Class 9 - Maths - Statistics

Exercise 14.1

Question 1:

Give 5 scenarios of day to day life from which any type of data can be collected.

Solution:

Daily life activities from where data can be collected are as follows:

(i) Daily household expenditure

(ii) Rainfall happening in your area

(iii) Electricity bill of monthly

(iv) Voting results or survey that the companies conduct

(v) Mark sheet of results that the student gets

Question2:

Classify the above data in Question 1 as primary or secondary data.

Solution:

Primary Data : (i),(iii) and (v)

Secondary data: (ii) and (iv)

Exercise 14.2

Question1:

The blood group record of 30 pupil of standard 9th are recorded in a following data:

A, O, AB, B, O, O, O, A, B, B, O, A, O, A, O, AB, A, O, O, A, O, A, AB, B, O, A, A, B, O, B

Illustrate this report in a form of a frequency distribution table.

Give the rarest blood group of these pupils present in the classroom.

Solution:

The frequency refers to the number of students having same blood group. We will represent

the data in table: Most common Blood Group (Highest frequency): O

Rarest Blood Group (Lowest frequency): AB

Question2:

The distance (in km) of 40 engineers from their residence to their place of work were

found as follows:

5            3          10           20             25         11          13        7        12        31

19          10         12           17             18         11           32      17       16         2

7            9           7              8               3            5            12       15       18         3

12        14          2              9               6            15           15        7           6         12

Construct a grouped frequency distribution table with class size 5 for the data given above taking the first interval as 0-5 (5 not included).

What main features do you observe from this tabular representation?

Solution:

The given data is very large. So, we construct a group frequency of class size 5. Therefore, class

interval will be 0-5, 5-10, 10-15, 15-20 and so on. The data is represented in the table as: The classes in the table are not overlapping. Also, 36 out of 40 engineers have their house

below 20 km of distance.

Question 3:

The relative humidity (in %) of a certain city for a month of 30 days was as follows:

98.1      98.6     99.2      90.3     86.5     95.3     92.9     96.3     94.2     95.1

89.2      92.3     97.1      93.5     92.7     95.1     97.2     93.3     95.2     97.3

96.2      92.1     84.9      90.2     95.7     98.3     97.3     96.1     92.1     89

(i) Construct a grouped frequency distribution table with classes 84 - 86, 86 - 88, etc.

(ii) Which month or season do you think this data is about?

(iii) What is the range of this data?

Answer:

(i) A group of frequency distribution table of class size 2 has to be constructed. The class

intervals will be 84 – 86, 86 – 88 and 88 – 90 …..

By observing the data given above, the required table can be constructed as follows: (ii) It can be observe that the relative humidity is high. Therefore, the data is about a month of

rainy season.

(iii) Range of data = Maximum value – Minimum value

= 99.2 – 84.9

= 14.3

Question 4:

The heights of 50 students, measured to the nearest centimeters, have been found to be as follows:

161     150    154    165    168    161    154    162    150    151

162    164    171    165    158   154    156    172     160    170

153    159    161    170    162   165    166    168     165    164

154    152    153    156    158   162    160    161     173    166

161    159    162    167    168   159    158    153     154    159

(i) Represent the data given above by a grouped frequency distribution table, taking the class intervals as 160 - 165, 165 - 170, etc.

(ii) What can you conclude about their heights from the table?

Answer:

(i) A grouped frequency distribution table has to be constructed taking class intervals 160 –

165, 165 – 170, etc. By observing the data given above, the required table can be constructed

As follows: (ii) It can be concluded that more than 50% of the students are shorter than 165 cm.

Question 5:

A study was conducted to find out the concentration of sulphur dioxide in the air in parts per

million(ppm) of a certain city. The data obtained for 30 days is as follows:

0.03    0.08    0.08    0.09    0.04    0.17

0.16    0.05    0.02    0.06    0.18    0.20

0.11   0.08     0.12    0.13    0.22    0.07

0.08   0.01     0.10    0.06    0.09    0.18

0.11   0.07     0.05    0.07    0.01    0.04

(i) Make a grouped frequency distribution table for this data with class intervals as 0.00 - 0.04, 0.04 - 0.08, and so on.

(ii) For how many days, was the concentration of sulphur dioxide more than 0.11 parts per million?

