Tabular Representation Of Statistical Data
Class 9th Mathematics RD Sharma Solution
- What do you understand by the word statistics? (i) Singular form (ii) Plural…
- Describe some fundamental characteristics of statistics.
- What are: (i) Primary data? (ii) Secondary data? Which of the twothe primary or…
- Why do we group data?
- Explain the meaning of the following terms: (i) Variable (ii) Class-interval…
- The ages of ten students of a group are given below. The ages have been…
- The monthly pocket money of six friends is given below: Rs. 45, Rs. 30, Rs. 40,…
- Write the class-size in each of the following: (i) 0-4, 5-9, 10-14 (ii) 10-19,…
- The final marks in mathematics of 30 students are as follows: 53, 61, 48, 60,…
- The weights of new born babies (in kg.) in a hospital on a particular day are…
- The number of runs scored by a cricket player in 25 inning are as follows :…
- The class size of a distribution is 25 and the first class-interval is…
- Write the class size and class limits in each of the following: (i) 104, 114,…
- Following data gives the number of children in 40 families: 1, 2, 6, 5, 1, 5,…
- The marks scored by 40 students of class IX in mathematics are given below:…
- The heights (in cm) of 30 students of class IX are given below: 155, 158, 154,…
- The monthly wages of 30 workers in a factory are given below: 83.0, 835, 890,…
- The daily maximum temperatures (in degree Celsius) recorded in a certain city…
- Construct a frequency table with equal class intervals from the following data…
- The daily minimum temperature in degrees Celsius recorded in a certain Arctic…
- The blood groups of 30 students of class VIII are recoded as follows: A, B, O,…
- Three coins were tossed 30 times. Each time the number of heads occurring was…
- Thirty children were asked about the number of hours they watched T.V.…
- Tally marks are used to findA. Class intervals B. Range C. Frequency D. Upper limits…
- The difference between the highest and lowest values of the observations is calledA.…
- The difference between the upper and the lower class limits is calledA. Mid points B.…
- In the class intervals 10-20, 20-30, 20 is taken inA. The interval 10-20 B. The…
- In a frequency distribution, the mid value of a class is 15 and the class interval is…
- The mid-value of a class interval is 42. If the class size is 10, then the upper and…
- The number of times a particular item occurs in a given data is called itsA. Variation…
- The width of each of nine classes in a frequency distribution is 2.5 and the lower…
- Let l be the lower class limit of a class-interval in a frequency distribution and m be…
- The following marks were obtained by the students in a test: 81, 72, 90, 86, 85, 92,…
- Tally are usually marked in a bunch ofA. 3 B. 4 C. 5 D. 6
- Define cumulative frequency distribution.
- Explain the difference between a frequency distribution and a cumulative…
- The marks scored by 55 students in a test given below: Marks: 0-5 5-10 10-15…
- Following are the ages of 360 patients getting medical treatment in a hospital…
- The water bills (in rupees) of houses in a certain street for the period 1.1.98…
- The number of books in different shelves of a library are as follows: 30, 32,…
- Given below are the cumulative frequencies showing the weights of 685 students…
- The following cumulative frequency distribution table shows the daily…
- Given below is a cumulative frequency distribution table showing the ages of…
Exercise 22.1
Question 1.What do you understand by the word “statistics”?
(i) Singular form
(ii) Plural form
Answer:
The word ‘Statistics’ is used in both singular as well as its plural sense.
(i) Statistics may be defined as the science of collection, presentation, analysis and interpretation of numerical data.
(ii) Statistics means numerical facts or observations collected with definite purpose.
Question 2.
Describe some fundamental characteristics of statistics.
Answer:
Fundamental characteristics of Statistics are:
(i) A single observation does not form statistics. Statistics is a sum of total number of observations.
(ii) Statistics are expressed quantitatively not qualitatively.
(iii) Statistics are collected with definite purpose.
(iv) Statistics in an experiment are comparable and can be classified into various groups.
