)(10

1

xxi

i −∑=

212

)(10

1

axi

i −∑=

80

∑=

10

1i

i

h

x

2.1. Calculul mediei aritmetice simple (vezi coloana 1 din tabelul 3.4.):

16010

1600 === ∑n

xx i �PLO��OHL�YkQ] WRU�

2.2. 9HULILFDUHD�SULQFLSDOHORU�SURSULHW L�DOH�PHGLHL�DULWPHWLFH

a) maxmin xxx <<

1808,138 << x

b) 0)( =−∑ xxi (vezi tabelul 3.4., coloana 2)

Media fiind un cât exact, suma abaterilor pozitive s-a compensat cu suma abaterilor negative:(-54,7) + (54,7) = 0

c) ∑=

⋅=n

ii xnx

1

1600160,010 =⋅ mil. lei

d) axn

axx

n

ii

−=−

=′∑

=1

)(

Alegând arbitrar a = 138,8 - primul termen -�úL�FDOFXOkQG�YDORULOH�xi - a (vezi tabelul ������VH�RE LQH�

2,2110

212)(

1 ==−

=′∑

=

n

axx

n

ii

mil. OHL�YkQ] WRU

deci: axx =′−

160-21,2=138,8 de unde:

axx +′= =x ����������� ����PLO��OHL�YkQ] WRU

e) h

x

n

h

xi

=

∑

/XkQG�DUELWUDU�XQ�K� �����YH]L�WDEHOXO�������FRORDQD����VH�RE LQH�

810

80 ==

=′′∑

n

h

x

x

i

hxx ⋅′′= 160208 =⋅=x �PLO��OHL�YkQ] WRU

3. &DOFXOXO�PHGLHL�S WUDWLFH��DO�PHGLHL�DUPRQLFH�úL�DO�PHGLHL�JHRPHWULFH 3.1.�&DOFXOXO�PHGLHL�S WUDWLFH ( )px (vezi tabelul 3.5., coloana 3):

5,16010

18,25761310

2

== ∑n ip n

xx PLO��OHL�YkQ] WRU

;xx p > 160,5 > 160

&DOFXOXO�PHGLHL�DUPRQLFH��S WUDWLFH�úL�JHRPHWULFH

Tabelul 3.5.

Nr. crt.

Volumul desfacerilor

(mil. lei) ix

1 2

ix ixlg

0 1 2 3 4 1 138,8 0,0072046 19265,44 2,14239

2 146,0 0,0068000 21316,00 2,16425

3 150,0 0,0066670 22500,00 2,17605

4 152,3 0,0065659 23195,29 2,18270

5 158,2 0,0063211 25027,24 2,19921

6 163,1 0,0061312 26601,61 2,21245

7 165,0 0,0060606 27225,00 2,21748

8 170,4 0,0058685 29036,16 2,23045

9 176,2 0,0056753 31046,44 2,24601

10 180,0 0,0055550 32400,0 2,25527

Total 1600,0 0,0628986 257613,18 22,02640

3.2. Calculul mediei armonice ( hx ) (vezi tabelul 3.5., coloana 2)

98,1580628986,0

101

===∑

i

h

x

nx �PLO�OHL�YkQ] WRU

;xxh < 158,98 < 160,0

Deci: ph xxx <<

3.3. Calculul mediei geometrice ( gx ) (vezi tabelul 3.5., coloana 4):

n

n

iig xx ∏

=

=1

; n

xx

n

ii

g

∑== 1

lglg

20264,210

02640,22lg =gx

antilog 2,20264 = 159,5 => gx =�������PLO�OHL��YkQ] WRU

Deci: ag xx < ; 159,5 < 160

5HFDSLWXOkQG�YDORULOH�PHGLLORU�RE LQXWH�DYHP�

0,160== xxa PLO�OHL�YkQ] WRU� px =160,5 PLO�OHL�YkQ] WRU�

hx = 158,98 PLO�OHL�YkQ] WRU� gx = 159,50 PLO�OHL�YkQ] WRU

5HOD LD�GLQWUH�PHGLL�HVWH�

pagh xxxx <<<

158,98 < 159,5 < 160 < 160.5

$EDWHULOH�vQWUH�FHOH�SDWUX�P ULPL�PHGLL�ILLQG�IRDUWH�PLFL��VH�SRDWH�WUDJH�FRQFOX]LD�F �SHQWUX�DFHDVW �VHULH�RULFH�PHGLH�V-DU�IRORVL��HD�P VRDU �VXILFLHQW�GH�FRUHFW�WHQGLQ D�FHQWUDO �úL�GHFL�PHGLD�vQ�DFHVW�FD]�SRDWH�IL�R�YDORDUH�WLSLF �

4. &DOFXOXO�LQGLFDWRULORU�PHGLL�GH�SR]L LH 4.1. Calculul medianei (Me)

• locul medianei: 5,52

110

2

1)( =+=+= n

MeU

• valoarea medianei:

65,1602

1,1632,158

265 =+=

+=

xxMe mil. lei

4.2. Calculul modului )LLQG�VHULH�VLPSO �úL�QHDYkQG�WHUPHQL�FDUH�V �VH�UHSHWH��QX�HVWH�FD]XO� 4.3. Calculul cuartilelor Calculul cuartilelor presupune stabilirea locului cuartilelor U(Q2��úL�DSRL�FDOFXODUHD�YDORULORU�lor:

75,24

11

4

1)( 1 ==+= n

QU

1482

150146

232

1 =+=+

=xx

Q mil. lei

65,1602 == eMQ mil. lei

25,84

)1(3)( 3 =+= n

QU

3,1732

2,1764,170

298

3 =+=+

=xx

Q mil. lei

Potrivit celor trei cuartile, seria se poate structura astfel:

6WUXFWXUD�VHULHL�vQ�IXQF LH�GH�YDORULOH�FHORU�WUHL�FXDUWLOH

Tabelul 3.6.

,QWHUYDO�GH�YDULD LH Structura seriei (%)

138,80-148,00 25

148,00-160,65 25

160,65-173,30 25

173,30-180,00 25

5. &DOFXOXO�LQGLFDWRULORU�GH�YDULD LH�úL�DVLPHWULH PenWUX�FDOFXOXO�LQGLFDWRULORU�GH�YDULD LH�VH�IRORVHúWH�WDEHOXO������

&DOFXOXO�LQGLFDWRULORU�GH�YDULD LH

Tabelul 3.7.

Nr. crt. ix xx i − ( )2xxi −

2ix

100⋅−x

xx i

0 1 2 3 4 5 1 138,8 21,2 449,44 19265,44 - 13,25

2 146,0 14,0 196,00 21316,00 - 8,75

3 150,0 10,0 100,00 22500,00 - 6,25

4 152,3 7,7 59,29 23195,24 - 4,81

5 158,2 1,8 3,24 25027,24 - 1,13

6 163,1 3,1 9,61 26601,61 1,94

7 165,0 5,0 25,0 27225,00 3,13

8 170,4 10,4 108,16 29036,16 6,50

9 176,2 16,2 262,44 31046,44 10,13

10 180,0 20,0 400,00 32400,00 12,50

1600,0

∑=

n

iix

1

109,4

∑ − xx i

1613,18

( )2

1∑

=

−n

ii xx

257613,18

5.1. ,QGLFDWRULL�VLPSOL�DL�YDULD LHL • $PSOLWXGLQHD�YDULD LHL

Aa = xmax - xmin =180-138,8 = 41,2 mil. lei

%75,25100160

2,41minmax% =⋅=

−=

x

xxA

• Abaterile absolute�DOH�ILHF UHL�YDULDELOH�ID �GH�PHGLH��di):

xxd ii −= (vezi tabelul 3.7., coloana 2)

• Abaterile relative (di(%))

100(%) ⋅−

=x

xxd i

i (vezi tabelul 3.7., coloana 5)

5.2. ,QGLFDWRULL�VLQWHWLFL�DL�YDULD LHL • $EDWHUHD�PHGLH�OLQLDU ( )xd �úL�VH�IRORVHVF�GDWHOH�GLQ�WDEHOXO�������FRORDQD��

94,1010

4,1091 ==−

=∑

=

n

xxd

n

ii

x PLO��OHL�YkQ] WRU

• Dispersia ( )2xσ VH�FDOFXOHD] �FX�GDWHOH�GLQ�WDEHOXO�������FRORDQD 3:

( )318,161

10

18,16131

2

2 ==−

=∑

=

n

xxn

ii

xσ

'LVSHUVLD�VH�PDL�SRDWH�FDOFXOD�úL�FX�IRUPXOD�

21

2

2 xn

xn

ii

x −=∑

=σ

Datele necesare VH�J VHVF�vQ�WDEHOXO�������FRORDQHOH ��úL���

318,16116010

18,257613 22 =−=xσ

Deci s-D�DMXQV�OD�DFHODúL�UH]XOWDW�

• $EDWHUHD�PHGLH�S WUDWLF ( )∑ −

=n

xxix

2

σ sH�FDOFXOHD] �H[WU JkQG�U G FLQD�S WUDW �

din dispersie:

70,12318,1612 === xx σσ PLO��OHL�YkQ] WRU�

• &RHILFLHQ LL�GH�YDULD LH (v’, v)

%84,6100160

94,10100 =⋅=⋅=′

x

dv x

%84,7100160

70,12100 =⋅=⋅=

xv xσ

v'�úL�v �������GHFL�VH�SRDWH�DILUPD�F �VHULD�DUH�XQ�JUDG�PDUH�GH�RPRJHQLWDWH�úL�PHGLD�HVWH�UHSUH]HQWDWLY �SHQWUX�vQWUHDJD�VHULH�

6. 5HSUH]HQWDUHD�JUDILF ÌQ�DFHVW�FD]��YDULD LD�QHILLQG�FRQWLQX ��VH�SRDWH�IRORVL�XQ�JUDILF�SULQ�FRORDQH�úL�R�GLDJUDP �

UDGLDO �FX�]HFH�UD]H��YH]L�ILJXULOH������úL�������

180176,2173,4163,1158,2

152,3150146138,8

165

0

20

40

60

80

100

120

140

160

180

200

1 2 3 4 5 6 7 8 9 10

)LJXUD������5HSDUWL LD�VDODULD LORU�vQ�IXQF LH�GH�YROXPXO�GHVIDFHULORU��PLO��OHL�

mil.

lei

020406080

100120140160180

1

2

3

4

5

6

7

8

9

10

)LJXUD������5HSDUWL LD�VDODULD LORU�vQ�IXQF LH de volumul desfacerilor (mil. lei)

2. 3HQWUX�R�VRFLHWDWH�FRPHUFLDO �VH�FXQRVF�XUP WRDUHOH�GDWH�SULYLWRU�OD�YHFKLPHD�vQ�SURGXF LH��

vQVFULVH�vQ�WDEHOXO�������GDWH�FRQYHQ LRQDOH��

5HSDUWL LD�VDODULD LORU�GLQWU-R�VHF LH�GXS �YHFKLPH Tabelul 3.8.

