All Exams >
A Level >
Mathematics for A Level >
All Questions
All questions of Statistics for A Level Exam
What is the range of the following data?23, 45, 34, 21, 89, 45, 47, 91- a)70
- b)56
- c)71
- d)69
Correct answer is option 'A'. Can you explain this answer?
What is the range of the following data?
23, 45, 34, 21, 89, 45, 47, 91
a)
70
b)
56
c)
71
d)
69
|
Vikas Kapoor answered |
Maximum and minimum value of the data 23, 45, 34, 21, 89, 45, 47, 91 are 21 and 91.
Range = 91 – 21 = 70
Range = 91 – 21 = 70
The mean deviation of the following data 14, 15, 16, 17, 13 is:- a)4
- b)2.3
- c)3
- d)1.2
Correct answer is option 'D'. Can you explain this answer?
The mean deviation of the following data 14, 15, 16, 17, 13 is:
a)
4
b)
2.3
c)
3
d)
1.2
|
|
Vikas Kapoor answered |
Here N= 5 , sigma x = 75
so mean = 15
now taking deviation from mean,( By ignoring signs)
we get sigma deviation from mean = 6
Now applying the formula of mean deviation
M.D.= SIGMA deviation from mean/ n
so M.D.= 6/5
= 1.2
so mean = 15
now taking deviation from mean,( By ignoring signs)
we get sigma deviation from mean = 6
Now applying the formula of mean deviation
M.D.= SIGMA deviation from mean/ n
so M.D.= 6/5
= 1.2
The mean of 5 observations is 4.4 and their variance is 8.24. If three observations are 1,2 and 6 , the other two observations are- a)4 and 8
- b)5 and 7
- c)5 and 9
- d)4 and 9
Correct answer is option 'D'. Can you explain this answer?
The mean of 5 observations is 4.4 and their variance is 8.24. If three observations are 1,2 and 6 , the other two observations are
a)
4 and 8
b)
5 and 7
c)
5 and 9
d)
4 and 9
|
Gaurav Kumar answered |
Let the other nos. be a and b
then (x+y+1+2+6)/5 = 4.4
x + y = 13 ---------------------(1)
Variance = 8.24
41.2 = 19.88 + (x2 + 19.36 – 8.8x) + (y2 + 19.36 – 8.8y)
21.32 = x2 + y2 + 38.72 – 8.8(x + y)
x2 + y2 + 38.72 – 8.8(13) – 21.32 = 0
(using equation (1))
x2 + y2 – 97 = 0 …(2)
Squaring equation (1) both the sides,
we get (x + y)2 = (13)^2
x2 + y2 + 2xy = 169
97 + 2xy = 169
(using equation (2))
xy = 36 or x = 36/y (1)
⇒ 36/y + y = 13
y2 + 36 = 13y
y2 – 13y + 36 = 0
(y – 4)(y – 9) = 0
Either (y – 4) = 0 or (y – 9) = 0
⇒ y = 4 or y = 9
For y = 4 x = 36/y
= 36/4 = 12
For y = 9
x = 36/9
x = 4
Thus, remaining two observations are 4 and 9.
then (x+y+1+2+6)/5 = 4.4
x + y = 13 ---------------------(1)
Variance = 8.24
41.2 = 19.88 + (x2 + 19.36 – 8.8x) + (y2 + 19.36 – 8.8y)
21.32 = x2 + y2 + 38.72 – 8.8(x + y)
x2 + y2 + 38.72 – 8.8(13) – 21.32 = 0
(using equation (1))
x2 + y2 – 97 = 0 …(2)
Squaring equation (1) both the sides,
we get (x + y)2 = (13)^2
x2 + y2 + 2xy = 169
97 + 2xy = 169
(using equation (2))
xy = 36 or x = 36/y (1)
⇒ 36/y + y = 13
y2 + 36 = 13y
y2 – 13y + 36 = 0
(y – 4)(y – 9) = 0
Either (y – 4) = 0 or (y – 9) = 0
⇒ y = 4 or y = 9
For y = 4 x = 36/y
= 36/4 = 12
For y = 9
x = 36/9
x = 4
Thus, remaining two observations are 4 and 9.