Question 3.
What are:
(i) Primary data?
(ii) Secondary data?
Which of the two—the primary or the secondary data—is more reliable and why?
Answer:
The word ‘Data’ means information. Statistical data are of two types:
(i) Primary data: When an investigator collects data himself with a definite plan or design in his/her mind is called Primary data.
(ii) Secondary data: Data which are not originally collected rather obtained from published or unpublished sources are known as secondary data.
Question 4.
Why do we group data?
Answer:
The data obtained in original form are called raw data. Raw data does not give any useful information and is rather confusing. Data is grouped so that it becomes understandable and can be interpreted.
According to various characteristics groups are formed by us. After grouping the data, we are in a position to make calculation of certain values which will help us in describing and analysing data.
Question 5.
Explain the meaning of the following terms:
(i) Variable
(ii) Class-interval
(iii) Class-size
(iv) Class-mark
(v) Frequency
(vi) Class limits
(vii) True class limits
Answer:
(i) Variable: Any character that can vary from one individual to other is called variable.
(ii) Class-interval: In the data of each group into which raw data is considered is called a Class-interval.
(iii) Class-size: The difference between the upper class limit and the lower class limit is called the class size of the class.
Class-mark: The middle value of the class is called the class-mark.
Class mark = 
(iv) Frequency: The number of observations corresponding to class is called its frequency.
(v) Class-limit: Each class is bounded by two figures, called the class limits. The figures on the left side of the classes are called lower limits while the figures on the right sides are called the upper limits.
(vi) True class limits: If classes are inclusive e.g., 15-19, 20-24, 25-29, 30-34
Then, true lower limit of class = Lower limit of class – 0.5
And, true upper limit of class = Upper limit of class + 0.5
e.g., True limits of class 15-19 are 14.5-19.5
Question 6.
The ages of ten students of a group are given below. The ages have been recorded in years and months:
8-6, 9-0, 8-4, 9-3, 7-8, 8-11, 8-7, 9-2, 7-10, 8-8
(i) What is the lowest age?
(ii) What is the highest age?
(iii) Determine the range?
Answer:
The ages of 10 students of a group are given below:
8-6, 9-0, 8-4, 9-3, 7-8, 8-11, 8-7, 9-2, 7-10, 8-8
(i) Lowest age is 7 years 8 months.
(ii) Highest age is 9 years 3 months.
(iii) Range = Highest age – Lowest age
= 9 years 3 months – 7 years 8 months
= 1 year 7 months
Question 7.
The monthly pocket money of six friends is given below:
Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45
(i) What is the highest pocket money?
(ii) What is the lowest pocket money?
(iii) What the range?
(iv) Arrange the amounts of pocket money in ascending order.
Answer:
The monthly pocket money of 6 friends is given below:
Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45
(i) Highest pocket money = Rs. 50
(ii) Lowest pocket money = Rs. 25
(iii) Range = Rs. 50 – Rs. 25
= Rs. 25
(iv) The amount of pocket money in ascending order is:
Rs. 25, Rs. 30, Rs. 40, Rs. 45, Rs. 45, Rs. 50
Question 8.
Write the class-size in each of the following:
(i) 0-4, 5-9, 10-14
(ii) 10-19, 20-29, 30-39
(iii) 100-120, 120-140, 160-180
(iv) 0-0.25, 0.25-0.50, 0.50-0.75
(v) 5-5.01, 5.01-5.02, 5.02-5.03
Answer:
(i) Class size = Upper limit – Lower limit
= 4 – 0 = 4
(ii) Class size = Upper limit – Lower limit
= 19 – 10 = 9
(iii) Class size = Upper limit – Lower limit
= 120 – 100 = 20
(iv) Class size = Upper limit – Lower limit
= 0.25 – 0 = 0.25
(v) Class size = Upper limit – Lower limit
= 5.01 – 5 = 0.01
Question 9.
