STATISTICS STUDY NOTES UNIT I MEASURES OF CENTRAL TENDENCY DISCRETE SERIES. Direct Method. N Short-cut Method. X A f d N Step-Deviation Method

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1 STATISTICS STUDY OTES UIT I MEASURES OF CETRAL TEDECY IDIVIDUAL SERIES ARITHMETIC MEA: Direct Method X X Short-cut Method X A d Step-Deviation Method X A d i MEDIA: th Size of term MODE: Either by inspection or the value that occurs largest number of times EMPIRICAL RELATIO: DISCRETE SERIES Direct Method X f X Short-cut Method X A f d Step-Deviation Method X A f d i Size of th term Grouping Method determines that value around which most of the frequencies are concentrated. Mode = 3 Mean Median COTIUOUS SERIES Direct Method X f X Short-cut Method X A f d Step-Deviation Method X A f d i th Size of term Median = L c. f. i Mode = L f f 0 i f (f f ) 0 Problems. From the following data compute Arithmetic Mean Marks o. of students Marks Midvalue o. of students f x X f

2 =

3 Arithmetic Mean X f X Calculate Arithmetic Mean from the following data Marks o. of students The class intervals are unequal but still to simplify calculations we can take 5 as common factor. Marks Midvalue o. of students d f d x f (x - 45) / = Arithmetic Mean X A f d i A = 45, f d= - 44, = 50, I = 5 X Find the missing frequency from the following data Marks o. of Students The arithmetic mean is 34 marks. Let the missing frequency be denoted by X

4 Marks Midvalue f` f x x X 35X = 70 + X 00+35X X f x X 70 X 34 (70 + X) = X X = X 35X 34X = X = Calculate the Median and Mode from the following data Central size Frequencies Since we are given central values first we determine the lower and upper limits of the classes. The class interval is 0 and hence the first class would be 0 0. Class Midvalue f d f d c.f. x (x 55) / fd = - 54 Calculation of Median: Med = size of th term = 94 47

5 Median lies in the class Median = L / c. f. i f M = Calculate the median and mode of the data given below. Using then find arithmetic mean Marks o. of Students Marks f` c.f = 80 Calculation of Median: Med = size of th term = th item Median lies in the class 0 30 Median = L / c. f. i f M = 7.73 Mode lies in the class is 0 30 f f 0 5 Mode = L i f (f f ) 44 (5 0) 0

6 MEASURES OF DISPERSIO IDIVIDUAL OBERSERVATIOS DISCRETE& COTIUOUS SERIES QUARTILE DEVIATIO: Q.D. = Q 3 Q Coefficient of Q.D. = Q Q 3 Q Q 3 Quartile Deviation: Q.D. = Q Q 3 Coefficient of Q.D. = Q 3 Q Q Q 3 STADARD DEVIATIO: Actual Mean Method: Actual Mean Method: f ( X X ) f ( X X ) Assumed Mean Method: d d Step Deviation Method Assumed Mean Method: fd fd Step Deviation Method d d i fd fd i C.V. 00 X C.V. 00 X. Find the Mean and standard deviation from the following distribution Mid value o. of Students

7 Midvalue o. of Students d f d f d x f (x 4) / f = 50 fd = - 98 d 548 Mean X A f d i Standard deviation fd fd i Find the Standard deviation and Coefficient of Variation from the following data Marks o. of students Up to 0 Up to 0 30 Up to Up to Up to Up to 60 0 Up to 70 Up to Class Midvalue o. of Students d f d f d x f (x 35) /

8 = 30 fd =5 fd 753

9 Mean X A f d i Standard deviation fd fd i C.V X The scores of two batsmen A and B in ten innings during a certain season are: A B Find which of the two batsmen more consistent in scoring X X X X X Y Y Y Y Y X = 460 Batsman A: Y = 500 X X = 6500 Y Y = 5968 Mean X X (X 6500 X)

11 C.V X 46 Batsman B: Mean Y Y Y Y C.V Y 50 Since Coefficient of Variation is less in the case of Batsman B, we conclude that the Batsman B is more consistent. 4. Calculate the Quartile deviation and the coefficient of quartile deviation from the following data Marks o. of students Below 0 8 Below 40 0 Below Below Below Marks f` c.f = 80 Q is the size of / 4 th item. Q lies in the class 0 40 Q L / 4 c.f. i f Q3 is the size of 3 / 4 th item. Q3 lies in the class 60 80

12 Q L 3 / 4 c. f. i f 0 Q.D. Q Q Q Q 30 3 Coefficient of QD Q Q Calculate the Inter-Quartile range and the coefficient of quartile deviation from the following data Marks o. of students Above 0 50 Above 0 40 Above 0 00 Above Above Above Above Above 70 4 Above 80 0 Marks f` c.f = 50 Q is the size of / 4 th item. Q lies in the class 0 0 Q L / 4 c.f. i f 40 Q3 is the size of 3 / 4 th item. Q3 lies in the class Q 3 L 3 / 4 c.f. i f 40

13 Inter Quartile Range Q Q Coefficient of QD Q Q Q Q 75 3 MOMETS Moments about mean (X X) 0 (X X) 3 4 (X X) (X X) 3 4 In a Frequency distributiom f (X X) 0 f (X X) Moments about arbitrary origin (X A) ( X A) In a frequency distribution f (X A) f ( X A ) Moments about mean 3 f (X X) 3 4 f (X X) 4 3 (X A)3 4 (X A) f (X A) f (X A) ,

14 SKEWESS AD KURTOSIS Karl Pearson s Skewness = Mean - Mode Bowley s Skewness = Q3 + Q- Med Mean Mode Karl Pearson s coefficient of Skewness = Bowley s coefficient of Skewness = Q 3+ Q - Med Q 3 - Q, , Calculate the coefficient of skewness by Karl Pearson s method and the values of β and β from the following data Profits o. of (in lakhs) companies Solution : Class Midvalue o. of d f d f d f d 3 f d 4 x Students (x 35) f / = 00 fd 4 fd 54 fd 3 6 fd 4 490

15 X A f d i

16 Modal class Mode L f f 0 i f (f f ) 60 (0 ) fd fd i Karl Pearson s coefficien t of Skewness = fd 0.4 f d.54 f d f d , Mean Mode

18 . By measuring the quartiles find a measure of skewness for the following distribution Annual Sales o. of firms Less than 0 30 Less than 30 5 Less than Less than Less than Less than Less than Less than Less than Sales f` c.f = 680 Q lies in the class 0-30 Q L / 4 c.f. i f 95. Q3 lies in the class Q L 3 / 4 c.f. i f 5 Inter Quartile Range Q 3 Q Coefficient of QD Q 3 Q Q 3 Q 75 Median class This is a SAMPLE (Few pages have been extracted from the complete notes:-it s meant to show you

19 the topics covered in the full notes and as per the course outline Download more at our websites: To get the complete notes either in softcopy form or in Hardcopy (printed & Binded) form, contact us on: Call/text/whatsApp / Get news and updates by liking our page on facebook and follow us on Twitter Sample/preview is OT FOR SALE

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