Grupe dH�VDODULD L�GXS �YHFKLPH��DQL�

Sub 5

5-10 10-15 15-20 20-25 25-30 30-35 ���úL�peste

1XP UXO�VDODULD LORU 10 40 60 80 50 30 20 10

Se cere: 1. UHSUH]HQWDUHD�JUDILF �D�VHULHL�IRORVLQG�IUHFYHQ HOH�UHDOH�úL�FHOH�FXPXODWH� 2. vechimea medie a muncitorilor prin caOFXOXO�PHGLHL� DULWPHWLFH� FX� IUHFYHQ H� DEVROXWH��

UHODWLYH�úL�SULQ�FDOFXOXO�VLPSOLILFDW��VSHFLILFkQG�FH�SURSULHW L�DOH�PHGLHL�DULWPHWLFH�VWDX�OD�ED]D�FDOFXOXOXL�PHGLHL�DULWPHWLFH�FX�IUHFYHQ H�UHODWLYH�úL�SULQ�FDOFXOXO�VLPSOLILFDW�

3. LQGLFDWRULL�PHGLL�GH�SR]L ie; 4. LQGLFDWRULL�VLPSOL�úL�VLQWHWLFL�DL�YDULD LHL� 5. indicatorii de asimetrie; 6. PHGLD�úL�GLVSHUVLD�SHQWUX�PXQFLWRULL cu o vechime mai mare decât vechimea medie. Rezolvare

1. 5HSUH]HQWDUHD�JUDILF �D�VHULHL

• +LVWRJUDPD�úL�SROLJRQXO�IUHFYHQ HORU��YH]L�ILJXUD 3.3.); • &XUED�FXPXODWLY �D�IUHFYHQ HORU��YH]L�ILJXUD�������

0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25 30 35 40 45

vechime (ani)

� ������ ��� ��

)LJXUD������'LVWULEX LD�VDODULD LORU�vQ�IXQF LH�GH�YHFKLPH

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30 35 40

vechime (ani)

QX

PU�

VDOD

ULD

L

)LJXUD������'LVWULEX LD�VDODULD LORU�vQ�IXQF LH�GH�YHFKLPH

2. Calculul vechimii medii În acest scop se vor folosi datele din tabelul 3.9.

Tabelul 3.9.

1XP UXO�VDODULD LORU

)UHFYHQ H absolute cumulate

Grupe de VDODULD L�GXS �

vechime (ani) abs. (ni) ( )

*%in

Centrul de

interval (xi)

ii nx ⋅ ⋅ix ( )*

%in h

axi −

a=17,5 h=5

ii nh

ax⋅

−

cres-F WRU

descres-F WRU

0 1 2 3 4 5 6 7 8 9 0-5

5-10 10-15 15-20 20-25 25-30 30-35 35-40

10 40 60 80 50 30 20 10

3,33 13,33 20,00 26,67 16,67 10,00 6,67 3,33

2,5 7,5

12,5 17,5 22,5 27,5 32,5 37,5

25 300 750

1400 1125 825 650 375

8,325 99,975

250,000 466,725 375,075 275,000 216,775 124,875

-3 -2 -1 0 1 2 3 4

-30 -80 -60

0 50 60 60 40

10 50

110 190 240 270 290 300

300 290 250 190 110

60 30 10

Total

300

∑=

k

iin

1

100

( )∑=

k

iin

1

*%

- 5450

∑=

k

iii nx

1

1816,75

∑=

k

iii nx

1

*(%)

-

40

∑=

⋅−k

ii

i nh

ax

1

- -

2.1. Calculul mediei aritmetice pe baza: • frecven HORU�DEVROXWH

2,18167.18300

5450

1

1 ≅===

∑

∑

=

=k

ii

k

iii

n

nx

x ani/salariat

• IUHFYHQ HORU�UHODWLYH:

( )2,18

1001

*%

≅=∑

=

k

iii nx

x ani/salariat

&DOFXOXO�PHGLHL�IRORVLQG�IUHFYHQWHOH�UHODWLYH�VH�ED]HD] �SH�SURSULHWDWHD�

;

1

1 x

c

nc

nx

k

i

i

k

i

ii

=∑

∑

=

= în acest caz ∑=

=k

iinc

1

2.2. Calculul simplificat al mediei aritmetice:

2,181667,185,175300

40

1

1 ≅=+⋅=+⋅

−

=

∑

∑

=

=ah

n

nh

ax

xk

ii

k

ii

i

ani/salariat

Calculul mediei prin acest procedeu s-D�ED]DW�SH�GRX �SURSULHW L�DOH�PHGLHL�DULWPHWLFH�úL�anume:

( )ax

n

naxx

k

ii

k

iii

−=⋅−

=′

∑

∑

=

=

1

1 , de unde:

( );

1

1 an

naxx

k

ii

k

iii

+⋅−

=∑

∑

=

=

h

x

n

nh

x

xk

ii

k

ii

i

==′′

∑

∑

=

=

1

1 , de unde: hn

nh

x

xk

ii

k

ii

i

⋅=∑

∑

=

=

1

1

&kQG� VXQW� LQWHUYDOH� HJDOH� HVWH� FRQYHQDELO� V se ia a egal cu centrul intervalului cu IUHFYHQ D�PD[LP ��DGLF �a ������úL�SHQWUX�h�R�YDORDUH�HJDO �FX�P ULPHD� LQWHUYDOHORU��DGLF �h = 5 (vezi tabelul 3.9.).

3. &DOFXOXO�LQGLFDWRULORU�PHGLL�GH�SR]L LH� 3.1.�&DOFXOXO�PHGLDQHL��0H��úL�D�FHORUODOWH�PHGLL�GH�SR]L LH: • FX�IUHFYHQ H�DEVROXWH�

5,1502

1300

2

1)( 1 =+=

+=

∑=

k

iin

MeU , deci 15 < Me < 20

5,1780

1105,150515

2

11

1

1

0 =−+=−

+

+=∑

∑ −

=

=

m

m

ii

k

ii

n

nn

hxMe ani

25,752

301

4

1)( 1

1 ==+

=∑

=

k

iin

QU , deci 10< Q1 <15

125,1260

5025,75510

14

1 1

1101 =−+=

−

+

+=∑∑

−

==

u

u

ii

k

ii

n

nn

hxQ ani

5,172 == eMQ ani

75,2253014

31

4

3)(

13 =⋅=

+= ∑

=

k

iinQU deci 20 < Q3 < 25

525,2350

19075,225520

14

3 1

1103 =−+=

−

+

+=∑∑

−

==

v

v

ii

k

ii

n

nn

hxQ ani

• FX�IUHFYHQ H�UHODWLYH�

( )

5,502

1100

2

1

)( 1

*%

=+=+

=∑=

k

ii

n

MeU , deci 15 < Me < 20

( )

( )

( )59,17

67,26

66,365,50515

2

1

*%

1

1

*1

*%

0

%

=−

+=−

+

+=∑

∑ −

=

=

m

m

i

k

ii

n

n

n

hxMei

ani

( )

25,252

101

4

1

)( 1

*

1

%

==+

=∑

=

k

ii

n

QU , deci 10 < Q1 < 15

( ) ( )

( )148,12

20

66,1625,25510

14

1

*%

1

1

*

1

*

01

%%

=−

+=

−

+

+=∑∑

−

==

u

u

i

k

i

n

nn

hxQii

ani

59,172 == eMQ ani

( ) 75,751014

31

4

3)(

1

*3 %

=⋅=

+= ∑

=

k

ii

nQU , deci 20 < Q3 < 25

( ) ( )

( )725,23

67,16

33,6375,75520

14

3

*%

1

1

*%

1

*%

03 =−

+=

−

+

+=∑∑

−

==

v

v

ii

k

ii

n

nn

hxQ ani

'LIHUHQ HOH�GLQWUH�LQGLFDWRULL�FDOFXOD L�FX�IUHFYHQ H�DEVROXWH�úL�FHL�FX�IUHFYHQ H�UHODWLYH�VXQW�

QHVHPQLILFDWLYH�úL�SURYLQ�GLQ�URWXQMLUHD�IUHFYHQ HORU�UHODWLYH� 3.2. Calculul modului (Mo): • FX�IUHFYHQ H�DEVROXWH�

Locul Mo�vO�FRQVWLWXLH�LQWHUYDOXO�FX�IUHFYHQ D�PD[LP �������Mo < 20.

173020

20515

21

10 =

++=

∆+∆∆

⋅+= hxMo ani

• cu frecven H�UHODWLYH�

27,17)67,1667,26()2067,26(

2067,26515

21

10 =

−+−−+=

∆+∆∆

⋅+= hxMo ani

9DORULOH�WHQGLQ HL�FHQWUDOH�FRQVLGHUDWH�FD�P ULPL�WLSLFH�FH�GHILQHVF�DFHDVW �VHULH��SHQWUX�

FDOFXOXO�FX�IUHFYHQ H�DEVROXWH��VXQW� x =18,2 ani; Me=17,5 ani; Mo=17 ani.