A batsman scores runs in 10 innings as 38,70,48,34,42,55,63,46,54 and 44 , then the mean score is- a)4.94
- b)49.4
- c)494
- d)0.494
Correct answer is option 'B'. Can you explain this answer?
A batsman scores runs in 10 innings as 38,70,48,34,42,55,63,46,54 and 44 , then the mean score is
a)
4.94
b)
49.4
c)
494
d)
0.494
|
Mayuri Agrawal answered |
38+70+48+34+42+55+63+46+54 +44=494
494/10
=49.4
494/10
=49.4
The mean deviation about the mean for the following data:5, 6, 7, 8, 6, 9, 13, 12, 15 is:- a)1.5
- b)3.2
- c)2.89
- d)5
Correct answer is option 'C'. Can you explain this answer?
The mean deviation about the mean for the following data:
5, 6, 7, 8, 6, 9, 13, 12, 15 is:
a)
1.5
b)
3.2
c)
2.89
d)
5
|
|
Vikas Kapoor answered |
let, X =5,6,7,8,9,13,12,15.
(5+6+7+8+9+13+12+15)÷9 = 9.
and hence a = 9.
the mean deviation about the mean is summation of |X-a|÷ the total number
i.e , |X-a| = 4,3,2,1,3,0,4,3,6 and the total no. is 9.
hence summation of |X-a| = 26,
the mean deviation is 26 ÷ 9 = 2.89 ans
(5+6+7+8+9+13+12+15)÷9 = 9.
and hence a = 9.
the mean deviation about the mean is summation of |X-a|÷ the total number
i.e , |X-a| = 4,3,2,1,3,0,4,3,6 and the total no. is 9.
hence summation of |X-a| = 26,
the mean deviation is 26 ÷ 9 = 2.89 ans
If the coefficient of variation between x and y is 0.28, covariance between x and y is 7.6, and the variance of x is 9, then the S.D. of the y series is- a)10.05
- b)10.1
- c)9.05
- d)9.8
Correct answer is option 'C'. Can you explain this answer?
If the coefficient of variation between x and y is 0.28, covariance between x and y is 7.6, and the variance of x is 9, then the S.D. of the y series is
a)
10.05
b)
10.1
c)
9.05
d)
9.8
|
Defence Exams answered |
Correlation coefficient = cov (x,y)/ (std deviation (x) ×std deviation (y))
Correlation coefficient = 0.28
cov (x,y) = 7.6
variance of x is 9.
=> std deviation (x) = √variance of X = √9 = 3
=> 0.28 = 7.6 / ( 3 * std deviation (y))
=> std deviation (y) = 7.6 / ( 3 * 0.28)
=> std deviation (y) = 9.05
standard deviation of Y series = 9.05
Correlation coefficient = 0.28
cov (x,y) = 7.6
variance of x is 9.
=> std deviation (x) = √variance of X = √9 = 3
=> 0.28 = 7.6 / ( 3 * std deviation (y))
=> std deviation (y) = 7.6 / ( 3 * 0.28)
=> std deviation (y) = 9.05
standard deviation of Y series = 9.05
For a given data, the standard deviation is 20.If 3 is added to each observation , what is the new variance of the resulting observations?- a)20
- b)23
- c)17
- d)60
Correct answer is option 'A'. Can you explain this answer?
For a given data, the standard deviation is 20.If 3 is added to each observation , what is the new variance of the resulting observations?
a)
20
b)
23
c)
17
d)
60
|
Sarita Yadav answered |
If a three, is added to each number in a set of data, the mean will be increased by 3 and the standard deviation will be unaltered (since the spread of the data will be unchanged).
