 # NCERT Solutions For Class 10 Maths Chapter 14-Statistics

NCERT Solutions for Class 10 Maths Chapter 14 Statistics are provided here, which can be downloaded for free, in PDF format. The Solutions for chapter 14 Statistics are prepared by our experts who have done specialization in Maths.

All the solved questions of Statistics are with respect to the NCERT syllabus and guidelines, to help students solve each exercise questions, and prepare for the CBSE Board exam. Using these solutions as a reference tool will be helpful for the students to score good marks. Students can also get the exercise-wise solutions for Class 10th Maths all chapters and practice solving the problems.

### Download PDF of NCERT Solutions for Class 10 Maths Chapter 14  – Statistics                      ### Class 10 Maths Chapter 14 Exercise 14.1 Page: 270

1. A survey was conducted by a group of students as a part of their environment awareness program, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.

 Number of Plants 0-2 2-4 4-6 6-8 8-10 10-12 12-14 Number of Houses 1 2 1 5 6 2 3

Which method did you use for finding the mean, and why?

Solution: In order to find the mean value,
we will use direct method because the numerical value of fi and xi are small.
Find the midpoint of the given interval using the formula.
Midpoint (xi) = (upper limit + lower limit)/2

 No. of plants (Class interval) No. of houses Frequency (fi) Mid-point (xi) fixi 0-2 1 1 1 2-4 2 3 6 4-6 1 5 5 6-8 5 7 35 8-10 6 9 54 10-12 2 11 22 12-14 3 13 39 Sum fi = 20 Sum fixi = 162

The formula to find the mean is: Mean = x̄ = ∑fi xi /∑f
= 162/20
= 8.1
Therefore, the mean number of plants per house is 8.1

2. Consider the following distribution of daily wages of 50 workers of a factory.

 Daily wages (in Rs.) 100-120 120-140 140-160 160-180 180-200 Number of workers 12 14 8 6 10

Find the mean daily wages of the workers of the factory by using an appropriate method.
Solution: Find the midpoint of the given interval using the formula.

Midpoint (xi) = (upper limit + lower limit)/2

In this case, the value of mid-point (xi) is very large,so let us assume the mean value, A = 150 and class interval is h = 20.

So, u= (xi – A)/h = u= (xi – 150)/20

Substitute and find the values as follows:

 Daily wages (Class interval) Number of workers frequency (fi) Mid-point (xi) ui = (xi – 150)/20 fiui 100-120 12 110 -2 -24 120-140 14 130 -1 -14 140-160 8 150 0 0 160-180 6 170 1 6 180-200 10 190 2 20 Total Sum fi = 50 Sum fiui = -12

So, the formula to find out the mean is:
Mean = x̄ = A + h∑fiui /∑f
=150 + (20 × -12/50)
= 150 – 4.8
= 145.20
Thus, mean daily wage of the workers = Rs. 145.20

3. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency f.

 Daily Pocket Allowence(in c) 11-13 13-15 15-17 17-19 19-21 21-23 23-35 Number of children 7 6 9 13 f 5 4

Solution: To find out the missing frequency, use the mean formula. Here, the value of mid-point (xi)  mean x̄ = 18

 Class interval Number of children (fi) Mid-point (xi) fixi 11-13 7 12 84 13-15 6 14 84 15-17 9 16 144 17-19 13 18 = A 234 19-21 f 20 20f 21-23 5 22 110 23-25 4 24 96 Total fi = 44+f Sum fixi = 752+20f

The mean formula is Mean = x̄ = ∑fixi /∑f

= (752+20f)/(44+f)

Now substitute the values and equate to find the missing frequency (f)

⇒ 18 = (752+20f)/(44+f)

⇒ 18(44+f) = (752+20f)

⇒ 792+18f = 752+20f

⇒ 792+18f = 752+20f

⇒ 792 – 752 = 20f – 18f

⇒ 40 = 2f

⇒ f = 20

So, the missing frequency, f = 20.

4. Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarized as follows. Find the mean heart beats per minute for these women, choosing a suitable method.

 Number of heartbeats per minute 65-68 68-71 71-74 74-77 77-80 80-83 83-86 Number of women 2 4 3 8 7 4 2

Solution:
From the given data, let us assume the mean as A = 75.5
x= (Upper limit + Lower limit)/2
Class size (h) = 3
Now, find the ui and fi ui as follows:

 Class Interval Number of women (fi) Mid-point (xi) ui = (xi – 75.5)/h fiui 65-68 2 66.5 -3 -6 68-71 4 69.5 -2 -8 71-74 3 72.5 -1 -3 74-77 8 75.5 0 0 77-80 7 78.5 1 7 80-83 4 81.5 3 8 83-86 2 84.5 3 6 Sum fi= 30 Sum fiui = 4

Mean = x̄ = A + h∑fiui /∑f
= 75.5 + 3×(4/30) 75.5 + 4/10
= 75.5 + 0.4
= 75.9
Therefore, the mean heart beats per minute for these women is 75.9

5. In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.

 Number of mangoes 50-52 53-55 56-58 59-61 62-64 Number of boxes 15 110 135 115 25

Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?
Solution: Since, the given data is not continuous so we add 0.5 to the upper limit and subtract 0.45 from the lower limit as the gap between two intervals are 1

Here, assumed mean (A) = 57 Class size (h) = 3 Here, the step deviation is used because the frequency values are big.

 Class Interval Number of boxes (fi) Mid-point (xi) di = xi – A fidi 49.5-52.5 15 51 -6 90 52.5-55.5 110 54 -3 -330 55.5-58.5 135 57 = A 0 0 58.5-61.5 115 60 3 345 61.5-64.5 25 63 6 150 Sum fi = 400 Sum fidi = 75

The formula to find out the Mean is:

Mean = x̄ = A +h ∑fidi /∑f

= 57 + 3(75/400)

= 57 + 0.1875

= 57.19

Therefore, the mean number of mangoes kept in a packing box is 57.19

6. The table below shows the daily expenditure on food of 25 households in a locality. Find the mean daily expenditure on food by a suitable method.

 Daily expenditure(in c) 100-150 150-200 200-250 250-300 300-350 Number of households 4 5 12 2 2

Solution: Find the midpoint of the given interval using the formula.

Midpoint (xi) = (upper limit + lower limit)/2

Let is assume the mean (A) = 225 Class size (h) = 50

 Class Interval Number of households (fi) Mid-point (xi) di = xi – A ui=di/50 fiui 100-150 4 125 -100 -2 -8 150-200 5 175 -50 -1 -5 200-250 12 225 0 0 0 250-300 2 275 50 1 2 300-350 2 325 100 2 4 Sum fi = 25 Sum fiui = -7

Mean = x̄ = A +h∑fiui /∑fi

= 225 + 50(-7/25)

= 225 – 14

= 211

Therefore, the mean daily expenditure on food is 211

7. To find out the concentration of SO2 in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below:

 Concentration of SO2 ( in ppm) Frequency 0.00 – 0.04 4 0.04 – 0.08 9 0.08 – 0.12 9 0.12 – 0.16 2 0.16 – 0.20 4 0.20 – 0.24 2

Find the mean concentration of SO2 in the air.

Solution: To find out the mean, first find the midpoint of the given frequencies as follows:

 Concentration of SO2 (in ppm) Frequency (fi) Mid-point (xi) fixi 0.00-0.04 4 0.02 0.08 0.04-0.08 9 0.06 0.54 0.08-0.12 9 0.10 0.90 0.12-0.16 2 0.14 0.28 0.16-0.20 4 0.18 0.72 0.20-0.24 2 0.20 0.40 Total Sum fi = 30 Sum (fixi) = 2.96

The formula to find out the mean is
Mean = x̄ = ∑fixi /∑fi
= 2.96/30
= 0.099 ppm
Therefore, the mean concentration of SO2 in air is 0.099 ppm.