4. Calculul indicatorilor�GH�YDULD LH (vezi tabelul 3.10.). 4.1. Calculul indicatorilor simpli

ÌQ�DFHVW�FD]�QX�VXQW�FRQFOXGHQ L�

4.2. Calculul indicatorilor sintetici • $EDWHUHD�PHGLH�OLQLDU �� id ) �VH�RE LQH�GLQ�vQVXPDUHD�YDORULORU�GLQ�FRORDQD����YH]L�WDEHOXO�

3������I U �V �VH� LQ �VHDPD�GH�VHPQ�

5,6300

1956

1

1 ==⋅−

=∑

∑

=

=k

ii

k

iii

x

n

nxxd ani/salariat

• Dispersia ( 2xσ ):

- FDOFXOXO�RELúQXLW�

( )89,67

300

20367

1

1

2

2 ==⋅−

=∑

∑

=

=k

ii

k

iii

x

n

nxxσ

- calculul simplificat:

84,67)5,172,18(25300

820)( 222

1

1

2

2 =−−⋅=−−⋅⋅

−

=∑

∑

=

= axhn

nh

ax

k

ii

k

ii

i

xσ

• $EDWHUHD�PHGLH�S WUDWLF �� xσ )

2,884,672 === xx σσ ani/salariat

• &RHILFLHQ LL�GH�YDULD LH��v’�úL�v) - GDF �VH�SRUQHúWH�GH�OD�DEDWHUHD�PHGLH�OLQLDU �

%7,351002,18

5,6100 =⋅=⋅=′

x

dv x

- GDF �VH�SRUQHúWH�GH�OD�DEDWHUHD�PHGLH�S WUDWLF �

%451002,18

2,8100 =⋅=⋅=

xv xσ

Interpretând valoarea coeficientului de YDULD LH�VH�SRDWH�DILUPD�F �PHGLD�QX�HVWH�VXILFLHQW�GH�

UHSUH]HQWDWLY ��FD�XUPDUH�D�IDSWXOXL�F �VHULD�QX�HVWH�VXILFLHQW�GH�RPRJHQ ��'H�DOWIHO��ILLQG�YRUED�GH�WR L�VDODULD LL��HUD�GH�DúWHSWDW�FD�YHFKLPHD�PHGLH�V �DLE �R�GLVSHUVLH�PDUH��VDODULD LL�vQ�DQVDPEOXl lor V �DSDU LQ �OD�vQWUHJXO�LQWHUYDO�GH�YDULD LH�D�YHFKLPLL�

&DOFXOXO�LQGLFDWRULORU�VLQWHWLFL�DL�YDULD LHL

Tabelul 3.10.

1XP UXO�

muncitorilor

Grupe de VDODULD L�

GXS �

vechime (ani) abs. (ni) ( )*

%in

Centrul de

interval (xi)

xxi −

( ) ii nxx ⋅− ( ) ii nxx .2−

a = 17,5 h = 5

ii nh

ax⋅

− 2

0 1 2 3 4 5 6 7 0-5

5-10 10-15 15-20 20-25 25-30 30-35 35-40

10 40 60 80 50 30 20 10

3,33 13,33 20,00 26,67 16,67 10,00

6,67 3,33

2,5 7,5

12,5 17,5 22,5 27,5 32,5 37,5

-15,7 -10,7 -5,7 -0,7 4,3 9,3

14,3 19,3

-157 -428 -342

-56 215 279 286 193

2464,9 4579,6 1949,4

39,2 924,5

2594,7 4089,8 3724,9

90 160

60 0

60 120 180 160

Total

300

∑=

k

iin

1

100

( )∑=

k

iin

1

*%

-

-

-10 ( )∑ ⋅− ii nxx

1) 20367,0

( )∑ − ii nxx .2

820

∑=

⋅

−k

ii

i nh

ax

1

2

1) 'HRDUHFH� PHGLD� QX� HVWH� XQ� FkW� H[DFW� úL� D� IRVW� URWXQMLW �� QX� YHULILF � SURSULHWDWHD��( )∑ ⋅− ii nxx = 0 .�'LIHUHQ D�ILLQG�IRDUWH�PLF ��-10), rotunjirea medie�QX�YD�LQIOXHQ D�GHFkW�

într-R�P VXU �IRDUWH�PLF �P ULPHD�FHORUODO L�LQGLFDWRUL�

5. Calculul indicatorilor de asimetrie ÌQ� DFHVW� VFRS�YRP�FDOFXOD� FRHILFLHQ LL� GH� DVLPHWULH� SURSXúL�GH�3HDUVRQ� (Cas� úL�&¶as�� úL�Yule:

146,02,8

172,18)( =−=−=x

as

MoxC

σ

256,02,8

)5,172,18(3)(3 =−=−=′x

as

MexC

σ

12

12

qq

qqAS

+−

= , unde q1=Me - Q1; q2 =Q3 – Me

q1 = 17,5 – 12,125 = 5,375 q2 = 33,575 – 17,5 = 6,075

061,0375,5075,6

375,5075,6 =+−=AS

&HL�WUHL�LQGLFDWRUL�GH�DVLPHWULH�FDOFXOD L�LQGLF �R�DVLPHWULH�PRGHUDW �SR]LWLY �

6. 0HGLD�úL�GLVSHUVLD�FDUDFWHULVWicii alternative • Media (p)

N

Mp = ,

unde: M� �QXP UXO�PXQFLWRULORU�FX�R�YHFKLPH�!� x = 18,2 ani; M =110; N �QXP UXO�WRWDO�DO�PXQFLWRULORU��1� �����

3667,0300

110 ===N

Mp

• Dispersia ( 2pσ )

2322,0)3667,01(3667,0)1(2 =−=−= pppσ

3. Într-R�VRFLHWDWH�FRPHUFLDO ��FDUH�DUH�����GH�PXQFLWRUL�V-D�RUJDQL]DW�R�REVHUYDUH�VWDWLVWLF �úL�

s-DX�RE LQXW�XUP WRDUHOH�GDWH�

5HSDUWL LD�PXQFLWRULORU�vQ�IXQF LH�GH�SURGXF LD�RE LQXW ��EXF L�

Tabelul 3.11.

Grupe de muncitori GXS �SURGXF LD�RE LQXW ��EXF��

1XP U�PXQFLWRUL

Sub 120 10 120-140 18 140-160 23 160-180 38 180-200 51 200-220 40 220-240 15 ����úL�SHVWH 5 Total 200

5HSDUWL LD�PXQFLWRULORU�vQ�IXQF LH�GH�YHFKLPH��DQL�

Tabelul 3.12.

Grupe de muncitori GXS �YHFKLPH (ani)

Structura muncitorilor vQ�IXQF LH�GH�YHFKLPH����

Sub 2 4,2 2-4 8,5 4-8 12,9 8-12 30,2 12-18 26,2 18-24 10,2 ���úL�SHVWH 7,8 Total 100,0

Se cere: 1. V �VH�FDUDFWHUL]H]H�VWDWLVWLF�FHOH�GRX �VHULL�IRORVLQG�PHWRGD�JUDILF ��LQGLFDWRULL�WHQGLQ HL

FHQWUDOH��PHGLLOH�GH�VWUXFWXU ��LQGLFDWRULL�GH�YDULD LH úL asimetrie; 2. V �VH�SUHFL]H]H�GXS �FDUH�GLQ�YDULDELOH�HVWH�PDL�RPRJHQ �FROHFWLYLWDWHD�GH�PXQFLWRUL�DL�

VRFLHW LL�FRPHUFLDOH�UHVSHFWLYH� Rezolvare 1. 6HULD�SULYLQG� UHSDUWL LD�PXQFLWRULORU� vQ� IXQF LH�GH�SURGXF LD� LQGLYLGXDO �HVWH�SUH]HQWDW �

FX�IUHFYHQ H�DEVROXWH�úL�HVWH�JUXSDW �SH���LQWHUYDOH�GH�YDULD LH�HJDOH� 3HQWUX�FDUDFWHUL]DUHD� VWDWLVWLF �D�DFHVWHL� VHULL� GH� IUHFYHQ H� VH� YD� IDFH� UHSUH]HQWDUHD� JUDILF �

SULQ�KLVWRJUDP ��SROLJRQXO�úL�FXUED�FXPXODWLY �D�IUHFYHQ HORU��YH]L�ILJ�������úL�������IRORVLQG�FRO���úL�7 din tabelul 3.13. Graficul de concentrare se face analog cu cel din figura 1.4. (vezi capitolul 1).

)LJXUD������'LVWULEX LD�PXQFLWRULORU�vQ�IXQF LH�GH�SURGXF LD�]LOQLF ��EXF L�

0

10

20

30

40

50

60

80 100 120 140 160 180 200 220 240 260 280���� ���� � ��� ������� �

� ���� ���� � ��

10

1823

38

51

40

15

5

0

25

50

75

100

125

150

175

200

225

100 120 140 160 180 200 220 240 260SURGXF LD��EXF��

nr. m

unci

tori

)LJXUD������'LVWULEX LD�PXQFLWRULORU�vQ�IXQF LH�GH�SURGXF LD�]LOQLF ��EXF L� )LLQG�LQWHUYDOH�HJDOH��SHQWUX�PHGLH�úL�GLVSHUVLH�VH�YRU�XWLOL]D�IRUPXOHOH�GH�FDOFXO�VLPSOLILFDW�

ah

n

nh

ax

xk

ii

k

ii

i

+⋅

−

=

∑

∑

=

=

1

1, respectiv 22

1

1

2

2 )( axhn

nh

ax

k

ii

k

ii

i

x −−⋅⋅

−

=∑

∑

=

=σ

Pentru calculul indicatorilor se va folosi tabelul 3.13.

&DOFXOXO�LQGLFDWRULORU�FDUH�FDUDFWHUL]HD] �UHSDUWL LD�FHORU�����GH�PXQFLWRUL� vQ�IXQF LH�GH�P ULPHD�SURGXF LHL�]LOQLFH�RE LQXWH

Tabelul 3.13.

)UHFYHQ H�FXPXODWH

Grupe de muncitori

GXS �

SURGXF LD RE LQXW �

(buc.)

