Tiêu đề XI - maths - chapter 15- STATISTICS(123-139).pdf ✅

NARAYANAGROUP 123 STATISTICS JEE-MAIN-SR-MATHS VOL-I STATISTICS SYNOPSIS FREQUENCY DISTRIBUTION  Class Limits:The starting and end vlaues of each class are called the lower limit and upper limit respectively of that class. Ex. 1) The lower limit of the class 0-9 is 0 2) The upper limit of the class 50-59 is 59  Class boundaries : The average of the upper limit of a class and the lower limit of the next class is called the upper boundary of that class. The upper boundary of a class becomes the lower boundary of the next class. These boundaries are called True class limits. Ex. 1) 1-10, 11-20, 21-30 ..... are the classes, the lower boundary of the class 11-20 is 10 11 10.5 2   2) 60-69, 70-79, 80-89, 90-99 .......... are the classes, the upper boundary of the class 70-79 is 79 80 79.5 2    Class interval (or) the size of the class : The difference between the lower limits or the upper limits of two consecutive classes is called the Class- interval (or) the size of the class. Ex. The class interval in the frequency distribution with the classes 1-8, 9-16, 17-24 ... length of class = 9-1 = 8  Mid value of the class : Mid value of class 1-10 is 1 10 5.5 2    For over lapping classes 0-10, 10-20, 20-30 etc the class mark of the class 0-10 is 0 10 5 2   3) For non over lapping class 0-19, 20-39, 40-59,...... etc the class mark of the class 20-39 is 20 39 29.5 2          Measures of Central Tendency: One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire data. Such a value is called the central value or an average. The following are the important types of averages: 1. Arithmetic Mean 2. Geometric Mean 3. Harmonic mean 4. Median 5. Mode We consider these measures in three cases (i) Individual series (i.e. each individual observation is given) (ii) discrete series (i.e the observations along with number of times a particular observation called the frequency is given) (iii) continuous series (i.e. the class intervals along with their frequencies are given) Arithmetic Mean  Individual Series : If 1 2 , ......... n x x x are the values of the variable x , then the arithmetic mean usually denoted by x or or E x    is given by 1 2 1 ..... 1 n n i i x x x x x n n       Note: ( ) A.M. ( )     i x A x A n where A is the assumed average. (For individual series)  Discrete Series : If a variable takes values 1 2 , ......... n x x x with corresponding frequencies f1 , f 2 ..... fn then the arithmetic mean x is given by 1 1 2 2 1 2 1 ..... 1 .... n n n i i n i f x f x f x x f x f f f N         , where 1 n i i N f     Continuous Series : In case of a set of data with class intervals, we cannot find the exact value of the mean because we do not know the exact values of the variables. We, therefore, try to obtain an approximate value of the mean. The method of approximate is to replace all the observed values belonging to a class by mid-value of the class. If x1 , x2 ... xn are the mid values of the class intervals having corresponding frequencies f1 , f2 ... fn then we apply the same formula as in discrete series. 1 1 1 , n n i i i i i x f x N f N      

NARAYANAGROUP 125 STATISTICS JEE-MAIN-SR-MATHS VOL-I W.E-4: If the mean of 9 observations is 100 and mean of 6 observations is 80, then the mean of 15 observations is Sol. 1 1 2 2 n x and n x     9, 100 6, 80 1 1 2 2 1 2 n x n x x n n    9 100 6 80 92 9 6       W.E-5: If a variate X is expressed as a linear function of two variates U and V in the form X aU bV   then the mean X of X is Sol. we have      X a U b V 1 1 1 X X a U b V . . n n n          X aU bV W.E-6:If the arithmetic mean of the numbers 1 2 3 , , .... n x x x x is x , then the arithmetic mean of the numbers 1 2 3 , , ,.... n ax b ax b ax b ax b     , where a, b are two constants, would be Sol. Required mean       1 2 .... n