8. A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.

 Number of days 0-6 6-10 10-14 14-20 20-28 28-38 38-40 Number of students 11 10 7 4 4 3 1

Solution: Find the midpoint of the given interval using the formula.

Midpoint (xi) = (upper limit + lower limit)/2

 Class interval Frequency (fi) Mid-point (xi) fixi 0-6 11 3 33 6-10 10 8 80 10-14 7 12 84 14-20 4 17 68 20-28 4 24 96 28-38 3 33 99 38-40 1 39 39 Sum fi = 40 Sum fixi = 499

The mean formula is,
Mean = x̄ = ∑fixi /∑fi
= 499/40
= 12.48 days

Therefore, the mean number of days a student was absent = 12.48.

9. The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.

 Literacy rate (in %) 45-55 55-65 65-75 75-85 85-98 Number of cities 3 10 11 8 3

Solution: Find the midpoint of the given interval using the formula.

Midpoint (xi) = (upper limit + lower limit)/2

In this case, the value of mid-point (xi) is very large,

so let us assume the mean value, A = 70 and class interval is h = 10.

So, u= (xi – A)/h

= u

= (xi – 70)/10

Substitute and find the values as follows:

 Class Interval Frequency (fi) (xi) di = xi – a ui = di/h fiui 45-55 3 50 -20 -2 -6 55-65 10 60 -10 -1 -10 65-75 11 70 0 0 0 75-85 8 80 10 1 8 85-95 3 90 20 2 6 Sum fi  = 35 Sum fiui  = -2

So, Mean = x̄ = A + (∑fiui /∑fi) х h

= 70 + (-2/35) х 10

= 69.42

Therefore, the mean literacy part

= 69.42

### Class 10 Maths Chapter 14 Exercise 14.2 Page: 275

1. The following table shows the ages of the patients admitted in a hospital during a year:

 Age (in years) 5-15 15-25 25-35 35-45 45-55 55-65 Number of patients 6 11 21 23 14 5

Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Solution: To find out the modal class, let us the consider the class interval with high frequency First find the midpoint using the formula,

xi =(upper limit +lower limit)/2

 Class Interval Frequency (fi) Mid-point (xi) fixi 5-15 6 10 60 15-25 11 20 220 25-35 21 30 630 35-45 23 40 920 45-55 14 50 700 55-65 5 60 300 Sum fi = 80 Sum fixi = 2830

The mean formula is
Mean = x̄ = ∑fixi /∑fi
= 2830/80
= 35.37 years

Therefore, the mean of the given data = 35.37 years

2. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components:

 Lifetime (in hours) 0-20 20-40 40-60 60-80 80-100 100-120 Frequency 10 35 52 61 38 29

Determine the modal lifetimes of the components. 3. The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure:

 Expenditure Number of families 1000-1500 24 1500-2000 40 2000-2500 33 2500-3000 28 3000-3500 30 3500-4000 22 4000-4500 16 4500-5000 7 Calculation for mean: First find the midpoint using the formula,

xi =(upper limit +lower limit)/2

Let us assume a mean, A be 2750

 Class Interval fi xi di = xi – a ui = di/h fiui 1000-1500 24 1250 -1500 -3 -72 1500-2000 40 1750 -1000 -2 -80 2000-2500 33 2250 -500 -1 -33 2500-3000 28 2750 0 0 0 3000-3500 30 3250 500 1 30 3500-4000 22 3750 1000 2 44 4000-4500 16 4250 1500 3 48 4500-5000 7 4750 2000 4 28 fi = 200 fiui = -35

The formula to calculate the mean,
Mean = x̄ = a + (∑fiui /∑fi) х h

Substitute the values in the given formula
= 2750 + (-35/200) х 500
= 2750 – 87.50
= 2662.50 So, the mean monthly expenditure of the families
= Rupees 2662.50
4. The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures

 No of Students per teacher Number of states / U.T 15-20 3 20-25 8 25-30 9 30-35 10 35-40 3 40-45 0 45-50 0 50-55 2 Calculation of mean:

Find the midpoint using the formula, xi =(upper limit +lower limit)/2

 Class Interval Frequency (fi) Mid-point (xi) fixi 15-20 3 17.5 52.5 20-25 8 22.5 180.0 25-30 9 27.5 247.5 30-35 10 32.5 325.0 35-40 3 37.5 112.5 40-45 0 42.5 0 45-50 0 47.5 0 50-55 2 52.5 105.5 Sum fi = 35 Sum fixi = 1022.5

`
Mean = x̄ = ∑fixi/∑fi
= 1022.5/35
= 29.2
Therefore, mean = 29.2
5. The given distribution shows the number of runs scored by some top batsmen of the world in one- day international cricket matches.

 Run Scored Number of Batsman 3000-4000 4 4000-5000 18 5000-6000 9 6000-7000 7 7000-8000 6 8000-9000 3 9000-10000 1 10000-11000 1

Find the mode of the data. 6. A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarized it in the table given below. Find the mode of the data:

 Number of cars Frequency 0-10 7 10-20 14 20-30 13 30-40 12 40-50 20 50-60 11 60-70 15 70-80 8 ### Class 10 Maths Chapter 14 Exercise 14.3 Page: 287

1. The following frequency distribution gives the monthly consumption of an electricity of 68 consumers in a locality. Find the median, mean and mode of the data and compare them.

 Monthly consumption(in units) No. of customers 65-85 4 85-105 5 105-125 13 125-145 20 145-165 14 165-185 8 185-205 4

Solution: Find the cumulative frequency of the given data as follows:

 Class Interval Frequency Cumulative frequency 65-85 4 4 85-105 5 9 105-125 13 22 125-145 20 42 145-165 14 56 165-185 8 64 185-205 4 68 N=68

From the table, it is observed that, n = 68 and
hence n/2=34
Hence, the median class is 125-145 with cumulative frequency = 42
Where, l = 125, n = 68, cf = 22, f = 20, h = 20 Calculate the Mean:

 Class Interval fi xi di=xi-a ui=di/h fiui 65-85 4 75 -60 -3 -12 85-105 5 95 -40 -2 -10 105-125 13 115 -20 -1 -13 125-145 20 135 0 0 0 145-165 14 155 20 1 14 165-185 8 175 40 2 16 185-205 4 195 60 3 12 Sum fi= 68 Sum fiui= 7

x̄ = a + h ∑fiui /∑fi

=135+20(7/68)

Mean=137.05

In this case, mean, median and mode are more/less equal in this distribution.

2. If the median of a distribution given below is 28.5 then, find the value of an x &y.

 Class Interval Frequency 0-10 5 10-20 x 20-30 20 30-40 15 40-50 y 50-60 5 Total 60 3. The Life insurance agent found the following data for the distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to the  persons whose age is 18 years onwards but less than the 60 years.

 Age(in years) Number of policy holder Below 20 2 Below 25 6 Below 30 24 Below 35 45 Below 40 78 Below 45 89 Below 50 92 Below 55 98 Below 60 100

Solution:

 Class interval Frequency Cumulative frequency 15-20 2 2 20-25 4 6 25-30 18 24 30-35 21 45 35-40 33 78 40-45 11 89 45-50 3 92 50-55 6 98 55-60 2 100 4. The lengths of 40 leaves in a plant are measured correctly to the nearest millimetre, and the data obtained is represented as in the following table:

 Length(in mm) Number of leaves 118-126 3 127-135 5 136-144 9 145-153 12 154-162 5 163-171 4 172-180 2

Find the median length of leaves.
Solution: Since the data are not continuous reduce 0.5 in the lower limit and add 0.5 in the upper limit.