1XP U�

muncitori (ni)

Centre de

interval (xi)

h

axi −

a=190 h=20

ii nh

xx⋅

−

ii nh

ax⋅

− 2

&UHVF WRU Descres-

F WRU

0 1 2 3 4 5 6 7 sub 120 10 110 -4 -40 160 10 200 120-140 18 130 -3 -54 162 28 190 140-160 23 150 -2 -46 92 51 172 160-180 38 170 -1 -38 38 89 149 180-200 51 190 0 0 0 140 111 200-220 40 210 1 40 40 180 60 220-240 15 230 2 30 60 195 20 ����úL�SHVWH 5 250 3 15 45 200 5 Total 200 -93 597 - -

7,18019020200

93 =+⋅−=x buc./muncitor

51,1107)1907,180(20200

597 222 =−−⋅=xσ

28,3351,1107 ==xσ buc./muncitor

%42,181007,180

28,33100 =⋅=⋅=

xv xσ

3H� ED]D� FRHILFLHQWXOXL� GH� YDULD LH� VH� SRDWH� WUDJH� FRQFOX]LD� F � PHGLD� HVWH� UHSUH]HQWDWLY �

pentru cele mai multe valori individuale ale serieL�GHRDUHFH�DP�RE LQXW�XQ�FRHILFLHQW�PLF�GH�YDULD LH�����������FHHD�FH�vQVHDPQ �vQ�DFHODúL�WLPS�F �GDWHOH�VXQW�GHVWXO�GH�RPRJHQH�vQWUH�HOH�

'LQ�DFHODúL�WDEHO�VH�SRW�FDOFXOD�úL�FHLODO L�GRL�LQGLFDWRUL�DL�WHQGLQ HL�FHQWUDOH��YDORDUH�PRGDO �úL�YDORDUH�PHGLDQ �

,QWHUYDOXO�FX�IUHFYHQ �PD[LP ��FXSULQGH�úL�PRGXO�

180 < Mo < 200, de unde:

83,190)4051()3851(

)3851(20180

21

10 =

−+−−+=

∆+∆∆

⋅+= hxMo EXF L

Locul medianei în serie:

5,1002

1200

2

1)( 1 =+=

+=

∑=

k

iin

MeU

ÌQ� LQWHUYDOXO�DO�FLQFLOHD�VXPD�IUHFYHQ HORU�FXPXODWH�HVWH�PDL�PDUH�GHFkW� ORFXO�PHGLDQHL� vQ�

seULH��DGLF � 180 < Me < 200

5,18451

895,10020180

2

11

1

1

0 =−+=−

+

+=∑

∑ −

=

=

m

m

ii

k

ii

n

nn

hxMe EXF L

Deci, x , Me�úL�Mo�VH�J VHVF�vQ�DFHODúL�LQWHUYDO��VHULD�HVWH�XúRU�DVLPHWULF �úL�SXWHP�YHULILFD�

UHOD LD� 3( x -Me) = x -Mo 3(180,700-184,5) =-11,4 x -Mo = 180,7-190,83 =-10,13

ÌQVHDPQ �F �SHQWUX�VHULD�QRDVWU ��FX�R�GLIHUHQ �QHVHPQLILFDWLY ��VH�YHULILF �UHOD LD�GLQWUH�FHL�WUHL�LQGLFDWRUL�DL�WHQGLQ HL�FHQWUDOH�H[LVWHQW �vQ�FD]XO�UHSDUWL LLORU�PRGHUDW�DVLPHWULFH�

Pentru stabilirea gradului de asimetrie vom avea:

30,028,33

13,10

28,33

83,1907,180)(−==

−=

−=

xas

MoxC

σ

34,028,33

4,11

28,33

)5,1847,180(3)(3 −==−=−=′x

as

MexC

σ

12

12

qq

qqAS

+−

= , unde q1=Me - Q1; q2 =Q3 – Me

25,504

201

4

1)( 1

1 ==+

=∑

=

k

iin

QU

140 < Q1 < 160

34,15923

2825,5020140

14

1 1

1101 =−+=

−

+

+=∑∑

−

==

u

u

ii

k

ii

n

nn

hxQ EXF L

75,1502014

31

4

3)(

13 =⋅=

+= ∑

=

k

iinQU

200 < Q3 < 220

375,20540

14075,15020200

14

3 1

1103 =−+=

−

+

+=∑∑

−

==

v

v

ii

k

ii

n

nn

hxQ EXF L

q1 = 184,5-159,34 = 25,16 q2 = 205,375-184,5 = 20,875

093,016,25875,20

16,25875,20 −=+−=AS

În toate cazurile s-DX� RE LQXW� YDORUL� QHJDWLYH� UHODWLY�PLFL� SHQWUX� FRHILFLHQ LL� GH� DVLPHWULH��

FHHD�FH�vQVHDPQ �F �SUHGRPLQ �YDORULOH�PDL�PDUL�úL�FD�DWDUH�LQGLFDWRULL�WHQGLQ HL�FHQWUDOH�VXQW�PDL�UHSUH]HQWDWLYL� SHQWUX� WHUPHQLL� FDUH� SUH]LQW � DEDWHUL� QHJDWLYH� vQ� FRPSDUD LH� FX� FHL� FDUH� SUH]LQW �abateri pozitive ( x < Me < Mo).

2. 6HULD� SULYLQG� UHSDUWL LD� PXQFLWRULORU� GXS � YHFKLPH� HVWH� SH� LQWHUYDOH� QHHJDOH� úL� FX�

IUHFYHQ H�UHODWLYH� 3HQWUX�UHSUH]HQWDUHD�JUDILF �HVWH�QHFHVDU�V �VH�IDF �R�VHULH�GH�FDOFXOH�SUHJ WLWRDUH�

&DOFXOXO�IUHFYHQ HORU�UHGXVH�úL�D�IUHFYHQ HORU�FXPXODWH�pentru seria referitoare la vechime

Tabelul 3.14. )UHFYHQ H�

relative cumulate

*UXSH�GXS �

vechime (ani)

Structura muncitorilor

(n*i(%))

0 ULPHD�

intervalului (hi)

5DSRUWXO�ID �GH�

primul interval

1h

hK i

i =

)UHFYHQ H�

reduse

i

i

K

n *(%)

FUHVF -tor

descres-F WRU

A 1 2 3 4 5 6 0-2 4,2 2 1,0 4,2 4,2 100,0

2-4 8,5 2 1,0 8,5 12,7 95,8

4-8 12,9 4 2,0 6,5 25,6 87,3

8-12 30,2 4 2,0 15,1 55,8 74,4

12-18 26,2 6 3,0 8,7 82,0 44,2

18-24 10,2 6 3,0 3,4 92,2 18,0

24-30 7,8 6 3,0 2,6 100 7,8

Total 100 - 15 - - -

Folosind datele din tabelul 3.14. se constituie: histograma, polLJRQXO� IUHFYHQ HORU� úL� FXUED�

FXPXODWLY ��YH]L�ILJ�������úL�������

)LJXUD������5HSDUWL LD�PXQFLWRULORU�vQ�IXQF LH�GH�YHFKLPH

0

20

40

60

80

100

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

vechime (ani)

stru

ctur

a m

unci

tori

lor

(%)

)vJXUD������5HSDUWL LD�PXQFLWRULORU�vQ�IXQF LH�GH�YHFKLPH

4,2

8,5

12,9

30,2

26,2

10,27,8

0

2

4

6

8

10

12

14

16

0 4 8 12 16 20 24 28 32

vechime (ani)

stru

ctur

a m

unci

tori

lor

(%)

ùL� vQ� DFHVW� FD]� WUHEXLH� FDOFXOD L� LQGLFDWRULL� FDUH� FDUDFWHUL]HD] � VWDWLVWLF� DFHDVW � VHULH�� )LLQG�LQWHUYDOH� QHHJDOH� WUHEXLH� FDOFXODWH� FHQWUHOH� GH� LQWHUYDO� úL� DSRL� VH� YD� DSOLFD� R� IRUPXO � GH� FDOFXO�VLPSOLILFDW�� 'H� UHJXO �� VH� RSWHD] � QXPDL� SHQWUX� IRORVLUHD� XQHL� VLQJXUH� SURSULHW L� bazate pe PLFúRUDUHD�WHUPHQLORU�FX�R�FRQVWDQW ��a" (vezi tabelul 3.15.).

&DOFXOXO�LQGLFDWRULORU�WHQGLQ HL�FHQWUDOH��D�PHGLLORU�GH�SR]L LH�úL�D�LQGLFDWRULORU�GH�YDULD LH�

într-o serie cu intervale neegale

Tabelul 3.15.