Answer:

(i) Taking class intervals as 0.00-0.04, 0.04-0.08 and so on, a grouped frequency table can be

Constructed as follows: The number of days for which the concentration of SO2 is more than 0.11 is the number of

Days for which the concentration is in between 0.12 – 0.16, 0.16 – 0.20, 0.20 – 0.24

(ii) Required number of days = 2 + 4 + 2 = 8

Therefore, for 8 days, the concentration of SO2 is more than 0.11 ppm.

Question 6:

Three coins were tossed 30 times simultaneously. 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, the required frequency distribution table can be constructed as follows: Question 7:

The value of π upto 50 decimal places is given below:

3.14159265358979323846264338327950288419716939937510

(i) Make a frequency distribution of the digits from 0 to 9 after the decimal point.

(ii) What are the most and the least frequently occurring digits?

Answer:

(i) By observing the digit after decimal point, the required frequency distribution table can be

constructed as follows: (ii) It can be observed from the above table that the least frequency is 2 of digit 0 and the maximum frequency is 8 of digit 3 and 9.

Therefore, the most frequently occurring digits are 3 and 9 and the least frequently occurring digit is 0.

Question 8:

Thirty children were asked about the number of hours they watched TV 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) Our class interval will be 0 – 5, 5 – 10, 10 – 15, etc.

The grouped frequency distribution table can be constructed as follows: (ii) The number of children who watched TV for 15 or more hours a week is 2 (i.e. the number

of children in class interval 15 - 20)

Question 9:

A company manufactures car batteries of a particular type. The lives (in years) of 40 such batteries were recorded as follows:

2.6    3.0     3.7     3.2     2.2     4.1     3.5     4.5

3.5    2.3    3.2     3.4     3.8      3.2     4.6     3.7

2.5    4.4    3.4     3.3     2.9      3.0     4.3     2.8

3.5    3.2    3.9     3.2     3.2     3.1    3.7      3.4

4.6    3.8    3.2     2.6     3.5     4.2    2.9      3.6

Construct a grouped frequency distribution table for this data, using class intervals of size 0.5 starting from the interval 2 - 2.5.

Answer:

A grouped frequency table of class size 0.5 has to be constructed, starting from class interval

2 – 2.5. Therefore the class intervals will be 2 – 2.5, 2.5 – 3.0, 3.0 – 3.5, etc.

By observing the data given above, the required grouped frequency distribution table can be Constructed as follows: Exercise 14.3

Question 1:

A survey conducted by an organisation for the cause of illness and death among the women between the ages 15 - 44 (in years) worldwide, found the following figures (in %): (i) Represent the information given above graphically.

(ii) Which condition is the major cause of women’s ill health and death worldwide?

(iii) Try to find out, with the help of your teacher, any two factors which play a major role in the cause in (ii) above being the major cause.

Answer:

(i) By representing cause on x-axis and family fatality rate on y-axis and choosing an appro-

priate scale (1 unit = 5% for y-axis), the graph of the information given above can be

Constructed as follows:

All the rectangle bars are of the same width and have equal spacing between them.

(ii) Reproductive health condition is the major cause of women’s ill health and dead world

wide as 31.8% of women are affected by it.

(iii) The factors are as follows:

1. Lack of medical facilities
2. Lack of correct knowledge of treatment. Question 2:

The following data on the number of girls (to the nearest ten) per thousand boys in different sections of Indian society is given below. (i) Represent the information above by a bar graph.

(ii) In the classroom discuss what conclusions can be arrived at from the graph.

Answer:

(i) By representing section (variable) on x-axis and number of girls per thousand boys on y-axis,

the graph of the information given above can be constructed by choosing an appropriate scale

(1 unit = 100 girls for y-axis) Here, all the rectangle bars are of the same length and have equal spacing in between them.

(ii) It can be observed that maximum number of girls per thousand boys (i.e., 970) is for ST and

minimum number of girls per thousand boys (i.e., 910) is for urban. Also, the number of girls

per thousand boys is greater in rural areas than that in urban areas, backward districts than

that in non-backward districts, SC and ST than that in non SC/ST.

Question 3:

Given below are the seats won by different political parties in the polling outcome of a state assembly elections: (i) Draw a bar graph to represent the polling results.