The final marks in mathematics of 30 students are as follows:
53, 61, 48, 60, 78, 68, 55, 100, 67, 90
75, 88, 77, 37, 84, 58, 60, 48, 62, 56
44, 58, 52, 64, 98, 59, 70, 39, 50, 60
(i) Arrange these marks in the ascending order, 30 to 39 one group, 40 to 49 second group, etc.
Now answer the following:
(ii) What is the highest score?
(iii) What is the lowest score?
(iv) What is the range?
(v) If 40 is the pass mark how many have failed?
(vi) How many have scored 75 or more?
(vii) Which observations between 50 and 60 have not actually appeared?
(vii) How many have scored less than 50?
Answer:
(i)
(ii) The highest score is 100.
(iii) The lowest score is 37
(iv) Range = Highest score – Lowest score
= 100 – 37
= 63
(v) If the pass marks are 40 2 students are failed.
(vi) 8 people have scored 75 or more.
(vii) The observation between 50 and 60 that are not actually appeared are 51, 54, 57
(viii) Number of students that have scored less than 50 are 5
Question 10.
The weights of new born babies (in kg.) in a hospital on a particular day are as follows:
2.3, 2.2, 2.1, 2.7, 2.6, 3.0, 2.5, 2.9, 2.8, 3.1, 2.5, 2.8, 2.7, 2.9, 2.4
(i) Rearrange the highest weight.
(ii) Determine the highest weight.
(iii) Determine the lowest weight.
(iv) Determine the range.
(v) How many babies were born on that day?
(vi) How many babies weigh below 2.5 kg?
(vii) How many babies weigh more than 2.8 kg?
(viii) How many babies weigh 2.8 kg?
Answer:
(i) The arrangement according to the highest weight is:
3.1, 3.0, 2.9, 2.8, 2.7, 2.7, 2.6, 2.5, 2.4, 2.4, 2.3, 2.2, 2.1
(ii) The highest weight is 3.1 kg.
(iii) The lowest weight is 2.1 kg.
(iv) Range = Highest weight – Lowest weight
= 3.1 – 2.1
= 1.0 kg
(v) Number of babies that were born on that day were 15.
(vi) 4 babies weigh below 2.5 kg.
(vii) 4 babies weigh more than 2.8 kg.
(viii) 2 babies weigh 2.8 kg.
Question 11.
The number of runs scored by a cricket player in 25 inning are as follows :
26, 35, 94, 48, 82, 105, 53, 0, 39, 42, 71, 0, 64, 15, 34, 67, 0, 42, 124, 84, 54, 48, 139, 64, 47
Rearrange these runs in ascending order.
(ii) Determine the player, is highest score.
(iii) How many times did the player not score a run?
(iv) How many centuries dis he score?
(v) How many times did he score more than 50 runs?
Answer:
The number of runs scored by a player in 25 inning are:
26, 35, 94, 48, 82, 105, 53, 0, 39, 42, 71, 0, 64, 15, 34, 67, 0, 42, 124, 84, 54, 48, 139, 64, 47
(i) Runs in ascending order are:
0, 0, 0, 15, 26, 34, 35, 39, 42, 42, 47, 48, 48, 53, 54, 64, 64, 67, 71, 82, 84, 94, 105, 124, 139
(ii) The highest number = 139
(iii) The player did not scored any run 3 times.
(iv) He scored 3 centuries.
(v) He scored more than 50 runs 12 times.
Question 12.
The class size of a distribution is 25 and the first class-interval is 200-224. There are seven class-intervals:
(i) Write the class-intervals.
(ii) Write the class-marks of each interval.
Answer:
Given, Class size = 25
First class interval = 200-224
(i) Seven class intervals are: 200-240, 225-249, 250-274, 275-299, 300-324, 325-349, 350-374
(ii) Class marks of 200-224: 
= 
= 212
Class marks of 225-249 = 
= 
= 237
Class marks of 250-274 = 
= 
= 262
Class marks of 275-299 = 
= 
= 287
Class marks of 300-324 = 
= 
= 312
Class marks of 325-349 = 
= 
= 337
Class marks of 350-374 = 
= 
= 362
Question 13.