Intervale GH�YDULD LH

)UHFYHQ �

reODWLY

( *(%)in )

Centrul intervalului

( ix )

(xi - a) a = 10

*(%))( ii nax ⋅− *

(%)2)( ii nax ⋅− )UHFYHQ H�UHODWLYH�

cumulative

A 1 2 3 4 5 6 0 - 2

4,2

1

-9

-37,8

340,2

4,2 2 - 4 8,5 3 -7 -59,5 416,5 12,7

4 - 8 12,9 6 -4 -51,6 206,4 25,6 8 - 12 30,2 10 0 0 0 5,5,8 12 - 18 26,2 15 5 31 655 82,0 18 - 24 10,2 21 11 112,2 1234,3 92,2 24 - 30

7,8

27

17

132,6

2554,2

100,0 Total 100 - - 226,9 5113,5 -

( )

3,1210100

9,226

100

*(%) ≅+=

⋅−= ∑ ii nax

x ani/muncitor

( )( ) 84,45)103,12(

100

5,5113

10022

*(%)

2

2 =−−⋅=−−−

=∑

axnax ii

xσ

77,684,45 ==xσ ani/muncitor

%551003,12

77,6100 =⋅=⋅=

xv xσ

8 < Mo < 12

25,11)2,262,30()9,122,30(

)9,122,30(48

21

10 =

−+−−+=

∆+∆∆

⋅+= hxMo ani

Locul medianei în serie:

5,502

1100

2

1)( 1 =+=

+=

∑=

k

iin

MeU

8 <Me < 12

( )298,11

2,30

6,265,5048

2

1

*%

1

1

*1

*

0

(%)

(%)

=−

+=−

+

+=∑

∑ −

=

=

m

i

i

n

n

n

hxMe

m

i

k

i

ani

25,254

101)( 1 ==QU

4 < Q1 <8

89,79,12

7,1225,25441 =−+=Q ani

298,112 == eMQ ani

75,751014

3)( 3 =⋅=QU

12 < Q3 < 18

568,162,26

8,5575,756123 =−+=Q ani

155,077,6

25,113,12)( =−=−=x

as

MoxC

σ

44,077,6

)298,113,12(3)(3 =−=−=′x

as

MexC

σ

q1 = Me-Q1=11,298 – 7,89 = 3,488 q2 = Q3 – Me = 16,568 – 11,298 = 5,27

203,0488,32,25

488,32,5

12

12 =+

−=+−

=qq

qqAS

$QDOL]kQG� UH]XOWDWHOH� VH� FRQVWDW � F � VHULD� DUH� FDUDFWHU� QHRPRJHQ� GHRDUHFH� FRHILFLHQWXO� GH�

YDULD LH� WRWDO �HVWH�GH������5HIHULQGX-QH� OD�YHFKLPH�� DFHDVW � VLWXD LH�HUD�GH�DúWHSWDW�� VHULD� DYkQG�LQWHUYDOH�QHHJDOH�GH�YDULD LH�úL�IUHFYHQ HOH�GLVWULEXLQGX-se neuniform.

&DOFXOkQG�FRHILFLHQ LL�GH�DVLPHWULH�VH�FRQVWDW �R�DVLPHWULH spre valorile mici (Mo < Me < x ). ÌQ� ILQDO�� VH� SRDWH� VSXQH� F � VHULD� SULYLQG� GLVWULEX LD� SURGXF LHL� �vQ� EXF L�� HVWH� PXOW� PDL�

RPRJHQ ��FDUDFWHUL]DW �GH� LQGLFDWRUL�FX�YDORUL�PLFL�SHQWUX�YDULD LH�úL�DVLPHWULH�úL�HVWH�QHRPRJHQ �SHQWUX�VHULD�SULYLQG�YHFKLPHD�PHGLH��FHHD�FH�QH�SHUPLWH�V �WUDJHP�FRQFOX]LD�F �vQ�PRG�FRUHFW� vQ�vQWUHSULQGHUHD� UHVSHFWLY �SURGXF LLOH� LQGLYLGXDOH�RE LQXWH�QX�GHSLQG� vQ�SULQFLSDO�GH�YHFKLPH�FL�GH�DO L�IDFWRUL�HVHQ LDOL�VSHFLILFL�

4. Într-R�VRFLHWDWH�FRPHUFLDO ��V-DX�vQUHJLVWUDW�FX�SULYLUH�OD�XQ�SURGXV�GH�ED] �GHVWLQDW�YkQ] ULL�

SULQ�XQLW LOH�SURSULL��GDWHOH�QHFHVDUH�GHWHUPLQ ULL�QLYHOXOXL�PHGLX�DO�SURGXFWLYLW LL�PXQFLL�(xi)�úL�DO�consumului specific GH� WLPS�GH�PXQF � �yi���'DWHOH� VH�SUH]LQW � vQWU-R�VHULH�ELGLPHQVLRQDO � vQ� FDUH�FKHOWXLHOLOH�WRWDOH�GH�WLPS�GH�PXQF ��zi��UHSUH]LQW �IUHFYHQ HOH�FRPXQH�SHQWUX�FHOH�GRX �YDULDELOH�

Tabelul 3.16.

Produs Nivelul SURGXFWLYLW LL�

muncii orare �PLL�OHL�RU �

(xi)

Consumul specific de timp GH�PXQF ��RUH-om la 1000 lei

SURGXF LH� (yi)

Cheltuielile totale de timp

GH�PXQF (mii ore-om)

(zi)

A 1 2 3

A 10,0 0,100 11624 B 6,25 0,160 6444 C 7,47 0,134 100 Total 8,65 0,116 18168

Se cere:

1. calculul nivelului mediu al�SURGXFWLYLW LL�PXQFLL�( w = x ); 2. FDOFXOXO�FRQVXPXOXL�PHGLX�GH�WLPS�GH�PXQF �( t = y ).

1RW ��V �VH�VSHFLILFH�FH�IHO�GH�PHGLL�V-DX�XWLOL]DW�úL�GH�FH�" Rezolvare 'DF � FHOH� GRX � FDUDFWHULVWLFL� VXQW�FRQVLGHUDWH� LQWHUGHSHQGHQWH� vQWUH� HOH�� VH� SXQH�SUREOHPD�

DOHJHULL�FHOXL�PDL�SRWULYLW�WLS�GH�PHGLH�SHQWUX�D�S VWUD�DFHHDúL�UHOD LH�GH�LQYHUV �SURSRU LRQDOLWDWH�úL�vQWUH�P ULPLOH�PHGLL�FDOFXODWH�

ÌQWUH� FHOH� GRX � YDULDELOH� DSDU� UHOD LLOH��i

i yx

1= �� SHQWUX� FDUH� VH� YD� RE LQH� R� UHSDUWL LH�

ELGLPHQVLRQDO �FX�IUHFYHQ H�FRPXQH� ( )ii yxn .

3ULPD� VHULH� GH� GLVWULEX LH� D� QLYHOXULORU� SURGXFWLYLW LL� PXQFLL� HVWH� IRUPDW � GLQ� GDWHOH�FRORDQHORU���úL����LDU�FHD�GH-a doua sHULH�GH�GLVWULEX LH�-�D�FRQVXPXULORU�VSHFLILFH�GH�WLPS�GH�PXQF �-� HVWH� IRUPDW �GLQ�GDWHOH� FRORDQHORU� �� úL� ���$VWIHO� SUH]HQWDWH�� FHOH�GRX � VHULL� DX� DFHOHDúL�SRQGHUL�SHQWUX�DPEHOH�FDUDFWHULVWLFL��DGLF �QXP UXO�WRWDO�GH�RP-ore lucrate.

3HQWUX�D�S VWUD�UHOD LD�GH�LQYHUV �SURSRU LRQDOLWDWH�GLQWUH�FHOH�GRX �YDULDELOH�úL�vQWUH�P ULPLOH�PHGLL� SHQWUX� FDOFXOXO� QLYHOXOXL� PHGLX� DO� SURGXFWLYLW LL� PXQFLL� YRP� IRORVL� PHGLD� DULWPHWLF �SRQGHUDW � LDU� SHQWUX� FDOFXOXO� FRQVXPXOXL�PHGLX� GH� WLPS� GH�PXQF � VH� YD� DSOLFD�PHGLD� DUPRQLF �SRQGHUDW �

1. &DOFXOXO�QLYHOXOXL�PHGLX�DO�SURGXFWLYLW LL�PXQFLL

65,818168

10047,7644425,61162410

1

1 =⋅+⋅+⋅===∑

∑

=

=k

iyx

k

iyxi

ii

ii

n

nxxw mii lei/ore-om

2. &DOFXOXO�FRQVXPXOXL�PHGLX�GH�WLPS�GH�PXQF

116,0100

134,0

16444

16,0

111624

10,0

118168

1

1

1 =⋅+⋅+⋅

===∑

∑

=

=k

iyx

i

k

iyx

ii

ii

ny

nyt ore-om/1000 lei

9HULILFDUHD�UHOD LHL�GLQWUH�FHOH�GRX �PHGLL��vQ�PLL�OHL�

;1

yx = ;62,8

116,0

1 =

'LIHUHQ D� SRDWH� SURYHQL� GLQ� URWXQMLUHD� FDOFXOXOXL�� 'HFL�� PHGLD� DUPRQLF � SRDWH� VXEVWLWXL�PHGLD� DULWPHWLF � FkQG� VH� FXQRVF� SURGXVHOH� GH� IUHFYHQ � úL� QX� IUHFYHQ HOH� UHDOH�� 8Q� DVWIHO� GH�exemplu poate fi cel al indiceluL�DUPRQLF�DO�SUH XULORU��YH]L�FDSLWROXO��,QGLFL���

5. 6H�FXQRVF�XUP WRDUHOH�GDWH�FX�SULYLUH�OD�GLVWULEX LD�YkQ] WRULORU�GLQWU-un complex comercial

vQ�IXQF LH�GH�YHFKLPHD�vQ�PXQF �úL�YDORDUHD�YkQ] ULORU�UHDOL]DWH�vQWU-R�V SW PkQ �

Tabelul 3.17.

