Title 03-Statistics (Part-1)(1).pdf ✅

STATISTICS 3 CHAPTER CONTENTS • Class Mark • Cummulative Frequency • Mean • Median • Mode • Ogive Curve ➢ IMPORTANT POINTS  The word data means information (its exact dictionary meaning is: given facts). Statistical data are of two types : (i) Primary data (ii) Secondary data  When an investigator collects data himself with a definite plan or design in his (her) mind, it is called Primary data.  Data which are not originally collected rather obtained from published or unpublished sources are known as Secondary data.  After collection of data, the investigator has to find ways to condense then in tabular form in order to study their silent features. Such an arrangement is called Presentation of data.  Raw data when put in ascending or descending order of magnitude is called an array or arranged data.  The number of times an observation occurs in the given data is called frequency of the observation.  Classes/class intervals are the groups in which all the observations are divided.  Suppose class-interval is 10-20, then 10 is called lower limit and 20 is called upper limit of the class  Mid-value of class-interval is called Class-mark Class-mark = 2 lower limit + upper limit Class-mark = lower limit + 2 1 (difference between the upper and lower limits)  If the frequency of first class interval is added to the frequency of second class and this sum is added to third class and so on then frequencies so obtained are known as Cumulative Frequency (c.f.).  There are two types of cumulative frequencies (a) less than, (b) greater than ❖ EXAMPLES ❖ Ex.1 Given below are the ages of 25 students of class IX in a school. Prepare a discrete frequency distribution. 15, 16, 16, 14, 17, 17, 16, 15, 15, 16, 16, 17, 15, 16, 16, 14, 16, 15, 14, 15, 16, 16, 15, 14, 15. Sol. Frequency distribution of ages of 25 students Age Tally marks Frequency 14 4 15 8 16 10 17 3 Total 25 Ex.2 Form a discrete frequency distribution from the following scores:- Sol. 15, 18, 16, 20, 25, 24, 25, 20, 16, 15, 18, 18, 16, 24, 15, 20, 28, 30, 27, 16, 24, 25, 20, 18, 28, 27, 25, 24, 24, 18, 18, 25, 20, 16, 15, 20, 27, 28, 29, 16.
Frequency Distribution of Scores Variate Tally marks Frequency 15 4 16 6 18 6 20 6 24 5 25 5 27 3 28 3 29 1 30 1 Total 40 Ex.3 The water tax bills (in rupees) of 30 houses in a locality are given below. Construct a grouped frequency distribution with class size of 10. 30, 32, 45, 54, 74, 78, 108, 112, 66, 76, 88, 40, 14, 20, 15, 35, 44, 66, 75, 84, 95, 96, 102, 110, 88, 74, 112, 14, 34, 44. Sol. Here the maximum and minimum values of the variate are 112 and 14 respectively.  Range = 112 – 14 = 98. It is given that the class size is 10, and Class size Range = 10 98 = 9.8 So, we shoule have 10 classes each of size 10. The minimum and maximum values of the variate are 14 and 112 respectively. So we have to make the classes in such a way that first class includes the minimum value and the last class includes the maximum value. If we take the first class as 14-24 it includes the minimum value 14. If the last class is taken as 104-114, then it includes the maximum value 112. Here, we form classes by exclusive method. In the class 14-24, 14 is included but 24 is excluded. Similarly, in other classes, the lower limit is included and the upper limit is excluded. In the view of above discussion, we construct the frequency distribution table as follows: Bill (in rupees) Tally marks Frequency 14-24 4 24-34 2 34-44 3 44-54 3 54-64 1 64-74 2 74-84 5 84-94 3 94-104 3 104-114 4 Total 30 Ex.4 The marks obtained by 40 students of class IX in an examination are given below : 18, 8, 12, 6, 8, 16, 12, 5, 23, 2,16, 23, 2, 10, 20, 12, 9, 7, 6, 5, 3, 5, 13, 21, 13, 15, 20, 24, 1, 7, 21, 16, 13, 18, 23, 7, 3, 18, 17, 16. Present the data in the form of a frequency distribution using the same class size, one such class being 15-20 (where 20 is not included) Sol. The minimum and maximum marks in the given raw data are 0 and 24 respectively. It is given that 15-20 is one of the class intervals and the class size is same. So, the classes of equal size are 0-5, 5-10, 10-15, 15-20 and 20-25 Thus, the frequency distribution is as given under : Frequency Distribution of Marks Marks Tally marks Frequency 0-5 6 5-10 10 10-15 8 15-20 8 20-25 8 Total 40 Ex.5 The class marks of a distribution are : 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97, 102 Determine the class size, the class limits and the true class limits.