 Class Interval Frequency Cumulative frequency 117.5-126.5 3 3 126.5-135.5 5 8 135.5-144.5 9 17 144.5-153.5 12 29 153.5-162.5 5 34 162.5-171.5 4 38 171.5-180.5 2 40

So, the data obtained are:
n = 40 and n/2 = 20
Median class = 144.5-153.5
then, l = 144.5,
cf = 17,
f = 12 &
h = 9 =144.5+(9/4)

=146.75 mm

Therefore, the median length of the leaves = 146.75 mm.
5. The following table gives the distribution of a life time of 400 neon lamps.

 Lifetime(in hours) Number of lamps 1500-2000 14 2000-2500 56 2500-3000 60 3000-3500 86 3500-4000 74 4000-4500 62 4500-5000 48

Find the median lifetime of a lamp.
Solution:

 Class Interval Frequency Cumulative 1500-2000 14 14 2000-2500 56 70 2500-3000 60 130 3000-3500 86 216 3500-4000 74 290 4000-4500 62 352 4500-5000 48 400

Data: n = 400 &n/2
= 200 Median class
= 3000 – 3500
Therefore, l = 3000,
cf = 130,
f = 86 &
h = 500 =3000+(35000/86)

=3000+406.97

=3406.97

Therefore, the median lifetime of the lamps = 3406.97 hours
6. In this 100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in English alphabets in the surnames was obtained as follows:

 Number of letters 1-4 4-7 7-10 10-13 13-16 16-19 Number of surnames 6 30 40 16 4 4

Determine the number of median letters in the surnames. Find the number of mean letters in the surnames and also, find the size of modal in the surnames.
Solution: To calculate median:

 Class Interval Frequency Cumulative Frequency 1-4 6 6 4-7 30 36 7-10 40 76 10-13 16 92 13-16 4 96 16-19 4 100 Calculate the Mean:

 Class Interval fi xi fixi 1-4 6 2.5 15 4-7 30 5.5 165 7-10 40 8.5 340 10-13 16 11.5 184 13-16 4 14.5 51 16-19 4 17.5 70 Sum fi = 100 Sum fixi = 825

Mean = ∑fi x i/∑fi
Mean = 825/100 = 8.25
Therefore mean = 8.25

7. The distributions of below give a weight of 30 students of a class. Find the median weight of a student.

 Weight(in kg) 40-45 45-50 50-55 55-60 60-65 65-70 70-75 Number of students 2 3 8 6 6 3 2

Solution:

 Class Interval Frequency Cumulative frequency 40-45 2 2 45-50 3 5 50-55 8 13 55-60 6 19 60-65 6 25 65-70 3 28 70-75 2 30 ### Class 10 Maths Chapter 14 Exercise 14.4 Page: 293

1. The following distribution gives the daily income of 50 workers if a factory. Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive.

 Daily income in Rupees 100-120 120-140 140-160 160-180 180-200 Number of workers 12 14 8 6 10

Solution Convert the given distribution table to a less than type cumulative frequency distribution, and we get

 Daily income Frequency Cumulative Frequency Less than 120 12 12 Less than 140 14 26 Less than 160 8 34 Less than 180 6 40 Less than 200 10 50

From the table plot the points corresponding to the ordered pairs such as (120, 12), (140, 26), (160, 34), (180, 40) and (200, 50) on graph paper and the plotted points are joined to get a smooth curve and the obtained curve is known as less than type ogive curve 2. During the medical check-up of 35 students of a class, their weights were recorded as follows:

 Weight in kg Number of students Less than 38 0 Less than 40 3 Less than 42 5 Less than 44 9 Less than 46 14 Less than 48 28 Less than 50 32 Less than 52 35

Draw a less than type ogive for the given data.
Hence obtain the median weight from the graph and verify the result by using the formula.