(ii)Which political party won the maximum number of seats?

Answer:

(i) By taking polling results on x-axis and seats won as y-axis and choosing an appropriate scale (1 unit = 10 seats for y-axis), the required graph

of the above information can be constructed as follows: Here, the rectangle bars are of the same length and have equal spacing in between them.

(ii)Political party A won maximum number of seats.

Question 4:

The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table: (i) Draw a histogram to represent the given data.

(ii) Is there any other suitable graphical representation for the same data?

(iii)Is it correct to conclude that the maximum numbers of leaves are 153 mm long? Why?

Answer:

(i) It can be observed that the length of leaves is represented in a discontinuous class interval having a difference of 1 in between them.

Therefore, 1/2 = 0.5 has to be added to each upper class limit and also have to subtract 0.5 from the lower class limits so as to make the class intervals continuous.  Taking the length of leaves on x-axis and the number of leaves on y-axis, the histogram of this

information can be drawn as above.

Here, 1 unit on y-axis represents 2 leaves.

(ii) Other suitable graphical representation of this data is frequency polygon.

(iii) No, as maximum number of leaves (i.e., 12) has their length in between 144.5 mm and

153.5 mm. It is not necessary that all have their lengths as 153 mm.

Question 5:

The following table gives the life times of neon lamps: (i) Represent the given information with the help of a histogram.

(ii)How many lamps have a lifetime of more than 700 hours?

Answer:

(i) By taking life time (in hours) of neon lamps on x-axis and the number of lamps on y-axis, the histogram of the given information can be drawn as follows. Here, 1 unit on y-axis represents 10 lamps.

(ii) It can be concluded that the number of neon lamps having their lifetime more than 700 is the sum of the number of neon lamps having their lifetime as

700 − 800, 800 − 900, and 900 – 1000.

Therefore, the number of neon lamps having their lifetime more than 700 hours is 184.(74 + 62 + 48 = 184)

Question 6:

The following table gives the distribution of students of two sections according to the mark obtained by them: Represent the marks of the students of both the sections on the same graph by two frequency

polygons. From the two polygons compare the performance of the two sections.

Answer:

We can find the class marks of the given class intervals by using the following formula.

Class mark = (Upper class limit + Lower class limit)/2 Taking class marks on x-axis and frequency on y-axis and choosing an appropriate scale (1 unit = 3 for y-axis), the frequency polygon can be drawn as follows. It can be observed that the performance of students of section A is better than the students of section B in terms of good marks.

Question 7:

The runs scored by two teams A and B on the first 60 balls in a cricket match are given below: Represent the data of both the teams on the same graph by frequency polygons.

[Hint: First make the class intervals continuous.]

Answer:

It can be observed that the class intervals of the given data are not continuous.

There is a gap of 1 in between them. Therefore, 1/2 = 0.5 has to be added to the upper class limits and 0.5 has to be subtracted from the lower class limits.

Also, class mark of each interval can be found by using the following formula.

Class mark = (Upper class limit + Lower class limit)/2

Continuous data with class mark of each class interval can be represented as follows. By taking class marks on x-axis and runs scored on y-axis, a frequency polygon can be constructed as follows. Question 8:

A random survey of the number of children of various age groups playing in park was found as follows: Draw a histogram to represent the data above.

Answer:

Here, it can be observed that the data has class intervals of varying width. The proportion of children per 1 year interval can be calculated as follows:   Taking the age of children on x-axis and proportion of children per 1 year interval on y-axis, the histogram can be drawn as follows:

Question 9:

100 surnames were randomly picked up from a local telephone directory and a frequency distribution of the number of letters in the English alphabet in the surnames was found as follows: (i) Draw a histogram to depict the given information.

(ii)Write the class interval in which the maximum number of surname lie.

Answer:

(i) Here, it can be observed that the data has class intervals of varying width. The proportion of the number of surnames per 2 letters interval can be calculated as follows. By taking the number of letters on x-axis and the proportion of the number of surnames per 2 letters interval on y-axis and choosing an appropriate scale

(1 unit = 4 students for y axis), the histogram can be constructed as follows. (ii) The class interval in which the maximum number of surnames lies is 6 − 8 as it has 44 surnames in it i.e., the maximum for this data.

Exercise 14.4

Question 1:

The following number of goals was scored by a team in a series of 10 matches:

2, 3, 4, 5, 0, 1, 3, 3, 4, 3

Find the mean, median and mode of these scores.

Answer:

The number of goals scored by the team is

2, 3, 4, 5, 0, 1, 3, 3, 4, 3

Mean of data = Sum of all observations/Total number of observations

= (2 + 3 + 4 + 5 + 0 + 1 + 3 + 3 + 4 + 3)/10

= 28/10

= 2.8 goals

Arranging the number of goals in ascending order,

0, 1, 2, 3, 3, 3, 3, 4, 4, 5

The number of observations is 10, which is an even number. Therefore,

Median = (5th term + 6th term)/2

= (3 + 3)/2

= 6/2

= 3

Mode of data is the observation with the maximum frequency in data.

Therefore, the mode score of data is 3 as it has the maximum frequency as 4 in the data.

Question 2:

In a mathematics test given to 15 students, the following marks (out of 100) are recorded:

41, 39, 48, 52, 46, 62, 54, 40, 96, 52, 98, 40, 42, 52, 60

Find the mean, median and mode of this data.

Answer:

The marks of 15 students in mathematics test are

41, 39, 48, 52, 46, 62, 54, 40, 96, 52, 98, 40, 42, 52, 60

Mean of data = Sum of all observations/Total number of observations

= (41 + 39 + 48 + 52 + 46 + 62 + 54 + 40 + 96 + 52 + 98 + 40 + 42 + 52 + 60)/15

= 822/15

= 54.8

Arranging the scores obtained by 15 students in an ascending order,

39, 40, 40, 41, 42, 46, 48, 52, 52, 52, 54, 60, 62, 96, 98

As the number of observations is 15 which is odd, therefore, the median of data will

be (15+1)th/2 = 16th/2 = 8th observation whether the data is arranged in an ascending or

descending order.

Therefore, median score of data = 52

Mode of data is the observation with the maximum frequency in data. Therefore, mode of this

data is 52 having the highest frequency in data as 3.

Question 3:

The following observations have been arranged in ascending order. If the median of the data is 63, find the value of x.

29, 32, 48, 50, x, x + 2, 72, 78, 84, 95

Answer:

It can be observed that the total number of observations in the given data is 10 (even

number). Therefore, the median of this data will be the mean of 10/2

i.e., 5th and (10/2)+1 i.e., 6th term.

Therefore, median of data = (5th observation + 6th observation)/2

=> 63 = (x + x + 2)/2

=> 63 * 2 = 2x + 2

=> 126 = 2x + 2

=> 2x = 126 – 2

=> 2x = 124

=> x = 124/2

=> x = 62

So, the value of x is 62

Question 4:

Find the mode of 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, and 18.

Answer:

Arranging the data in an ascending order,

14, 14, 14, 14, 17, 18, 18, 18, 22, 23, 25, 28

It can be observed that 14 has the highest frequency, i.e. 4, in the given data.

Therefore, mode of the given data is 14.

Question 5:

Find the mean salary of 60 workers of a factory from the following table:

Answer:

Using the table, the mean can be calculated as follows.

Mean = Σfi xi /Σfi

Mean salary = 305000/60

= 5083.33

Therefore, mean salary of 60 workers is Rs 5083.33.

Question 6:

Give one example of a situation in which

(i)The mean is an appropriate measure of central tendency.

(ii) The mean is not an appropriate measure of central tendency but the median is an appro-priate measure of central tendency.

Answer:

When any data has a few observations such that these are very far from the other obser-

vations in it, it is better to calculate the median than the mean of the data as median gives a

better estimate of average in this case.

(i) Consider the following example − the following data represents the heights of the members

of a family.

154.9 cm, 162.8 cm, 170.6 cm, 158.8 cm, 163.3 cm, 166.8 cm, 160.2 cm

In this case, it can be observed that the observations in the given data are close to each other.

Therefore, mean will be calculated as an appropriate measure of central tendency.

(ii)The following data represents the marks obtained by 12 students in a test.

48, 59, 46, 52, 54, 46, 97, 42, 49, 58, 60, 99

In this case, it can be observed that there are some observations which are very far from other

observations. Therefore, here, median will be calculated as an appropriate measure of central

tendency.

.