Hence, variance of the new data = 20
Hence, variance of the new data = 20
The two lines of regression are 2x - 7y + 6 = 0 and 7x – 2y +1 = 0. What is correlation coefficient between x and y ?- a)4/49
- b)- 2/7
- c)2/7
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
The two lines of regression are 2x - 7y + 6 = 0 and 7x – 2y +1 = 0. What is correlation coefficient between x and y ?
a)
4/49
b)
- 2/7
c)
2/7
d)
none of these
|
Hansa Sharma answered |
ρ = (b(xy) * b(yx))
But sign of ρρ is same as sign of b(xy), b(yx)
Therefore, ρ = 2/7
But sign of ρρ is same as sign of b(xy), b(yx)
Therefore, ρ = 2/7
The mean deviation about the mean for the following data 3, 7, 8, 9, 4, 6, 8, 13, 12, 10 is:- a)5
- b)3
- c)2
- d)2.4
Correct answer is option 'D'. Can you explain this answer?
The mean deviation about the mean for the following data 3, 7, 8, 9, 4, 6, 8, 13, 12, 10 is:
a)
5
b)
3
c)
2
d)
2.4
|
|
Gaurav Kumar answered |
Arrange data in ascending order,
3,4,6,7,8,8,9,10,12,13
No. of observations = 10
Median = n/2 => 10/2 = 5h observation.
5th observation is 8
Now we calculate mean deviation about median, i.e;
=> ∑∣xi−M∣/10
= {|3-8| +|4-8| +|6-8| +|7-8| +|8-8| +|8-8| +|9-8| +|10-8| +|12-8| +|13-8| }/10
= { 5 + 4 + 2 + 1 + 0 + 0 + 1 + 2 + 4 + 5}/10
= 24/10 => 2.4
3,4,6,7,8,8,9,10,12,13
No. of observations = 10
Median = n/2 => 10/2 = 5h observation.
5th observation is 8
Now we calculate mean deviation about median, i.e;
=> ∑∣xi−M∣/10
= {|3-8| +|4-8| +|6-8| +|7-8| +|8-8| +|8-8| +|9-8| +|10-8| +|12-8| +|13-8| }/10
= { 5 + 4 + 2 + 1 + 0 + 0 + 1 + 2 + 4 + 5}/10
= 24/10 => 2.4
The standard deviation of first 10 multiples of 4 is:- a)7
- b)8
- c)11.5
- d)14
Correct answer is option 'C'. Can you explain this answer?
The standard deviation of first 10 multiples of 4 is:
a)
7
b)
8
c)
11.5
d)
14
|
Krishna Iyer answered |
First 10 multiples of 4 are 4,8,12...40.
This is an A.P.
sum=n/2(a+l)
= 10/2(4+40)
∴ sum=220.
Mean, u=sum/n
= 220/10 = 22
D1 = 4-22 = -18
D2 = 8-22 = -16
D3 = 12-22 = -10
D4 = 16- 22 = -8
Similarly we subtract multiple of 4 by 22 upto 10 terms we get
-18, -14, -10, -8………...18
S.D. = σ2 = ∑(D2)/n
=[ (-18)2 ,(-14)2, (-10)2, (-6)2 + (-2)2 +(6)2 + (10)2 + (14)2 + (18)2]/10
Solving this, we get
σ = 11.5
This is an A.P.
sum=n/2(a+l)
= 10/2(4+40)
∴ sum=220.
Mean, u=sum/n
= 220/10 = 22
D1 = 4-22 = -18
D2 = 8-22 = -16
D3 = 12-22 = -10
D4 = 16- 22 = -8
Similarly we subtract multiple of 4 by 22 upto 10 terms we get
-18, -14, -10, -8………...18
S.D. = σ2 = ∑(D2)/n
=[ (-18)2 ,(-14)2, (-10)2, (-6)2 + (-2)2 +(6)2 + (10)2 + (14)2 + (18)2]/10
Solving this, we get
σ = 11.5
If the mean of the squares of first n natural numbers be 11, then n is equal to- a)13
- b)5
- c)- 13/2
- d)11
Correct answer is option 'B'. Can you explain this answer?
