Question 1: What do you understand by the word “statistics” in (i) singular form (ii) plural form?

The word “statistics” is used in both its singular as well as its plural senses.

In singular sense, statistics maybe defined as the science of collection, presentation, analysis and interpretation of numerical data.

In plural sense, statistics means numerical facts or observations collected with definite purpose.

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Question 2: Describe some fundamental characteristics of statistics.

Some of the important characteristics of statistics are given below:

• Statistics are aggregates of facts.
• Statistics are numerically expressed.
• Statistics are affected to a marked extent by multiplicity of causes.
• Statistics are enumerated or estimated according to a reasonable standard of accuracy.
• Statistics are collected for a predetermined purpose.
• Statistics are collected in a systemic manner.
• Statistics must be comparable to each other

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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?

Statistical data are of two types (i) primary data (ii) secondary data.

Primary Data: Primary data is data that is collected by a researcher from first-hand sources, using methods like surveys, interviews, or experiments. It is collected with the research project in mind, directly from primary sources.

Secondary Data: Secondary data is data gathered from studies, surveys, or experiments that have been run by other people or for other research.

Typically, a researcher will begin a project by working with secondary data. This allows time to formulate questions and gain an understanding of the issues being dealt with before the costly and time-consuming operation of collecting primary data.

Primary data is more reliable as it is collected for a specific objective.

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Question 4: Why do we group data?

We group data to analyze it better. Data can be defined as groups of information that represent the qualitative or quantitative attributes of a variable or set of variables, which is the same as saying that data can be any set of information that describes a given entity.

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Question 5: Explain the meaning of the following terms: (i) variate (ii) class-integral (iii) class-size (iv) class-mark (v) frequency (vi) class limits (vii) true class limits

• variate:  a variate is a quantity which may take any of the values of a specified set with a specified relative frequency or probability. The variate is therefore often known as a random variable. It is to regarded as defined, not merely by a set of permissible values like an ordinary mathematical variable, but by an associated frequency (probability) function expressing how often those values appear in the situation under discussion.
• class-integral: Mathematically it is defined as the difference between the upper class limit and the lower class limit. Class Interval= Upper Class limit – Lower class limit. In statistics, the data is arranged into different classes and the width of such class is called class interval.
• 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. The class width is the difference between the upper or lower class limits of consecutive classes.
• class-mark: a value represented by the mid-value of a class interval.
• frequency: In statistics the frequency (or absolute frequency) of an event is the number of times the event occurred in an experiment or study.
• class limits: Difference between the true upper limit and true lower limit of a class interval is called the class size. … Range: The difference between the maximum value and the minimum value of the observation is called the range.

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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?    (iii) Determine the range?   (ii) What is the highest age?

First arrange the data in ascending order. This will give us the following arrangement.

$7 - 8, 7 - 11, 8 - 4, 8 - 6, 8 - 7, 8 - 8, 8 - 11, 9, 9 - 2, 9 - 3$

• Lowest age: $7 - 8$ (Seven Years Eight Months)
• Highest age: $9 - 3$ (Nine Years Three Months)
• Range: $1 - 7$ (One Year Seven Months)

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Question 7:  The monthly pocket money of six friends is: $Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45$ (i) What is the highest pocket money?      (ii) What is the lowest pocket money?      (iii) What is the range?      (iv) Arrange the amounts of pocket money in ascending order.

• Highest pocket money: $Rs. \ 50$
• Lowest pocket money $Rs. 25$
• Range: $50- 25 = 25$ Rs.
• Pocket money in ascending order: $Rs. \ \ 25, Rs. 30, Rs. \ 40, Rs. \ 45, Rs. \ 45, Rs. \ 50$

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Question 8:  Write the class-size in each of the following: (i) $0-4, 5-9, 10-14$   (ii) $10-19, 20-29, 30-49$ (iii) $100-120, 120-140, 160-180$     (iv) $0-0.25 ,0.25-0.50, 0.50-0.75$    (v) $5-5.01, 5.01-5.02, 5.02-5.03$

Class Size = Difference between the lower limits of two consecutive classes. Hence

• Class size $= 5-0 = 5$
• Class size $= 20-10 = 10$
• Class size $= 120 - 100 = 20$
• Class size $= 0.25 - 0 = 0.25$
• Class size $= 5.01 - 5.0 = 0.1$

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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?     (iv) What is the range?     (iii) What is the lowest score     (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?     (viii) How many have scored less than $50$?

