Lecture 3 - Conditional Frequency Distributions

LECTURE 3 n of modalities of one variable + n of modalities of the other ones Conditional frequency distributions Males Total Modality Frequency Modality Frequency Females Modality Frequency Total Modality Frequency Females Males 5 4 5 55 9 5 5 9 6 24 6 25 G 49 orgin 6 24 25 49 7 95 78 103 7 198 7 95 103 198 8 98 8 105 8 203 8 105 203 9 6 9 17 9 23 9 6 17 23 10 10 2 10 3 10 3 12 1 12 12 L tonderst given by 7 conditional distribution: GENDER HOURS. 7 ↓ 2 conditional distribution: 0 HOURS GENDER =2=Female and male > We call the variable we are studying (hours of sleep), X the variable through which we extract the statistical units to be considered in the analysis (gender) ➤ We define the variable Y Conditional on X=x and we write Y/x=x to express the restriction of Y to the modality X=x X x 14,2 1 > The distribution of the variable Y without conditioning with respect to au X modality is called the marginal distribution Relative frequencies By dividing an absolute frequency by the total number of statistical units we obtain the so-called relative frequencies.   ABSOLUTE FREQUENCIES RELATIVE FREQUENCIES TOTAL NUMBER OF OBSERVATION 4- Freq Rel F Rel. Fem Mocality FM Tol Medalty F M Tol [16:02 0.02 0.02 24-249 0.11 0.10 0.10 X400 03115 7 95-183-195 042 646641 (400.000 43 47430 3.100 08-105203 043 041 042 27 003 0.07 0.05 (750000 (1000 2000 10 12- 10 12 1 12 COD 0.00 0.00 They allow comparisons of distributions boks on sifferent bers of statistical under Frequency distribution: NOTATION y, modality i-th/interval (Ci-1; Ci) of the variable Y, for i= 1,2,...,k • absolute frequency: number of statistical units that present the modality/interval yi IN total number of observations (Nona+na+...+nx) •fi relative frequency (fi) modality/interval Freq Rel.F. EXAMPLE X f/N X-F 228 0.47 1 258 0.58 E N-486 /N Total N 1- Summation: N= ∑ni=n,+na+...+n/sum of all ni for i that goes from 1 to k/ Lproperties 1 ) 2 Σάψα Σψί 32aak-a+a+atta Questions: 1' Is it a data matrix? No 2. How many variable are described? 3 2.1 What type are they? Categorical Nominal 2.2 What is their support? 3. How many statistical units are there? 24 4. Do the data refers to a census or sample survey? ConsUS 'Support = list of the modalities observed L. For gender they are female, male Le For Admit they are admit, reject L. For Dept we are not sure these are the only modalities Data matrix Adm Gender Dep Adm 312 A Cumulative Frequency • the sum of the absolute relative frequencies for the classes Yi it makes sense if the variable modalities have a ranking: y, YzC... Yк \[F_{j}\cdot f_{1}+...+f_{j}=\sum_{n=1}^{j}f_{n}\] EXAMPLE Freq Freq Freq Freq Abs Rel. Rel. \[N_{\eta}=\sum_{n=1}^{n}n_{h}=n_{\eta}=9\] Modality Al Rel. Rel. \[=N_{2}=\sum_{n=1}^{3}n_{n}=n_{1}+n_{2}=9+49=58\] 0.119 5 9 9 0.019 0.019 \[=F_{\mu}=\sum_{i=1}^{n}f_{\mu}=f_{\mu}+f_{\mu}+f_{\partial}+f_{\mu}=0.019+0.101+0.4~07+0.418=0.944.\] 6 49 58 0.101 7 198 256 10.407 0.527 8 203 450 6418 0944 9 23 482 0.047 0.992 10 3 485 0.006 0.998 486 1.000 12 1 0.002 Empirical distribution function Lrepresentation of the trend of a quantitative variable EMPIRICAL DISTRIBUTION NUMBER OF OBSERVATION&Yy FUNCTION COMPUTED IN Y = TOTAL NUMBER OF OBSERVATION Let Y be the variable to analyze, the Empirical Cumulative Distribution Function computed on the sample \((y_{1},y_{2},...,y_{N})\) is the function ↓ ECDF \[F_{Y}(y)=\frac{(\pi)\{Y\le y\}}{N}\] number of statistical units that have ye a given specipic modality, divided by the total number of modality The domain of the function is given by R The suppors is the interval [0,1] EXAMPLE Starting from the distribution of the cumulative relative frequencies, the RCDF is defined as follows: T Freq. Freq Mod abs F cum. P hal-dosed 10 3 485 105/406 1012 12 1 486 \[x\ge12\] 124