If the mean of the squares of first n natural numbers be 11, then n is equal to
a)
13
b)
5
c)
- 13/2
d)
11
|
Learners Habitat answered |
The mean of the squares of the first n natural numbers is given by:
The sum of the squares of the first n natural numbers is:
Thus, the mean is:
We are given that the mean is 11. Therefore:
Multiply through by 6:
(n+1)(2n+1)=66
Expand:
2n2 +3n+1=66
Simplify:
2n2+3n−65=0
Solve this quadratic equation using the quadratic formula:
Here, a=2, b=3, and c=−65:
Calculate the two solutions:
Since n must be a positive integer, n=5.
The sum of the squares of the first n natural numbers is:
Thus, the mean is:
We are given that the mean is 11. Therefore:
Multiply through by 6:
(n+1)(2n+1)=66
Expand:
2n2 +3n+1=66
Simplify:
2n2+3n−65=0
Solve this quadratic equation using the quadratic formula:
Here, a=2, b=3, and c=−65:
Calculate the two solutions:
Since n must be a positive integer, n=5.
The Q.D. of the daily wages (in Rs) of 7 persons given below: 12,7,15,10,17,19,25 is- a)4.5
- b)9
- c)5
- d)14.5
Correct answer is option 'A'. Can you explain this answer?
The Q.D. of the daily wages (in Rs) of 7 persons given below: 12,7,15,10,17,19,25 is
a)
4.5
b)
9
c)
5
d)
14.5
|
Arshiya Shah answered |
Q.D. of Daily Wages of 7 Persons
To find the Q.D. (Quartile Deviation) of the daily wages of 7 persons given below: 12, 7, 15, 10, 17, 19, 25, we need to follow the steps given below:
Step 1: Arrange the data in ascending order.
7, 10, 12, 15, 17, 19, 25
Step 2: Find the median or the second quartile (Q2) of the data.
Median of the data = (n + 1)/2-th value = (7 + 1)/2-th value = 4th value
Hence, median = 15
Step 3: Find the first quartile (Q1) of the data.
Q1 = (n + 1)/4-th value = (7 + 1)/4-th value = 2nd value
Hence, Q1 = 10
Step 4: Find the third quartile (Q3) of the data.
Q3 = 3(n + 1)/4-th value = 3(7 + 1)/4-th value = 6th value
Hence, Q3 = 19
Step 5: Calculate the Quartile Deviation (Q.D.) using the formula:
Q.D. = (Q3 - Q1)/2
Substituting the values of Q1 and Q3, we get:
Q.D. = (19 - 10)/2 = 4.5
Therefore, the Q.D. of the daily wages of 7 persons given is 4.5.
Note: Quartile deviation is a measure of dispersion that gives an idea about the spread of data around the median. It is calculated as half of the difference between the third and first quartiles.
To find the Q.D. (Quartile Deviation) of the daily wages of 7 persons given below: 12, 7, 15, 10, 17, 19, 25, we need to follow the steps given below:
Step 1: Arrange the data in ascending order.
7, 10, 12, 15, 17, 19, 25
Step 2: Find the median or the second quartile (Q2) of the data.
Median of the data = (n + 1)/2-th value = (7 + 1)/2-th value = 4th value
Hence, median = 15
Step 3: Find the first quartile (Q1) of the data.
Q1 = (n + 1)/4-th value = (7 + 1)/4-th value = 2nd value
Hence, Q1 = 10
Step 4: Find the third quartile (Q3) of the data.
Q3 = 3(n + 1)/4-th value = 3(7 + 1)/4-th value = 6th value
Hence, Q3 = 19
Step 5: Calculate the Quartile Deviation (Q.D.) using the formula:
Q.D. = (Q3 - Q1)/2
Substituting the values of Q1 and Q3, we get:
Q.D. = (19 - 10)/2 = 4.5
Therefore, the Q.D. of the daily wages of 7 persons given is 4.5.