 Class Data 30-39 37, 39 40-49 44, 48, 48 50-59 50, 52, 53, 55, 56, 58, 58, 59 60-69 60, 60, 60, 61, 62, 64, 67, 68 70-79 70, 75, 77, 78 80-89 84, 88 90-99 90, 98 100-109 100

(ii) Highest score: $100$

(ii) Range $= 100 - 37 = 63$

(iii) Lowest score $= 37$

(v) If $40$ is the pass mark than $2$ have failed

(vi) No of students who have scored $75$ or more: $8$

(vii) $51, 54$, and $57$ have not shown up between $50$ and $60$.

(viii) 5 students have scored less than $50$

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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 weights in descending order     (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 $2.8 \ kg$?

• Arranged in descending order: $3.1, 3.0, 2.9, 2.9, 2.8, 2.8, 2.7, 2.7, 2.6, 2.5, 2.5, 2.4, 2.3, 2.2, 2.1$
• Height weight: $3.1 \ kg$
• Lowest weight: $2.1 \ kg$
• Range $= 3.1 - 2.1 = 1 \ kg$
• No of babies born on that day $= 15$
• Babies weighing below $2.5 \ kg = 4$
• Babies weighing more than $2.8 \ kg = 4$
• Babies weighing $2.8 \ kg = 2$

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Question 11:  The number of runs scored by a cricket player in 25 innings are as follows:    $26, 35,94, 48, 82, 105, 53, 0, 39, 42,71, 0, 64, 15, 34, 67, 0, 42, 124, \\ 84, 54, 48, 139, 64, 47$

(i) 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 did he score?     (v) How many times did he score more than $50$ runs?

1. Ascending order: $0, 0, 0, 15, 26, 34, 35, 39, 42, 42, 47, 48, 48, 53, 54, 64, \\ 64, 67, 71, 82, 84, 94, 105, 124, 139$
2. Highest score: $139$
• Player did not score $3$ times
1. The player scored $3$ centuries
2. The player scored more than $50$ runs $12$ times

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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.

 Class Interval Class Mark 200-224 212 225-249 237 250-274 262 275-299 287 300-324 312 325-349 337 350-374 362

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Question 13:  Write the class size and class limits in each of the following:     (i) $104, 114, 124, 134, 144, 154, 164$     (ii) $47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97, 102$     (iii) $12.5, 17.5, 22.5, 27.5, 32.5, 37.5, 42.5, 47.5$

(i) Class size $= 114-104 = 10$

 Class Interval Class Mark 99-109 104 109-119 114 119-129 124 129-139 134 139-149 144 149-159 154 159-169 164

(ii) Class size $= 52-47 = 5$

 Class Interval Class Mark 44.5-49.5 47 49.5-54.5 52 54.5-59.5 57 59.5-64.5 62 64.5-69.5 67 69.5-74.5 72 74.5-79.5 77 79.5-84.5 82 84.5-89.5 87 89.5-94.5 92 94.5-99.5 97 99.5-104.5 102

(iii) Class size $= 17.5-12.5=5$

 Class Interval Class Mark 10-15 12.5 15-20 17.5 20-25 22.5 25-30 27.5 30-35 32.5 35-40 37.5 40-45 42.5 45-50 47.5

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Question 14:  Following data gives the number of children in 41 families:    $1, 2, 6, 5, 1, 5, 1, 3, 2, 6, 2, 3, 4, 2, 0, 0, 4, 4, 3, 2, 2, 0, 0, 1, \\ 2, 2, 4, 3, 2, 1, 0, 5, 1, 2, 4, 3, 4, 1, 6, 2, 2$

Represent it in the form of a frequency distribution.

 No. of Children Tally Bars Frequency 0 |||| 5 1 |||| || 7 2 ||||  ||||  || 12 3 |||| 5 4 |||| | 6 5 ||| 3 6 ||| 3 Total: 41

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Question 15:  The marks scored by 40 students of class IX in mathematics are given below:    $81, 55, 68, 79, 85, 43, 29, 68, 54, 73, 47, 35, 72, 64, 95, 44, 50, 77, 64, 35, \\ 79, 52, 45, 54, 70, 83, 62, 64, 72, 92, 84, 76, 63, 43, 54, 38, 73, 68, 52, 54$

Prepare a frequency distribution with class size of 10 marks by exclusive method.