Write the class size and class limits in each of the following:
(i) 104, 114, 124, 134, 144, 154 and 164
(ii) 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97 and 102
(iii) 12.5, 17.5, 22.5, 27.5, 32.5, 37.5, 42.5, 47.5
Answer:
(i) 104, 114, 124, 134, 144, 154 and 164
Class size = 114 – 104 = 10
(ii) 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97 and 102
Class mark = 52 – 47 = 5
(iii) 12.5, 17.5, 22.5, 27.5, 32.5, 37.5, 42.5, 47.5
Class mark = 17.5 – 12.5 = 5
Question 14.
Following data gives the number of children in 40 families:
1, 2, 6, 5, 1, 5, 1, 3, 2, 6, 2, 3, 4, 2, 0, 0, 4, 4, 3, 2, 2, 0, 0, 1, 2, 2, 4, 3, 2, 1, 0, 5, 1, 2, 4, 3, 2, 1, 6, 2, 2.
Represent it in the form of a frequency distribution.
Answer:
Question 15.
The marks scored by 40 students of class IX in mathematics are given below:
81, 56, 68, 79, 85, 13, 29, 68, 54, 73, 47, 35, 72, 64, 95, 44, 50, 77, 64, 35, 79, 52, 45, 54, 70, 83, 62, 64, 72, 92, 84, 76, 63, 43, 54, 38, 73, 68, 52, 54.
Prepare a frequency distribution with class size of 10 marks.
Answer:
Question 16.
The heights (in cm) of 30 students of class IX are given below:
155, 158, 154, 158, 160, 148, 149, 150, 153, 159, 161, 148, 157, 153, 157, 162, 159, 151, 154, 156, 152, 156, 160, 152, 147, 155, 163, 155, 157, 153
Prepare a frequency distribution table with 160-164 as one of the class intervals.
Answer:
Question 17.
The monthly wages of 30 workers in a factory are given below:
83.0, 835, 890, 810, 835, 836, 869, 845, 898, 890, 820, 860, 832, 833, 855, 845, 804, 808, 812, 840, 885, 835, 836, 878, 840, 868, 890, 806, 840, 890
Represent the data in the form of a frequency distribution with class size 10.
Answer:
Question 18.
The daily maximum temperatures (in degree Celsius) recorded in a certain city during the month of November are as follows:
25.8, 24.5, 25.6, 20.7, 21.8, 20.5, 20.6, 20.9, 22.3, 22.7, 23.1, 22.8, 22.9, 21.7, 21.3, 20.5, 20.9, 23.1, 22.4, 21.5, 22.7, 22., 22.0, 23.9, 24.7, 22.8, 23.8, 24.6, 23.9, 21.1
Represent them as a frequency distribution table with class size 1°C.
Answer:
Question 19.
Construct a frequency table with equal class intervals from the following data on the monthly wages (in rupees) of 28 labourers working in a factory, taking one of the class intervals as 210-230 (230 not included) :
220, 268, 258, 242, 210, 268, 272, 242, 311, 290, 300, 320, 319, 304, 302, 318, 306, 292, 254, 278, 210, 240, 280, 316, 306, 215, 256, 236
Answer:
Question 20.
The daily minimum temperature in degrees Celsius recorded in a certain Arctic region are as follows:
-12.5, -10.8, -18.6, -8.4, -10.8, -4.2, -4.8, -6.7, -13.2, -11.8, -2.3, 1.2, 2.6, 0, 2.4, 0, 3.2, 2.7, 3.4, 0, -2.4, -2.4, 0, 3.2, 2.7, 3.4, 0, -2.4, -5.8, -8.9, -14.6, -12.3, -11.5, -7.8, -2.9.
Represent them as frequency distribution table taking -19.9 to -15 as the first class interval.
