CBSE Class 10th Maths Chapter 14 Statistics

 Mean= Average,

Mode: Value of observation(x) which occurs more often( i.e observation of max frequency).

Median: n/2 th or (n+1)/2 th observation for sum of freq if its even & observation for sum of freq odd observations

Exercise 14.1                                                                                                      

 

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?

Answer:

 

In order to find the mean value, we will use direct method because the numerical value of f_i and x_i are small. Find the midpoint of the given interval using the formula.

Midpoint (x_i) = (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 = ∑f_i x_i /∑f_i 

= 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. Answer:

 

Find the midpoint of the given interval using the formula.

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

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

So, u_i = (x_i - A)/h = u_i  = (x_i - 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∑f_iu_i /∑f_i =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.

To find out the missing frequency, use the mean formula.

Here, the value of mid-point (x_i)  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 = ∑f_ix_i /∑f_i = (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 heart beats per minute were recorded and summarized as follows. Find the mean heart beats per minute for these women, choosing a suitable method.

Number of heart beats 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

Answer:

 

From the given data, let us assume the mean as A = 75.5 x_i = (Upper limit + Lower limit)/2

Class size (h) = 3

Now, find the u_i  and f_iu_i 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∑f_iu_i /∑f_i 

= 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?

Answer:

 

Since, the given data is not continuous so we add 0.5 to the upper limit and subtract 0.5 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 ∑f_id_i /∑f_i 

= 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.

 

Find the midpoint of the given interval using the formula.

Midpoint (x_i) = (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∑f_iu_i /∑f_i

 = 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.

Answer:

 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 = ∑f_ix_i /∑f_i

= 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
Answer:

Find the midpoint of the given interval using the formula.

Midpoint (x_i) = (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 = ∑f_ix_i /∑f_i

= 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
Answer:

 

Find the midpoint of the given interval using the formula.

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

In this case, the value of mid-point (x_i) is very large, so let us assume the mean value, A = 70 and class interval is h = 10.

So, u_i = (x_i-A)/h = u_i = (x_i-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+(∑f_iu_i /∑f_i)×h

= 70+(-2/35)×10 

= 69.42

Therefore, the mean literacy part = 69.42

 

Exercise 14.2                                                                                                

 

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.

Answer:

 

greatest frequency = 23, so the modal class = 35 – 45,  l = 35, 

class width (h) = 10,  f1 = 23, 

f0 = 21 and f₂ = 14

The formula to find the mode is

Mode = l+ [(f1-f0)/(2f1-f0-f₂)]×h

Substitute the values in the formula, we get

Mode = 35+[(23-21)/(46-21-14)]×10 

Mode = 35+(20/11) =  35+1.8

Mode = 36.8 year

So the mode of the given data = 36.8 year Calculation of Mean:

First find the midpoint using the formula, x_i = (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 = ∑f_ix_i /∑f_i

= 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.

 

Answer:

l=60

From the given data the modal class is 60–80.  The frequencies are:

f1 = 61, f0 = 52, f₂ = 38 and h =20 The formula to find the mode is

Mode = l+ [(f1-f0)/(2f1-f0-f₂)]×h

 

Substitute the values in the formula, we get

Mode =60+[(61-52)/(122-52-38)]×20  

Mode = 60+((9 x 20)/32)

Mode = 60+(45/8) = 60+ 5.625

Therefore, modal lifetime of the components = 65.625 hours. 

 

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:

 

Given data:

Modal class = 1500-2000,

 l = 1500,  Frequencies:

f1 = 40 f0 = 24, f₂ = 33 and 

h = 500 Mode formula:

Mode = l+ [(f1-f0)/(2f1-f0-f₂)]×h

Substitute the values in the formula, we get

Mode =1500+[(40-24)/(80-24-33)]×500 

Mode = 1500+((16×500)/23)

Mode = 1500+(8000/23) = 1500 + 347.83

Therefore, modal monthly expenditure of the families = Rupees 1847.83  

Calculation for mean:

First find the midpoint using the formula, x_i =(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 +(∑f_iu_i /∑f_i)×h

Substitute the values in the given formula

= 2750+(-35/200)×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

 

Answer:

Given data:

Modal class = 30 – 35,  l = 30, 

Class width (h) = 5,  f1 = 10, f0 = 9 and f₂ = 3 Mode Formula:

Mode = l+ [(f1-f0)/(2f1-f0-f₂)]×h

Substitute the values in the given formula

Mode = 30+((10-9)/(20-9-3))×5 

Mode = 30+(5/8) = 30+0.625

Mode = 30.625 

Therefore, the mode of the given data = 30.625 

 

Calculation of mean:

Find the midpoint using the formula, x_i =(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 = ∑f_ix_i /∑f_i

= 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.