Maximum Value $= 95$   Minimum Value $= 29$   Given Class size $= 10$

No of classes $= \frac{66}{10} = 6.6$

Therefore, we should have $7$ classes

Exclusive method

 Class Interval Tally Bars Frequency 29-39 |||| 4 39-49 |||| 5 49-59 |||| ||| 8 59-69 |||| ||| 8 69-79 |||| || 7 79-89 |||| | 6 89-99 || 2 Total = 40

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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.

Maximum Value $= 163$   Minimum Value $= 147$   Given Class $= 160 - 164$

Inclusive method

 Class Interval Tally Bars Frequency 145-149 |||| 4 150-154 |||| |||| 9 155-159 |||| |||| || 12 160-164 |||| 5 Total = 30

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Question 17:  The monthly wages of 30 workers in a factory are given below:

$830, 835, 890, 810, 835, 836, 869, 845, 898, 890, 820, 860, 832, 833, 855, 845, 804, \\ 808, 812, 840, 885, 835, 836, 878, 840, 868, 890, 806, 840, 890$

Represent the data in the form of a frequency distribution with class size 10.

Maximum Value $= 898$   Minimum Value $= 804$   Given Class size $= 10$

No of classes $= \frac{94}{10} = 9.4$

Therefore, we should have $10$ classes

Exclusive method

 Class Interval Tally Bars Frequency 804-814 |||| 5 814-824 | 1 824-834 ||| 3 834-844 |||| ||| 8 844-854 || 2 854-864 || 2 864-874 || 2 874-884 | 1 884-894 |||| 5 894-904 | 1 Total = 30

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Question 18:  The daily maximum temperatures (in degree Celsius) recorded in a certain city during the month of November are as follows:

$25.8, 24.5, 25.6, 20.7, 21.8, 20.5, 20.6, 20.9, 22.3, 22.7, 23.1, 22.8, 22.9, 21.7, 21.3, \\ 20.5, 20.9, 23.1, 22.4, 21.5, 22.7, 22.8, 22.0, 23.9, 24.7, 22.8, 23.8, 24.6, 23.9, 21.1$

Represent them as a frequency distribution table with class size $1^o \ C$.

Maximum Value $= 25.8$   Minimum Value $= 20.5$   Given Class size $= 1$

No of classes $= \frac{5.3}{1} = 5.3$

Therefore, we should have $6$ classes

Exclusive method

 Class Interval Tally Bars Frequency 20.5-21.5 |||| ||| 8 21.5-22.5 |||| | 6 22.5-23.5 |||| ||| 8 23.5-24.5 ||| 3 24.5-25.5 ||| 3 25.5-26.5 || 2 Total = 30

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Question 19:  Construct a frequency table with equal class intervals from the following data on the monthly wages (in rupees) of 28 labor working in a factory, taking one of the class intervals as 210-230 (230 not included):

$220, 268, 258, 242, 210, 268, 272, 242, 311, 290, 300, 320, 319, 304, 302, 318, \\ 306, 292, 254, 278, 210, 240, 280, 316, 306, 215, 256, 236$

Exclusive method

 Class Interval Tally Bars Frequency 210-230 |||| 4 230-250 |||| 4 250-270 |||| 5 270-290 ||| 3 290-310 |||| || 7 310-330 |||| 5 Total = 28

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Question 20:  The daily minimum temperatures 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.

Exclusive method

 Class Interval Tally Bars Frequency -19.9 to -15 | 1 -15 to -10.1 |||| ||| 8 -10.1 to -5.2 |||| 5 -5.2 to -0.3 |||| || 7 -0.3 to 4.6 |||| |||| |||| 14 Total = 35

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Question 21:  The blood groups of 30 students of class VIII are recorded 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.

 Blood Group Tally Bars Frequency A |||| |||| 9 B |||| | 6 O |||| |||| || 12 AB ||| 3 Total = 30

Most common blood group: O

Most rare blood group: AB

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Question 22:  Three coins were tossed 30 times. Each time the number of heads occurring was noted down as follow: $0 ,1, 2, 2, 1, 2, 3, 1, 3, 0, 1, 3, 1, 1, 2, 2, 0, 1, 2, 1, 3, 0, 0, 1, 1, 2, 3, 2, 2, 0$

Prepare a frequency distribution table for the data given above.

 No of Heads Tally Bars Frequency 0 |||| | 6 1 |||| |||| 10 2 |||| |||| 9 3 |||| 5 Total = 30

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Question 23: 30 children were asked about the number of hours they watched TV programs 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?