Answer:
Since, first class interval is - 19.0 to 15
Frequency distribution with lower limit included an upper limit excluded is:
Question 21.
The blood groups of 30 students of class VIII are recoded as follows:
A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O
A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O
Represent this data in the form of a frequency distribution table. Find out which is the most common and which is the rarest blood group among these students.
Answer:
Here 9 students have blood groups A, 6 has B, 3 has AB and 12 has O.
As 12 students have the blood group O and 3 have their blood group AB. Clearly, the most common blood among these students is O and the rarest blood group among these students is AB.
Question 22.
Three coins were tossed 30 times. Each time the number of heads occurring was noted down as follows:
0 1 2 2 1 2 3 1 3 0
1 3 1 1 2 2 0 1 2 1
3 0 0 1 1 2 3 2 2 0
Prepare a frequency distribution table for the data given above.
Answer:
By observing the data given above following frequency distribution table can be constructed.
Question 23.
Thirty children were asked about the number of hours they watched T.V. programmes in the previous week. The results were found as follows:
1 6 2 3 5 12 5 8 4 8
10 3 4 12 2 8 15 1 17 6
3 2 8 5 9 6 8 7 14 12
(i) Make a grouped frequency distribution table for this data, taking class width 5 and one of the class intervals as 5-10.
(ii) How many children watched television for 15 or more hours a week?
Answer:
(i) Class intervals will be 0-5, 5-10, 10-15,…
The grouped frequency distribution table is as follows:
(ii) The number of children, who watch T.V. for 15 or more hours a week is 2 (i.e. no. of children in class interval 15-20)
Cce - Formative Assessment
Question 1.Tally marks are used to find
A. Class intervals
B. Range
C. Frequency
D. Upper limits
Answer:
Tally marks, also called hash marks, are a unary numeral system. They are a form of numeral used for counting. They are most useful in counting or tallying on going results, such as the score in a game or sport, as no intermediate results need to be erased or discarded.
Question 2.
The difference between the highest and lowest values of the observations is called
A. Frequency
B. Mean
C. Range
D. Class-intervals
Answer:
Range is the area of variation between upper and lower limits on a particular scale.
Question 3.
The difference between the upper and the lower class limits is called
A. Mid points
B. Class size
C. Frequency
D. Mean
Answer:
The difference between the upper and the lower limit is called a class size.
The lower limit for every class is the smallest value in that class. On the other hand, the upper limit for every class is the greatest value in that class.
Question 4.
In the class intervals 10-20, 20-30, 20 is taken in
A. The interval 10-20
B. The interval 20-30
C. Both intervals 10-20, 20-30
D. None of the intervals
Answer:
Since, 20 is included in the class interval 20-30.
Question 5.
In a frequency distribution, the mid value of a class is 15 and the class interval is 4. The lower limit of the class is
A. 10
B. 12
C. 13
D. 14
Answer:
Mid value = 
15 = 
30 = (x + x -4)
34 = 2x
x = 17
Lower class limit = 17 -4 = 13
Question 6.
The mid-value of a class interval is 42. If the class size is 10, then the upper and lower limits of the class are:
A. 47 and 37
B. 37 and 47
C. 37.5 and 47.5
D. 47.5 and 37.5
Answer:
Let the upper limit and lower limit of the class be x, y
Then = 42
Or x + y = 84
x - y = 10
Solving equations x = 47, y = 37
Question 7.
The number of times a particular item occurs in a given data is called its
A. Variation
B. Frequency
C. Cumulative frequency
D. Class-size
Answer:
The rate at which something occurs over a particular period of time or in a given sample is called its frequency.
Question 8.
The width of each of nine classes in a frequency distribution is 2.5 and the lower class boundary of the lowest class 10.6. Then the upper class boundary of the highest class is
A. 35.6
B. 33.1
C. 30.6
D. 28.1
Answer:
Upper class boundary of the highest class = (9 × 2.5) + 10.6
= 22.5 + 10.6
= 33.1
Question 9.