Sol. Here the class marks are uniformly spaced. So, the class size is the difference between any two consecutive class marks  Class size = 52 – 47 = 5 We know that, if a is the class mark of a class interval and h is its class size, then the lower and upper limits of the class interval are a – 2 h and a + 2 h respectively.  Lower limit of first class interval = 47 – 2 5 = 44.5 And, upper limit of first class interval = 47 + 2 5 = 49.5 So, first class interval is 44.5 – 49.5 Similarly, we obtain the other class limits as given under : 102 99.5 -104.5 97 94.5 - 99.5 92 89.5 - 94.5 87 84.5 - 89.5 82 79.5 - 84.5 77 74.5 - 79.5 72 69.5 - 74.5 67 64.5 - 69.5 62 59.5 - 64.5 57 54.5 - 59.5 52 49.5 - 54.5 47 44.5 - 49.5 Class marks Class limits Since the classes are exclusive (continuous) so the true class limits are same as the class limits. Ex.6 The class marks of a distribution are 26, 31, 36, 41, 46, 51, 56, 61, 66, 71. Find the true class limits. Sol. Here the class marks are uniformly spaced. So, the class size is the difference between any two consecutive class marks.  Class size = 31 – 26 = 5. If a is the class mark of a class interval of size h, then the lower and upper limits of the class interval are a – 2 h and a + 2 h respectively. Here h = 5  Lower limit of first class interval = 26 – 2 5 = 23.5 And, upper limit of first class interval = 26 + 2 5 = 28.5  First class interval is 23.5 – 28.5. Thus, the class intervals are: 23.5 – 28.5, 28.5 – 33.5, 33.5 – 38.5,38.5 – 43.5, 43.5 – 48.5, 48.5 – 53.5 Since the classs are formed by exclusive method. Therefore, these limits are true class limits. ➢ CUMULATIVE FREQUENCY A table which displays the manner in which cumulative frequencies are distributed over various classes is called a cumulative frequency distribution or cumulative frequency table. There are two types of cumulative frequency. (1) Less than type (2) Greater than type ❖ EXAMPLES ❖ Ex.7 Write down less than type cumulative frequency and greater than type cumulative frequency. Frequency 140 – 145 10 145 – 150 12 150 – 155 18 155 – 160 35 160 – 165 45 165 – 170 38 170 – 175 22 175 – 180 20 Height (in cm)
Sol. We have Height (in cm) 140–145 145–150 150–155 155–160 160–165 165–170 170–175 175–180 Frequency 10 12 18 35 45 38 22 20 Height Less than type 145 150 155 160 165 170 175 180 Cumulative frequency 10 22 40 75 120 158 180 200 Height Greater than type 140 145 150 155 160 165 170 175 Cumulative frequency 200 190 178 160 125 80 42 20 Ex.8 The distances (in km) covered by 24 cars in 2 hours are given below : 125, 140, 128, 108, 96, 149, 136, 112, 84, 123, 130, 120, 103, 89, 65, 103, 145, 97, 102, 87, 67, 78, 98, 126 Represent them as a cumulative frequency table using 60 as the lower limit of the first group and all the classes having the class size of 15. Sol. We have, Class size = 15 Maximum distance covered = 149 km. Minimum distance covered = 65 km.  Range = (149 – 65) km = 84 km. So, number of classes = 6       = 5.6 15 84  Thus, the class intervals are 60-75, 75-90, 90-105, 105-120, 120-135, 135-50. The cumulative frequency distribution is as given below : Class Tally Frequency Cumulative interval marks frequency 60-75 2 2 75-90 4 6 90-105 6 12 105-120 2 14 120-135 6 20 135-150 4 24 Ex.9 The following table gives the marks scored by 378 students in an entrance examination : 90 -100 6 80 - 90 12 70 -80 12 60 - 70 39 50 - 60 85 40 -50 97 30 - 40 76 20 - 30 36 10 - 20 12 0 -10 3 Mark No. of students From this table form (i) the less than series, and (ii) the more than series. Sol.(i) Less than cumulative frequency table Less than 100 378 Less than 90 372 Less than 80 360 Less than 70 348 Less than 60 309 Less than 50 224 Less than 40 127 Less than 30 51 Less than 20 15 Less than 10 3 (Cumulativ e frequency) Number of students Marks obtained (ii) More than cumulative frequency table More than 89 6 More than 79 18 More than 69 30 More than 59 69 More than 49 154 More than 39 257 More than 29 327 More than 19 363 More than 9 375 More than 0 378 (Cumulativ e frequency) Number of students Marks obtained