6XEJUXSH�GH�YkQ] WRUL�GXS �YROXPXO�YkQ] ULORU��PLO��OHL� Grupe GH�YkQ] WRUL�

GXS �YHFKLPH��DQL) sub 190 190-200 200-210 210-220 220

úL�SHVWH

Total

A 1 2 3 4 5 6 sub 10 10-20 ���úL�SHVWH

5 - -

15 12

-

5 35 7

- 8

15

- - 8

25 55 30

Total 5 27 47 23 8 110

Se cere: 1. SROLJRQXO�IUHFYHQ HORU�SULYLQG�UHSDUWL LD�YkQ] WRULORU�GXS �YROXPXO�YkQ] ULORU�SH�WRWDO�úL�

pe grupe de vechime; 2. UHSUH]HQWDUHD�JUDILF �D�QXP UXOXL�YkQ] WRULORU�GXS �YHFKLPH�SH�FHOH trei grupe, folosind

JUDILFXO�SULQ�S WUDW� 3. calculul mediiloU�SH�JUXSH�GH�YHFKLPH�úL�SH�WRWDO� 4. LQGLFDWRULL�VLQWHWLFL�DL�YDULD LHL�SH�ILHFDUH�JUXS �úL�SH�WRWDO� 5. LQWHUSUHWDUHD�JUDGXOXL�GH�RPRJHQLWDWH�SH�JUXSH�úL�SH�WRWDO�� 6. verificarea regulii de adunare a dispersiilor; 7. ce indicatori statistici se pot calcula pe baza UHJXOLL�GH�DGXQDUH�D�GLVSHUVLLORU�úL�FXP�VH�

LQWHUSUHWHD] �VWDWLVWLF�DFHúWL�LQGLFDWRUL� 8. FDOFXOXO� úL� LQWHUSUHWDUHD� GLVSHUVLLORU� SHQWUX� FDUDFWHULVWLFD��YkQ] WRUL� FDUH� VH� DIO � SHVWH�

PHGLD�YkQ] ULORU�SH�WRWDO��

Rezolvare

1. 5HSUH]HQWDUHD�JUDILF �D�GLVWULEX LHL�YkQ] WRULORU��ILJXULOH������úL�������

0

5

10

15

20

25

30

35

40

45

50

170 180 190 200 210 220 230 240

)LJXUD������5HSDUWL LD�YkQ] WRULORU�vQWU-XQ�FRPSOH[�FRPHUFLDO�GXS �YROXPXO�GHVIDFHULORU

0

5

10

15

20

25

30

35

40

45

50

170 180 190 200 210 220 230 240

Total

gr. I

gr. II

gr. III

)LJXUD�������5HSDUWL LD�YkQ] WRULORU�GXS �YROXPXO�GHVIDFHULL�SH�WRWDO�úL�SH�JUXSH�GH�YHFKLPH

2. 5HSUH]HQWDUHD�JUDILF �D�QXP UXOXL�YkQ] WRULORU�GXS �YHFKLPH (vezi figura 3.11.).

)LJXUD�������1XP UXO�GH�YkQ] WRUL�SH�JUXSH�GH�YHFKLPH

55

25

30

sub 10 ani 10-20 ani ���DQL�úL�SHVWH

Scara: 1 cm = 2 persoane

1X

PU�

Yk

Q]

WRUL

9ROXPXO�YkQ] ULORU��PLO��OHL�

1X

PU�

YkQ

]WR

UL

9ROXPXO�YkQ] ULORU��PLO��OHL�

3. Calculul mediilor • PHGLLOH�GH�JUXS :

∑

∑

=

==m

jij

m

jijj

i

n

ny

y

1

1

19525

52051519551851 =⋅+⋅+⋅=y �PLO��OHL�YkQ] WRU

27,20455

821535205121952 =⋅+⋅+⋅=y �PLO��OHL�YkQ] WRU

33,21530

82251521572053 =⋅+⋅+⋅=y �PLO��OHL�YkQ] WRU

• PHGLD�JHQHUDO ( )y :

- independent:

∑

∑

=⋅

=⋅

=m

jj

m

jjj

n

ny

y

1

1

18,205110

82252321547205271955185 =⋅+⋅+⋅+⋅+⋅=y PLO��OHL�YkQ] WRU

- pe baza mediilor de JUXS �

∑

∑

=⋅

=⋅

=r

ii

r

iii

n

nyy

1

1

18,205110

3033,2155527,20425195 =⋅+⋅+⋅=y PLO��OHL�YkQ] WRU.

4. &DOFXOXO�LQGLFDWRULORU�VLQWHWLFL�DL�YDULD LHL� • GLVSHUVLLOH�GH�JUXS � ( )2

iσ :

( )

∑

∑

=

=−

=m

jij

m

jijij

i

n

nyy

1

1

2

2σ

4025

5 ) 195- (205 15) 195- (195 5195)- (185 22221 =⋅+⋅+⋅=σ

83,3555

8 ) 204,27 - (215 35) 204,27 - (205 12) 204,27 - (195 22222 =⋅+⋅+⋅=σ

89,4930

8 ) 215,33 - (225 15 ) 215,33 - (215 7 215,33 - (205 22 )223 =

⋅+⋅+⋅=σ

• PHGLD�GLVSHUVLLORU�SDU LDOH� ( )2σ :

63,40110

3089,495583,352540

1

1

2

2 =⋅+⋅+⋅==∑

∑

=⋅

=⋅

r

ii

r

iii

n

nσσ

• dispersia dintre grupe ( )2

/2

xyσδ = :

( )

06,52110

30)18,20533,215(55)18,20527,204(25)18,205195( 222

1

1

2

2

=⋅−+⋅−+⋅−

=−

=

∑

∑

=⋅

=⋅

r

ii

r

iii

n

nyy

δ

• GLVSHUVLD�WRWDO ( )2σ :

( )( ) ( ) ( )

( ) ( )69,92

110

818,2052252318,205215

110

4718,2052052718,205195518,205185

22

222

1

1

2

2

=⋅−+⋅−

+

+⋅−+⋅−+⋅−

=

−

=

∑

∑

=⋅

=⋅

m

jj

m

jjj

n

nyy

σ

5. Aprecierea gradului de omRJHQLWDWH�SH�JUXSH�úL�SH�WRWDO &DOFXOXO�FRHILFLHQ LORU�GH�YDULD LH�

• pe grupe

32,640211 === σσ PLO��OHL�YkQ] WRU

%24,3100195

32,6100

1

11 =⋅=⋅=

yv

σ

98,583,35222 === σσ PLO��OHL�YkQ] WRU

%93,210027,204

98,5100

2

22 =⋅=⋅=

yv

σ

06,789,49233 === σσ PLO��OHL�YkQ] WRU

%18,310033,215

06,7100

3

33 =⋅=⋅=

yv

σ

• pe total

62,969,922 === σσ PLO��OHL�YkQ] WRU

%69,410018,205

62,9100 =⋅=⋅=

yv

σ

&RPSDUkQG�UH]XOWDWHOH�RE LQXWH�VH�FRQVWDW �F �� • ILHFDUH�JUXS �OXDW �VHSDUDW�HVWH�PDL�RPRJHQ �GHFkW�FROHFWLYLWDWHD�JHQHUDO �GLQ�FDUH�D�IRVW�

H[WUDV � • grupD�D�GRXD�HVWH�PDL�RPRJHQ �GHFkW�FHOHODOWH�GRX � • YDORULOH�PLFL�DOH�FRHILFLHQ LORU�GH�YDULD LH�FDOFXOD L�SH�ILHFDUH�JUXS �úL�SH�WRWDO�DWHVW �XQ�

JUDG� GH� RPRJHQLWDWH� ULGLFDW� DO� JUXSHORU� úL� FROHFWLYLW LL� WRWDOH� úL� GHFL� XQ� JUDG� GH�UHSUH]HQWDWLYLWDWH�FRUHVSXQ] WRU�SHQWUX�PHGLLOH�FDUH�OH�FDUDFWHUL]HD] �

6. Verificarea regulii de adunare a dispersiilor

222 σδσ += 92,69 = 52,06 + 40,63

7. 3H�ED]D�UHJXOLL�GH�DGXQDUH�D�GLVSHUVLLORU�VH�SRW�FDOFXOD�DO L�GRL�LQGLFDWRUL�VWDWLVWLFL� • *UDGXO�GH�GHWHUPLQD LH ( 2

/ xyR )

%16,5610069,92

06,52100

2

22

/ =⋅=⋅=σδ

xyR

• *UDGXO�GH�QHGHWHUPLQD LH ( 2

/ xyK )

%84,4310069,92

63,40100

2

22

/ =⋅=⋅=σσ

xyK

6H� SRDWH� DILUPD� F � ������� GLQ� YDULD LD� WRWDO � D� YROXPXOXL� YkQ] ULORU� HVWH� H[SOLFDW � GH�

YDULD LD�SURGXV �GH�IDFWRUXO�GH�JUXSDUH��YHFKLPHD�– factor dominant deoarece 2xy

R > 50%), restul

GH��������ILLQG�LQIOXHQ D�UHODWLY �D�FHORUODO L�IDFWRUL�QHvQUHJLVWUD L�

8. &DOFXOXO� úL� LQWHUSUHWDUHD� GLVSHUVLLORU� SHQWUX� �YkQ] WRULL� FDUH� VH� DIO � SHVWH� YROXPXO�

PHGLX�DO�YkQ] ULORU�SH�WRWDO" *�Calculul mediilor

• PHGLLOH�SH�JUXS : i

ii n

mw = ,

de unde:

025

01 ==w

1455,055

82 ==w

7667,030

233 ==w

• media pe total: 2818,0110

31 ===n

mw

*�Calculul dispersiilor

• GLVSHUVLLOH�GH�JUXS :

( )ii

wwiw −⋅= 12σ ,

de unde:

( ) ( ) 00101 112

1=−⋅=−⋅= ww

wσ

( ) ( ) 1243,01455,011455,01 222

2=−=−⋅= ww

wσ

( ) ( ) 1789,07667,017667,01 332

3=−=−⋅= ww

wσ

• PHGLD�GLVSHUVLLORU�SDU LDOH ( 2wσ ):

1109,0110

301789,0551243,0250

1

1

2

2 =⋅+⋅+⋅=⋅

⋅=

∑

∑

=

=r

ii

r

iiw

w

n

ni

σσ

• dispersia dintre grupe ( 2

wδ ):

( ) ( ) ( )

( )0914,0

110

302818,07667,0

110

552818,01455,0252818,00

2

22

1

1

2

2

=⋅−+

+⋅−+⋅−=⋅−

=∑

∑

=

=r

ii

r

iii

w

n

nwwδ

• dispersia total ( 2

wσ ):

( ) ( ) 2023,02818,012818,012 =−⋅=−= wwwσ

5HJXOD�DGXQ ULL�GLVSHUVLLORU�VH�S VWUHD] �úL�vQ�FD]XO�FDUDFWHULVWLFLL�DOWHUQDWLYH� 222www δσσ +=

0,2023 = 0,1109 + 0,0914

6. 3HQWUX� XQ� DJHQW� HFRQRPLF� FH� GHVI úRDU � WUHL� DFWLYLW L� V-au înregistrat datele privind

veniturile salariale ale personalului pentru luna mai 2001 (vezi tabelul 3.18.).

5HSDUWL LD�VDODULD LORU�SH�DFWLYLW L��vQ�IXQF LH�GH�P ULPHD�YHQLWXULORU�VDODULDOH�vQ�OXQD�PDL�����

Tabelul 3.18. �vQ���ID �GH�WRWDOXO�JUXSHL�

Grupe duS �VDODULL�

(zeci mii lei)

Activitatea I

Activitatea II

Activitatea III

A 1 2 3 sub 250

10

-

- 250 - 270 12 - -

270 - 290 25 20 - 290 - 310 28 35 - 310 - 330 15 20 20 330 - 350 10 15 47 350 - 370 - 10 23 ����úL�Seste

-

-

10 Total 100 100 100

1XP UXO� GH� VDODULD L�D� vQUHJLVWUDW� XUP WRDUHD� VWUXFWXU �� DFWLYLWDWHD� O� - 20%; activitatea II -

-�����úL�DFWLYLWDWHD�,,,�- 30%. Se cere: 1. V �VH�DSOLFH�UHJXOD�GH�DGXQDUH�D�GLVSHUVLLORU�úL�V �VH�VSHFLILFH�GDF �IDFWRUXO�Ge grupare

este semnificativ sau nu; 2. V �VH�DUDWH�JUDGXO�GH�RPRJHQLWDWH�SH�FHOH� WUHL�DFWLYLW L� úL�SH� WRWDO� úL� V � VH� LQWHUSUHWH]H�

UH]XOWDWHOH�RE LQXWH�SHQWUX�FDUDFWHUL]DUHD�YDULD LHL� 3. V �VH�FDOFXOH]H�QXP UXO�GH�VDODULD L��IUHFYHQ HOH�DEVROXWH��úWLLQG�F �SH�WRWDO�QXP UXO�ORU�D�

IRVW�GH�����úL�V �VH�YHULILFH�SH�ED]D�IUHFYHQ HORU�DEVROXWH�GLVSHUVLD�WRWDO � Rezolvare 1. )LLQG� LQWHUYDOH� HJDOH�� PHGLLOH� úL� GLVSHUVLLOH� VH� SRW� GHWHUPLQD� FX� IRUPXOHOH� GH� FDOFXO�

VLPSOLILFDW��FX�IUHFYHQ H�UHODWLYH�

ah

nh

ay

y

m

jij

j

i +⋅

−

=∑

=

1001

*(%)

( )221

*(%)

2

2

100ayh

nh

aym

jij

j

i −−⋅

−

=∑

=σ

1.1. Pentru activitatea I� VH� YD�H[WUDJH� VHULD� úL� VH� YRU�FDOFXOD�� PHGLD��GLVSHUVLD��abaterea PHGLH�S WUDWLF �(vezi tabelul 3.19.).

&DOFXOXO�PHGLHL�úL�GLVSHUVLHL�VDODULLORU�SHQWUX�DFWLYLWDWHD�,

Tabelul 3.19.

Grupe GXS �

salarii (zeci mii lei)

Structura VDODULD LORU

( )*(%)1 jn

Centrul de interval ( )jy

h

ay j −

a = 300 h = 20

*(%)1 j

j nh

ay⋅

− *

(%)1

2

jj nh

ay⋅

−

A 1 2 3 4 5 230-250 250-270 270-290 290-310 310-330 330-350

10 12 25 28 15 10

240 260 280 300 320 340

-3 -2 -1 0 1 2

-30 -24 -25

0 15 20

90 48 25 0

15 40

Total 100 - - -44 218

200,29130020100

441 =+⋅−=y zeci mii lei/salariat

( ) 56,7943002,29120100

218 2221 =−−⋅=σ

187,2856,7941 ==σ zeci mii lei/salariat

%68,91002,291

187,281 =⋅=v

1.2. Pentru activitatea II (vezi calculele în tabelul 3.20.)

&DOFXOXO�PHGLHL�úL�GLVSHUVLHL�VDODULLORU�SHQWUX�DFWLYLWDWHD�D�,,,-a

Tabelul 3.20. *UXSH�GXS �

salarii (zeci mii lei)

Structura VDODULD LORU

( )*(%)2 jn

Centrul de interval ( )jy

h

ay j −

a = 300 h = 20

*(%)2 j

jn

h

ay⋅

− *

(%)2

2

j

jn

h

ay⋅

−

A 1 2 3 4 5 270-290 290-310 310-330 330-350 350-370

20 35 20 15 10

280 300 320 340 360

-1 0 1 2 3

-20 0

20 30 30

20 0

20 60 90

Total 100 - - 60 190

31230020100

602 =+⋅=y ]HFL�PLL�OHL�VDODULDW��DGLF ���������OHi/salariat

( ) 61630031220100

190 2222 =−−⋅=σ

82,246162 ==σ zeci mii lei/salariat

%95,7100312

82,242 =⋅=v

1.3. Pentru activitatea a III-a vezi calculele în tabelul 3.21.

&DOFXOXO�PHGLHL�úL�GLVSHUVLHL�VDODULLORU�SHQWUX�DFWLYLWDWHD�D����-a Tabelul 3.21.

*UXSH�GXS �

salarii (zeci mii lei)

Structura VDODULD LORU

( )*(%)3 jn

Centrul de interval ( )jy

h

ay j −

a = 300 h = 20

*(%)2 j

j nh

ay⋅

− *

(%)3

2

j

jn

h

ay⋅

−

A 1 2 3 4 5 310-330 330-350 350-370 3���úL�SHVWH

20 47 23 10

320 340 360 380

-1 0 1 2

-20 0

23 20

20 0

23 40

Total 100 - - 23 83

6,34434020100

233 =+⋅=y zeci mii lei/salariat sau 3446000 lei/salariat

( ) 84,31034084,31020100

83 2223 =−−⋅=σ

6,1784,3103 ==σ zeci mii lei/salariat

%11,510084,344

06,173 =⋅=v

1.4. &XQRVFkQG� PHGLLOH� SDU LDOH� úL� SRQGHULOH� VDODULD LORU� DFWLYLW LORU� UHVSHFWLYH� VH� SRDWH�

calcula media pe total:

( ) ( ) ( ) ( )62,317

100

306,34450312202,291

1001

*%

=⋅−⋅+⋅

==∑=

r

iii ny

y zeci mii lei/salariat

1.5.�$SOLFkQG�UHJXOD�DGXQ ULL�GLVSHUVLLORU�� 222 σδσ += VH�FRQVWDW �F �SXWem calcula doar

GRX �GLQWUH�HOH��XUPkQG�FD�GLVSHUVLD�WRWDO �V �ILH�FDOFXODW �SH�ED]D�UHOD LHL�GH�PDL�VXV�

164,560100

30)(310,84 50)(616 20)(794,56

1001

*(%)

2

2 =⋅+⋅+⋅

==∑

=

r

iii nσ

σ

( )

24,378100

30)62,3176,344(50)62,317312(20)62,3172,291(

100

222

1

*(%)

2

2

=⋅−+⋅−+⋅−

=

=−

=∑=

r

iii nyy

δ

222 σδσ += = 560,164 + 378,24 = 938,404

1.6. &DOFXOXO�JUDGXOXL�GH�GHWHUPLQD LH�úL�QHGHWHUPLQD LH:

%31,40100404,938

276,378100

2

22

/ =⋅=⋅=σδ

xyR

%69,59100404,938

164,560100

2

22

/ =⋅=⋅=σσ

xyK

⇒> 2

/2

/ xyxy RK �IDFWRU�GH�JUXSDUH�QHUHSUH]HQWDWLY��FHHD�FH�vQVHDPQ �F �vQ�OXQD�QRLHPEULH�QX�

a existat decât într-R� SURSRU LH� GH� ������� GLIHUHQ H� VSHFLILFH� vQWUH� FHOH� WUHL� DFWLYLW L� vQ� cazul YHQLWXULORU�VDODULDOH��FHHD�FH�HUD�GH�DúWHSWDW�

2. 3HQWUX�DQDOL]D�RPRJHQLW LL��VH�YRU�IRORVL�LQGLFDWRULL�SUH]HQWD L�vQ�tabelul 3.22. (coloana 5).

5HSDUWL LD�SH�FHOH�WUHL�DFWLYLW L�D�VDODULLORU�PHGLL�úL�D�LQGLFDWRULORU�GH�YDULD LH Tabelul 3.22.

AFWLYLW L

Structura

personalului Salariile medii

(zeci mii lei)

Dispersiile

Abaterile medii S WUDWLFH

(zeci mii lei)

&RHILFLHQ LL GH�YDULD LH

A 1 2 3 4 5 I II III

20% 50% 30%

291,2 312,0 344,6

794,56 616,00 310,84

28,187 24,820 17,600

9,68% 7,95% 5,11%

Total 100% 317,62 938,404 30,63 9,64%

ÌQ�H[HPSOXO�DQDOL]DW�VH�SRW�WUDJH�XUP WRDUHOH�FRQFOX]LL� • DWkW� SH� vQWUHDJD� VRFLHWDWH� FRPHUFLDO � FkW� úL� SH� ILHFDUH� DFWLYLWDWH� VHULLOH� VXQW� RPRJHQH��

deoarece s-DX� RE LQXW� YDORUL�PLFL� SHQWUX� FRHILFLHQ LL� GH� YDULD LH��'DF � VH� LHUDUKL]HD] �VHULLOH�GXS �FULWHULXO�YHQLWXULORU�VDODULDOH��RUGLQHD�RPRJHQLW LORU�HVWH��DFWLYLWDWHD�D�,,,-a (5,11%); activitatea a II-D���������úL�PDL�SX LQ�RPRJHQ �SULPD�DFWLYLWDWHD���������FDUH�GHS úHúWH�FX�IRDUWH�SX LQ�JUDGXO�GH�RPRJHQLWDWH�SH total (9,64%);

• DYkQG� FRHILFLHQ L� PLFL� GH� YDULD LH��PHGLLOH� VXQW� vQ� WRDWH� FD]XULOH� FX� XQ� JUDG�PDUH� GH�UHSUH]HQWDWLYLWDWH� ID � GH� YDORULOH� LQGLYLGXDOH� GLQ� FDUH� DX� IRVW� FDOFXODWH�� GHRDUHFH� WR L�FRHILFLHQ LL�GH�YDULD LH�VXQW�VXE�����

3. Seriile din tabelul 3.1���VH�SRW�SUH]HQWD�úL�FX�IUHFYHQ H�DEVROXWH� ùWLLQG�F �QXP UXO�WRWDO�DO�VDODULD LORU�D�IRVW�GH�����úL�DYkQG�UHSDUWL LD�ORU�UHODWLY �SH�DFWLYLW L�

VH�RE LQH�

• activitatea I 80100

20400 =⋅ VDODULD L

• activitatea a II-a 200100

50400 =⋅ salaria L

• activitatea a III-a 120100

30400 =⋅ VDODULD L

$SOLFkQG� JUHXW LOH� VSHFLILFH� GLQ� ILHFDUH� DFWLYLWDWHD� OD� QXP UXO� GH� VDODULD L�� VH� FDOFXOHD] �IUHFYHQ HOH� DEVROXWH�� FDUH� VH� GHWHUPLQ � QXPDL� vQ� QXPHUH� vQWUHJL� UHIHULQGX-se la persoane (vezi tabelul 3.23).

5HSDUWL LD�VDODULD LORU�SH�FHOH�WUHL�DFWLYLW L�vQ�IXQF LH�GH�YHQLWXULOH�VDODULDOH Tabelul 3.23.

*UXSH�GXS �VDODULL�

(zeci mii lei) Activitatea I

Activitatea II

Activitatea III

Total

sub 250 8 - - 8

250-270 10 - - 10

270-290 20 40 - 60

290-310 22 70 - 92

310-330 12 40 24 76

330-350 8 30 56 94

350-370 - 20 28 48

����úL�SHVWH - - 12 12

Total 80 200 120 400

&DOFXOXO�PHGLHL�úL�GLVSHUVLHL�SH�WRWDO��SH�ED]D�IUHFYHQ HORU�DEVROXWH Tabelul 3.24.

*UXSH�GXS �VDODULL�

(zeci mii lei) 1XP UXO�

salDULD LORU ( )jn

Centrul de interval ( )jy

h

ay j −

a = 340 h = 20

jj nh

ay⋅

−

jj nh

ay⋅

− 2

A 1 2 3 4 5 sub 250 8 240 -5 -40 200

250-270 10 260 -4 -40 160

270-290 60 280 -3 -180 540 290-310 92 300 -2 -184 368

310-330 76 320 -1 -76 76

330-350 94 340 0 0 0

350-370 48 360 1 48 48

����úL�SHVWH 12 380 2 24 48

Total 400 - -448 1440

6,31734020400

448

1

1 =+⋅−=+⋅

−

=∑

∑

=

= ahn

nh

ax

yk

ij

k

ij

j

zeci mii lei/salariat

24,938)6,317340(400400

1440)( 222

1

1

2

2 =−−⋅=−−⋅⋅

−

=∑

∑

=

=ayh

n

nh

ay

k

ij

k

ij

j

yσ

Deci s-a ajuns la aceleDúL� YDORUL� úL� SHQWUX� PHGLH� úL� SHQWUX� GLVSHUVLH� FD� vQ� FD]XO� IRORVLULL�IUHFYHQ HORU�UHODWLYH�

PROBLEME PROPUSE

1. Pentru 40 de agen L�HFRQRPLFL�FH�DFWLYHD] �vQ�DFHODúL�GRPHQLX�GH�DFWLYLWDWH� s-au înregistrat

datele privind profitul realizat (mil. lei) în luna august: 40, 63, 59, 52, 62, 50, 45. 51, 54, 50, 48, 55, 59, 47, 57, 53, 46, 55, 64, 53, 42, 51, 53, 56, 52, 58, 49, 54, 60, 46, 41, 47, 42, 44, 63, 58, 49, 55, 50, 53.

Se cere:

1. V �VH�JUXSH]H�FHL����GH�DJHQ L�HFRQRPLFL�SH�LQWHUYDOH�GH�YDULD LH�HJDOH��GXS �SURILWXO�UHDOL]DW� 2. V �VH�UHSUH]LQWH�JUDILF�UHSDUWL LD�RE LQXW �OD�SXQFWXO�SUHFHGHQW� 3. V �VH�FDOFXOH]H�LQGLFDWRULL�WHQGLQ HL�FHQWUDOH�úL�V �VH�LQWHUSUHWH]H�UHOD LD�GLQWUH�HL� 4. V �VH�YHULILFH�UHSUH]HQWDWLYLWDWHD�PHGLHL� 5. V �VH�GHWHUPLQH�LQGLFDWRULL�FDUH�VHSDU �����GLQ�DJHQ LL�HFRQRPLFL�VLWXD L�OD�FHQWUXO�

UHSDUWL LHL� 6. V �VH�P VRDUH�JUDGXO�GH�DVLPHWULH�

2. 3HQWUX�R�XQLWDWH�HFRQRPLF �VH�FXQRVF�GDWHOH�

Tabelul 3.25.

Grupe de muncitori GXS �P ULPHD�

SURGXF LHL��PLL�EXF��

Nr. muncitori

Sub 10 2 10-12 6 12-14 10 14-16 20 16 -18 18 ���úL�SHVWH 6 Total 60

Se cere:

1. V �VH�UHSUH]LQWH�JUDILF�GLVWULEX LD�PXQFLWRULORU�GXS �P ULPHD�SURGXF LHL� 2. V �VH�YHULILFH�GDF �GLVWULEX LD�PXQFLWRULORU�GXS �P ULPHD�SURGXF LHL�HVWH�RPRJHQ � 3. V �VH�FDOFXOH]H�QLYHOXO�FHQWUDOL]DW�DO�SURGXF LHL�SHQWUX�PXQFLWRULL�FX�SURGXF LD�GH�FHO�

SX LQ����PLL�EXF� 4. V �VH�FDOFXOH]H�LQGLFDWRUXO�FDUH�VHSDU �����GLQ�PXQFLWRUL�GH�UHVWXO�PXQFLWRULORU�

3. 3HQWUX�����GH�VDODULD L�DL�XQHL�VRFLHW L�FRPHUFLDOH��VH�FXQRVF�GDWHOH�

Tabelul 3.26.

Grupe GH�VDODULD L��

GXS �YHFKLPH��DQL� 1XP U�

VDODULD L Timpul mediu nelucrat (min.)

Abaterea medie S WUDWLF �SULYLQG�

timpul nelucrat (min.)

A 1 2 3 sub 10 20 30 4,8 10-20 90 45 9,0 ���úL�SHVWH 40 55 6,6

Se cere: 1. V �VH�UHSUH]LQWH�JUDILF�VWUXFWXUD�VDODULD LORU�GXS �YHFKLPH� 2. V �VH�YHULILFH�RPRJHQLWDWHD�SH�JUXSH�úL�SH�WRWDO� 3. V �VH�GHWHUPLQH�vQ�FH�P VXU �YDULD LD�WLPSXOXL�QHOXFUDW�VH�GDWRUHD] �GHRVHELULORU�SULYLQG�

YHFKLPHD�vQ�PXQF �

4. Dintr-XQ�VRQGDM�VWDWLVWLF�GH�����SURSRU LRQDO�VWUDWLILFDW�V-DX�RE LQXW�GDtele:

Tabelul 3.27.

*UXSH�GH�DJHQ L�HFRQRPLFL�GXS �P ULPHD�SURILWXOXL� (mil. lei)

*UXSH�GH�DJHQ L�

HFRQRPLFL�GXS �

P ULPHD�FDSLWDOXOXL�

(mil. lei) sub 10 10-14 14-18 18-22 22-26 ���úL�

peste

Total

A 1 2 3 4 5 6 7 I 5 10 5 - - - 20 II - 10 15 25 15 5 70 III - - 10 20 15 5 50 Total 5 20 30 45 30 10 140

Se cere:

1. V �VH�UHSUH]LQWH�JUDILF�GLVWULEX LD�DJHQ LORU�HFRQRPLFL�GXS �P ULPHD�SURILWXOXL��SH�WRWDO� 2. V �VH�YHULILFH�UHJXOD�GH�DGXQDUH�D�GLVSHUVLLORU�úL�V �VH�DUDWH�GDF �IDFWRUXO�GH�JUXSDUH�HVWH�

semnificativ sau nu; 3. V �VH�FDUDFWHUL]H]H�RPRJHQLWDWHD�SH�ILHFDUH�JUXS �úL�SH�WRWDO��

4. V �VH�P VRDUH�JUDGXO�GH�DVLPHWULH�SHQWUX�JUXSD�D�,,-a; 5. V �VH�YHULILFH�UHJXOD�DGXQ ULL�GLVSHUVLLORU�SHQWUX�FDUDFWHULVWLFD�³DJHQ L�HFRQRPLFL�FX�

profitul mai mic decât 18 mil.lei.

5. 3HQWUX�XQ�HúDQWLRQ�GH�����GH�DJHQ L�HFRQRPLFL��VH�FXQRVF�GDWHOH�

Tabelul 3.28.

Grupe tipice de DJHQ L�HFRQRPLFL�

GXS �FLIUD� de afaceri

Structura DJHQ LORU�

economici (%)

Profitul mediu (mil.lei/ag.ec.)

Dispersia privind P ULPHD�

profitului

A 1 2 3 sub 20 50 14,4 11,52 20-30 30 19,68 11,28 ���úL�SHVWH 20 30,8 5,76

Se cere:

1. V �VH�SUHFL]H]H�vQ�FH�SURSRU LH�FLIUD�GH�DIDFHUL�LQIOXHQ HD] �YDULD LD�SURILWXOXL� 2. V �VH�SUHFL]H]H�FDUH�JUXS �HVWH�PDL�RPRJHQ �� 3. V �VH�UHSUH]LQWH�JUDILF�SURILWXO�WRWDO�vQ�IXQF LH�GH�IDFWRULL�GH�LQIOXHQ �