Solution: From the given data, to represent the table in the form of graph, choose the upper limits of the class intervals are in x-axis and frequencies on y-axis by choosing the convenient scale.

Now plot the points corresponding to the ordered pairs given by (38, 0), (40, 3), (42, 5), (44, 9),(46, 14), (48, 28), (50, 32) and (52, 35) on a graph paper an join them to get a smooth curve. The curve obtained is known as less than type ogive. Locate the point 17.5 on the y-axis and draw a line parallel to the x-axis cutting the curve at a point. From the point, draw a perpendicular line to the x-axis.
The intersection point perpendicular to x-axis is the median of the given data.
Now, to find the mode by making a table.

 Class interval Number of students(Frequency) Cumulative Frequency Less than 38 0 0 Less than 40 3-0=3 3 Less than 42 5-3=2 8 Less than 44 9-5=4 9 Less than 46 14-9=5 14 Less than 48 28-14=14 28 Less than 50 32-28=4 32 Less than 52 35-22=3 35 3. The following tables gives production yield per hectare of wheat of 100 farms of a village.

 Production Yield 50-55 55-60 60-65 65-70 70-75 75-80 Number of farms 2 8 12 24 38 16

Change the distribution to a more than type distribution and draw its ogive.
Solution: Converting the given distribution to a more than type distribution, we get

 Production Yield (kg/ha) Number of farms More than or equal to 50 100 More than or equal to 55 100-2 = 98 More than or equal to 60 98-8= 90 More than or equal to 65 90-12=78 More than or equal to 70 78-24=54 More than or equal to 75 54-38 =16

From the table obtained draw the ogive by plotting the corresponding points where the upper limits in x-axis and the frequencies obtained in the y-axis are (50, 100), (55, 98), (60, 90), (65, 78), (70, 54) and (75, 16) on this graph paper. The graph obtained is known as more than type ogive curve. ## NCERT Solutions Class 10 Maths Chapter 14- Statistics

Class 10 Maths Chapter 14, Statistics, is one of the most important of all the chapter present in the textbook. The weightage of this chapter in the final exam is around 11 to 12 marks. On average, there will be 3 questions which could be asked from this chapter and marks will be distributed in a manner of 3+4+4( it could vary as per question).

Topics covered in Chapter 14, Statistics are;

• Mean of Grouped Data
• Mode of Grouped Data
• Median of Grouped Data
• Graphical Representation of Cumulative Frequency Distribution

List of Exercises in class 10 Maths Chapter 14 :
Exercise 14.1 Solutions 9 Question ( 9 long)
Exercise 14.2 Solutions 6 Question ( 6 long)
Exercise 14.3 Solutions 7 Question ( 7 long)
Exercise 14.4 Solutions 3 Question ( 3 long)

NCERT solutions for class 10 maths chapter 14- Statistics are made available for students who want to obtain good marks in this chapter. The methods and procedure to solve the questions have been explained clearly in these NCERT Solutions, such that, students find it easy to understand the fundamentals, quickly.

The world is highly data-oriented, in fact, each and every field have a group of data which represents the information. Statistics is the branch of mathematics which deals with the representation of data in a meaningful way.

You will face many real-life scenarios where the fundamentals of statistics are used to represent a set of data in tabular form or in graphs or in pie charts. There are a number of methods you will learn from this chapter such as, step deviation methods, finding mode and median of grouped data, converting frequency distribution and the relation between, mode, mean and median methods, etc. 10th Class NCERT solutions are the best material to prepare for the board exam.

### Key Features of NCERT Solutions for Class 10 Maths Chapter 14- Statistics

• The solutions for the statistics chapter works as a reference for the students.
• It will help students to score marks against the questions asked from the statistics chapter.
• Students can prepare and do the revision for chapter 14 with this source.
• The questions of statistics are solved by subject experts.
• The content of the material is as per the CBSE syllabus and guidelines.