Let l be the lower class limit of a class-interval in a frequency distribution and m be the mid-point of the class. Then, the upper class limit f the class is:
A. 
B. 
C. 2m - l
D. m – 2l
Answer:
We know,
= Mid points
Now we substitute given values and get,
= m
Upper class limit = 2m - l
Question 10.
The following marks were obtained by the students in a test:
81, 72, 90, 86, 85, 92, 70, 71, 83, 89, 95, 85, 79, 62
The range of the marks is
A. 9
B. 17
C. 27
D. 33
Answer:
Maximum marks = 95
Minimum marks = 62
Range = Maximum marks - Minimum marks
= 95 – 62
= 33
Question 11.
Tally are usually marked in a bunch of
A. 3
B. 4
C. 5
D. 6
Answer:
After a bunch of 4 they are cut with a slant line symbolizing it as 5.
Exercise 22.2
Question 1.Define cumulative frequency distribution.
Answer:
A table which displays the manner in which cumulative frequencies are distributed over various classes is called cumulative frequency distribution or cumulative frequency distribution table.
Question 2.
Explain the difference between a frequency distribution and a cumulative frequency distribution.
Answer:
Frequency table or frequency distribution is a method to represent raw data in the form which, one can easily understand the information contained in a raw data, where as a table which plays the manner in which cumulative frequencies are distributed over various classes is called a cumulative frequency distribution.
Question 3.
The marks scored by 55 students in a test given below:
Prepare a cumulative frequency table.
Answer:
Question 4.
Following are the ages of 360 patients getting medical treatment in a hospital on a day:
Construct a cumulative frequency distribution.
Answer:
Question 5.
The water bills (in rupees) of houses in a certain street for the period 1.1.98 to 31.3.98 are given below:
56, 43, 32, 38, 56, 24, 68, 85, 52, 47, 35, 58, 63, 74, 27, 84, 69, 35, 44, 75, 55, 30, 54, 65, 45, 67, 95, 72, 43, 65, 35, 59
Tabulate the data and present the data as a cumulative frequency table using 70-79 as one of the class intervals.
Answer:
The minimum bill is Rupees 24
And, the maximum bill is Rupees 95
Range = Maximum bill – Minimum bill
= 95 – 24 = 71
Given, class interval is 70-79. So, class size = 79 – 70
= 9
Therefore, Number of classes = 
= 
= 7.80 = 8
The cumulative frequency distribution table is as follows:
Question 6.
The number of books in different shelves of a library are as follows:
30, 32, 28, 24, 20, 25, 38, 37, 40, 45, 16, 20
19, 24, 27, 30, 32, 34, 35, 42, 27, 28, 19, 34
38, 39, 42, 29, 24, 27, 22, 29, 31, 19, 27, 25
28, 23, 24, 32, 34, 18, 27, 25, 37, 31, 24, 23,
43, 32, 28, 31, 24, 23, 26, 36, 32, 29, 28, 21
Prepare a cumulative frequency distribution table using 45-49 as the last class0interval.
Answer:
Minimum number of book shelves is 16 and maximum number of book shelves is 45.
Range = Maximum number of book shelves – Minimum number of book shelves
= 45 – 16
= 29
Given, Class interval is 45-49. So, class size = 49 – 45
= 4
Therefore, No. of classes =
= 
= 7.25
Number of classes = 8
The cumulative distribution frequency is as follows:
Question 7.
Given below are the cumulative frequencies showing the weights of 685 students of a school. Prepare a frequency distribution table.
Answer:
Question 8.
The following cumulative frequency distribution table shows the daily electricity consumption (in KV) of 40 factories in an industrial state:
(i) Represent this as a frequency distribution table.
(ii) Prepare a cumulative frequency table.
Answer:
(ii)
Question 9.
Given below is a cumulative frequency distribution table showing the ages of people living in a locality:
Prepare a frequency distribution table.
Answer:
