Measures of Central Tendency. SOCY601 Alan Neustadtl

Transcription

1 Measures of Central Tendency SOCY601 Alan Neustadtl

2 Measures of Central Tendency A measure of central tendency s a sngle number used to represent the center of a group of data. Dfferent varables may possess dfferent numercal characterstcs. So dfferent measures of central tendency may better summarze the varable. The basc measures are the: mode medan mean Ths class of measures can be calculated on grouped or ungrouped data. The dfference s n how the data values are weghted.

3 The Mode The mode s the: most frequently occurrng value n a group or raw scores value of the group that contans the most cases n grouped data

4 The Mode Example Frequency Dstrbuton of Sex n the 2000 General Socal Survey Race f Whte Black Other Total 2, ,817

5 The Medan The medan s defned as the mddle value (case) of n values of objects arranged n order of sze. For an odd number of cases, the mddle case wll n + 1 be equal to the case. 2 For an even number of cases, the mddle case n n wll be halfway between the and the case.

6 The Medan Example Rank Name Equtable Lfe Insurance Morgan Guaranty Trust Chemcal Bank of New York Frst Natonal Bank Chase Manhattan Bank Interlockng Drectorates

7 The Medan Example Rank Name Equtable Lfe Insurance Morgan Guaranty Trust Chemcal Bank of New York Frst Natonal Bank Chase Manhattan Bank New York Lfe Interlockng Drectorates

8 The Medan Grouped Data Stated Lmts 2,000-2,900 3,000-3,900 4,000-4,900 5,000-5,900 6,000-6,900 7,000-7,900 True Lmts 1,950-2,950 2,950-3,950 3,950-4,950 4,950-5,950 5,950-6,950 6,950-7, Look for the nterval contanng the medan or the f F n 189 = = Number of Cases Less Than: $2,950 $3,950 $4,950 $5,950 $6,950 $7,950 n 2 case.

9 The Medan Grouped Data Stated Lmts True Lmts F F Number of Cases Less Than: 2,000-2,900 1,950-2, $2,950 3,000-3,900 2,950-3, $3,950 4,000-4,900 3,950-4, $4,950 5,000-5,900 4,950-5, $5,950 6,000-6,900 5,950-6, $6,950 7,000-7,900 6,950-7, $7,950 There are 51 cases n ths nterval. We dvde the nterval nto 51 equal sub-ntervals equal to $ $1, = $19.61

10 The Medan Grouped Data Then we smply count the sub-ntervals from the lower class lmt untl we come to the medan. We could also get ths number by subtractng 81 from 94.5, the locaton of the medan.

11 The Medan Grouped Data n F md = l + 2 f Where: l=lower lmt of the nterval contanng the medan F=cumulatve frequency correspondng to the lower lmt f=number of cases n the nterval contanng the medan =wdth of the nterval contanng the medan

12 The Mean For Populatons: µ = N N For Samples: = n n

13 The Mean n = = 1 n = n 1 = n

14 The Mean as the Center of Gravty

15 Propertes of the Mean The mean has the algebrac property that the sum of the devatons of each score from the mean wll always be zero. Symbolcally: n = 1 ( ) = 0

16 Propertes of the Mean The sum of the squared devatons of each score from the mean s less than the sum of the squared devatons from any other constant (number). Symbolcally: n = 1 ( ) 2 = mnmum

17 Proof that: ( ) = 0 Gven: ( ) By dstrbuton, we can rewrte ths expresson as: The mean s a constant. The sum of a constant s equal to n tmes that constant. So, we can rewrte ths expresson as: n

18 Proof that: ( ) = 0 The mean s a constant. The sum of a constant s equal to n tmes that constant. So, we can rewrte ths expresson as: We also know the basc defnton of the mean and can substtute t: n n n The n s cancel: n n = 0

19 Sum of Squared Devatons About the Mean The logc of ths proof s that f we subtract any number other than the mean from each value of, square that amount, and sum up these values for all values of, we wll get a number that s larger than f we had carred out the same procedure usng the mean of. Let's call ths other number -bar prme and start wth the orgnal expresson: However, we wll strp out the summaton and exponentaton, and substtute -bar prme for the mean: n ( ) = 1 2 ( )

20 Sum of Squared Devatons About the Mean Now we can add and smultaneously subtract the actual mean to and from ths expresson. Ths has no "net" effect on the expresson. Ths s equal to: ( ) = ( ) + ( ) Squarng both sdes of the expresson brngs us a step closer to the orgnal equaton: ( ) 2 = ( ) + ( ) 2 ( ) Because a+ b = a + 2 ab+ b, when expanded ths s equal to:

21 Sum of Squared Devatons About the Mean Because a+ b = a + 2 ab+ b, when expanded ths s equal to: ( ) ( ) 2 = ( ) ( )( ) + ( ) 2 Now, add the summaton symbol back n on both sdes of the expresson and wth a lttle bt of algebrac manpulaton we get: ( ) = ( ) + 2( )( ) + ( ) = ( ) + 2( ) ( ) + ( ) 2 2 = ( ) + ( ) 2 2 = ( ) + n( )

22 The Mean from Grouped Data Ungrouped Data n = Grouped Data = 1 = = 1 n n w The only dfference n these formulas s the weght, w. Wth ungrouped data, the weght s mplctly equal to one. w

23 The Mean from Grouped Data Example Type of PAC Contrbutons N Weghted Corporatons $37, ,875 $69,563, Labor $112, $41,832, Non-Connected $13, ,318 $18,231, T/M/H $62, $52,848, Cooperatve $54, $3,034, W/O Stock $27, $4,063, Sums $307, ,621 $189,574, Averages $51, $41,024.61

24 The Mean from Grouped Data Example Annual Income of Amercan Men n 1975 Annual Income Percent of Men 36% 23% 20% 11% 5% 3% 2% Mdpont Weghted by Mdpont average=7.69 = n = 1 w w

25 The Mean from Grouped Data Example mean 7.7

26 Percentles A percentle s the outcome or score below whch a gven percentage of observatons fall, Where: P L ( p )( n ) c W p = p + f p P =the score f the th percentle L p =the true lower lmt of the nterval contanng the th percentle p = the th percentle wrtten as a proporton (e.g. 75 th = 0.75) n=the total number of observatons c p =the cumulatve frequency up to but not ncludng the nterval contanng P f p =the frequency n the nterval contanng the th percentle W =the wdth of the nterval contanng P ; W=U p -L p

27 Percentles Stated Lmts True Lmts f F Number of Cases Less Than: 2,000-2,900 1,950-2, $2,950 3,000-3,900 2,950-3, $3,950 4,000-4,900 3,950-4, $4,950 5,000-5,900 4,950-5, $5,950 6,000-6,900 5,950-6, $6,950 7,000-7,900 6,950-7, $7,950 P 50 (.5)( 189) 81 = $4, ,000 = $5,

28 Percentles Stated Lmts True Lmts f F Number of Cases Less Than: 2,000-2,900 1,950-2, $2,950 3,000-3,900 2,950-3, $3,950 4,000-4,900 3,950-4, $4,950 5,000-5,900 4,950-5, $5,950 6,000-6,900 5,950-6, $6,950 7,000-7,900 6,950-7, $7,950 P 75 (.75)( 189) 132 = $5, ,000 = $5,