ax b ax b ax b n          1 2 .... n a x x x b ax b n        1 2 .... n x x x x n                W.E-7: If the mean of the numbers 27 ,31   x x , 89 x , 107 ,156   x x is 82, then the mean of 130 ,126 ,   x x 68 ,  x 50 ,1   x x is Sol. Given 27 31 89 107 156          82 5          x x x x x           82 5 410 5 410 410 5 0 x x x Therefore, the required mean is 130 126 68 50 1 5 x x x x x x           375 5 75 7  x   W.E-8: A student obtain 75%, 80% and 85% in three subjects. If the marks of another subject is added, then his average cannot be less than Sol. Marks obtained from three subjects out of 300 is 75 80 85 240    If the marks of another subject is added, then the marks will be  240 out of 400 minimum average marks 240 60% 4   [when marks in the fourth subject = 0] W.E-9: The mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively. The correct mean is Sol. Sum of 100 items = 49 × 100 = 4900 sum of items added = 60+ 70+80=210 new sum = 4900+210–110=5000  correct mean 5000 50 100   W.E-10: The mean weight per student in a group of seven students is 55kg. If the individual weights of six students are 52, 58, 55, 53, 56 and 54, then the weight of the seventh student is Sol. The total weight of seven students is 55×7 = 385kg The sum of the weights of six students is 52+58+55+53+56+54=328kg Hence, the weight of the seventh student is    385 328 57kg Geometric Mean  In case of individual series 1 2 , ............. n x x x G.M. = ( 1 2 ............. n x x x ) 1/n In case of discrete or continuous series G.M.   1 2 1/ 1 2 .... n N f f f n  x x x , where 1 n i i N f    W.E-11: The geometric mean of the numbers 2 3 3,3 ,3 ,....,3n is Sol.   1/ 2 . 3.3 ......3 n n   G M 1 2 .....  1 1 2 2 3 3 3 n n n n n n        
126 NARAYANAGROUP STATISTICS JEE-MAIN-SR-MATHS VOL-I  Harmonic Mean: The harmonic mean is based on the reciprocals of the value of the variable H.M. = 1 2 1 1 1 1 1 .... n n x x x          or 1 1 1 1 n H n x i i   (Incase of Individual series) and 1 1 1 1 n i i i f H N x    (in case of discrete series or continuous series) If x1 , x2 , ...xn > 0 then it is known that A.M G.M H.M   W.E-12: Find the harmonic mean of 1 2 3 , , ,....., 2 3 4 1 n n , occurring with frequencies 1, 2, 3,.....n, respectively. Sol. We know that, Harmonic mean f f x              1 1 2 3 .... 2 n n f n         and   1 2 3 .... 1/ 2 2 / 3 3 / 4 / 1 f n x n n        3 2 4 3  1 2 ... 2 3 n n n                2 3 4 .... 1 n n  Which is an arithmetic progression with a = 2 and d = 1. By the formula of sum of n term of an A.P, 2 1    2 f n a n d x                      we have 2 2 1 3    2 2 n n       n n  Harmonic mean       2 1 1 2 3 3 2 3 n n n n n n n n           Median  Individual Series : If the items are arranged in ascending or descending order of magnitude then the middle value is called median. In case of odd number of values Median = size of 1 2 n th  item. In case of even number of values Median = average of 2 n th and 2 2 n  th observation.  Discrete Frequency Distribution :Arrange the data in ascending or descending order. Find the cumulative frequencies. Apply the formula : Median = Size of        2 N 1 th item (N is odd) = 1 observation 2 2         th N 1 observation 2           th N (N is even) N =  i f = sum of frequencies  Continuous Frequency Distribution : Consider the cumulative frequency (c.f.). Find 2 N , where 1 n i i N f    . Find the cumulative frequency (c.f.) just more than N/2. The corresponding value of x is median.In case of continuous distribution, the class corresponding to c.f. just more than 2 N is called the median class and the median is obtained by Median = 2 h N l C f         Where l  the lower limit of the median class; f  the frequency of the median class; h  the width of the median class; C  the c.f. of the class preceding to the median class and 1 n i i N f   