 

Answer:

Given data:

Modal class = 4000 – 5000,

 l = 4000,  class width (h) = 1000,  f1 = 18, f0 = 4 and f₂ = 9 Mode Formula:

Mode = l+ [(f1-f0)/(2f1-f0-f₂)]×h 

 

Substitute the values

Mode = 4000+((18-4)/(36-4-9))×1000 

Mode = 4000+(14000/23) = 4000+608.695

Mode = 4608.695

Mode = 4608.7 (approximately)

Thus, the mode of the given data is 4608.7 runs 

 

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:

Class width (h) = 10, f1 = 20, f0 = 12 and f₂ = 11

Mode = l+ [(f1-f0)/(2f1-f0-f₂)]×h 

 

Substitute the values

Mode = 40+((20-12)/(40-12-11))×10 

Mode = 40 + (80/17) = 40 + 4.7 = 44.7

Thus, the mode of the given data is 44.7 cars 

 

Exercise 14.3                                                                                                

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.

 

Hence, the median class is 125-145 with cumulative frequency = 42

Where, l = 125, n = 68, C_f = 22, f = 20, h = 20

 

Median is calculated as follows:

Median= l+(n/2-cf)h/f 

=125+((34−22)/20) × 20

=125+12 = 137

Therefore, median = 137

 

To calculate the mode: Modal class = 125-145,  f₁=20, f₀=13, f₂=14 & h = 20 Mode formula:

Mode = l+ [(f₁-f₀)/(2f₁-f₀-f₂)]×h

 

Mode = 125 + ((20-13)/(40-13-14))×20

=125+(140/13)

=125+10.77

=135.77

Therefore, mode = 135.77

 

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 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

Answer: 

 

Given data, n = 60

Median of the given data = 28.5

Where, n/2 = 30

Median class is 20 – 30 with a cumulative frequency = 25+x

Lower limit of median class, l = 20,  C_f = 5+x, f = 20 & h = 10

Median= l+(n/2-cf)h/f 

Substitute the values

28.5=20+((30−5−x)/20) × 10

8.5 = (25 - x)/2

17 = 25-x

Therefore, x =8

Now, from cumulative frequency, we can identify the value of x + y as follows:

Since,

60=5+20+15+5+x+y

Now, substitute the value of x, to find y

60 = 5+20+15+5+8+y

y = 60-53 y = 7

Therefore, the value of x = 8 and y = 7.

 

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

             

 

Answer:

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

Given data: n = 100 and n/2 = 50

Median class = 35-45

Then, l = 35, c_f = 45, f = 33 & h = 5

Median= l+(n/2-cf)h/f

Median = 35+((50-45)/33) × 5

= 35 + (5/33)5 

= 35.75

Therefore, the median age = 35.75 years.

 

4.                   The lengths of 40 leaves in a plant are measured correctly to the nearest millimeter, and the data obtained is represented as in the following table:

Answer:

Since the data are not continuous reduce 0.5 in the lower limit and add 0.5 in the upper limit.

n = 40 and n/2 = 20 Median class = 144.5-153.5 then, l = 144.5,  cf = 17, f = 12 & h = 9

Median= l+(n/2-cf)h/f 

Median = 144.5+((20-17)/12)×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.

Answer:

Lifetime (in hours) Number of lamps

Therefore, l = 3000, C_f  = 130,  f = 86 & h = 500

 

Median= l+(n/2-cf)h/f 

Median = 3000 + ((200-130)/86) × 500

= 3000 + (35000/86) 

= 3000 + 406.97

= 3406.97

Therefore, the median life time 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.

Answer: 

Therefore, l = 7, C_f = 36, f = 40 & h = 3

Median= l+(n/2-cf)h/f 

Median = 7+((50-36)/40) × 3

Median = 7+42/40

Median=8.05

 

Calculate the Mode:

Modal class = 7-10,

Where, l = 7, f₁ = 40, f₀ = 30, f₂ = 16 & h = 3     

Mode = l+ [(f1-f0)/(2f1-f0-f₂)]×h 

Mode = 7+((40-30)/(2×40-30-16)) × 3

= 7+(30/34)

= 7.88

Therefore mode = 7.88

 

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 = x̄ = ∑f_i x_i /∑f_i 

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.

40-45 45-50 50-55 55-60 60-65 65-70 70-75

Median= l+(n/2-cf)h/f 

Median = 55+((15-13)/6)×5

Median=55 + (10/6) = 55+1.666

Median =56.67

Therefore, the median weight of the students = 56.67

 

Exercise 14.4                                                                                                

 

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

Answer

Convert the given distribution table to a less than type cumulative frequency distribution, 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 curve

2.During the medical check-up of 35 students of a class, their weights were recorded as follows:

 

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.

 

Answer:

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.

 

 

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. 

The class 46 – 48 has the maximum frequency, therefore, this is modal class

Here, l = 46, h = 2, f₁= 14, f₀= 5 and f₂ = 4 The mode formula is given as:

Now, Mode =  

 Mode = l+ [(f1-f0)/(2f1-f0-f₂)]×h = 46 + 0.95 = 46.95

Thus, mode is verified.

3.  The following tables gives production yield per hectare of wheat of 100 farms of a village.

            

 

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.