Note: Quartile deviation is a measure of dispersion that gives an idea about the spread of data around the median. It is calculated as half of the difference between the third and first quartiles.
The S.D. of the observations 22,26,28,20,24,30 is- a)3
- b)2.4
- c)3.42
- d)2
Correct answer is option 'C'. Can you explain this answer?
The S.D. of the observations 22,26,28,20,24,30 is
a)
3
b)
2.4
c)
3.42
d)
2
|
Lavanya Menon answered |
μ = (22 + 26 + 28 + 20 + 24 + 30)/6
= 150/6
= 25
x(i) = (xi - μ)2
x(22) = (22-25)2 = 9
x(26) = (26-25)2 = 1
x(28) = (28-25)2 = 9
x(20) = (20-25)2 = 25
x(24) = (24-25)2 = 1
x(30) = (30-25)2 = 25
(xi - μ)2 = 70
Standard deviation : [(xi - μ)2]/N
= (70/6)½
= 3.42
= 150/6
= 25
x(i) = (xi - μ)2
x(22) = (22-25)2 = 9
x(26) = (26-25)2 = 1
x(28) = (28-25)2 = 9
x(20) = (20-25)2 = 25
x(24) = (24-25)2 = 1
x(30) = (30-25)2 = 25
(xi - μ)2 = 70
Standard deviation : [(xi - μ)2]/N
= (70/6)½
= 3.42
For a given data, the variance is 15. If each observation is multiplied by 2, what is the new variance of the resulting observations?- a)15
- b)60
- c)30
- d)7.5
Correct answer is option 'B'. Can you explain this answer?
For a given data, the variance is 15. If each observation is multiplied by 2, what is the new variance of the resulting observations?
a)
15
b)
60
c)
30
d)
7.5
|
Poonam Reddy answered |
Variance = 15
New variance = 22*15
= 4*15
= 60
New variance = 22*15
= 4*15
= 60
The mean of the first n terms of the A.P. (a + d) + (a + 3d) + (a + 5d) +………..is- a)
- b)a + n2d
- c)a + nd
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
The mean of the first n terms of the A.P. (a + d) + (a + 3d) + (a + 5d) +………..is
a)
b)
a + n2d
c)
a + nd
d)
none of these
|
Manish Aggarwal answered |
The given arithmetic progression (AP) is:
(a+d),(a+3d),(a+5d),…
This sequence has a common difference of 2d.
Step 1: General term
The n-th term of this AP is given by:
Tn =a+(2n−1)d
Step 2: Sum of the first n terms
The sum of the first n terms of an AP is:
Here: First term = a+d
Last term = Tn =a+(2n−1)d
Simplify the terms:
Step 3: Mean of the first n terms
The mean of the first n terms is:
Substitute Sn:
(a+d),(a+3d),(a+5d),…
This sequence has a common difference of 2d.
Step 1: General term
The n-th term of this AP is given by:
Tn =a+(2n−1)d
Step 2: Sum of the first n terms
The sum of the first n terms of an AP is:
Here: First term = a+d
Last term = Tn =a+(2n−1)d
Simplify the terms:
Step 3: Mean of the first n terms
The mean of the first n terms is:
Substitute Sn:
The standard deviation for the following data:
- a)5
- b)4.21
- c)3.12
- d)None of the above
Correct answer is option 'C'. Can you explain this answer?
The standard deviation for the following data:
a)
5
b)
4.21
c)
3.12
d)
None of the above
|
|
Neha Sharma answered |
Answer: C
Solution: Variance= [summation (y^2×f) /N] -[ summation (yf) /N]^2
=(296/25) -(0/25) ^2
=11.84
standard deviation=√11.84=3.12
Solution: Variance= [summation (y^2×f) /N] -[ summation (yf) /N]^2
=(296/25) -(0/25) ^2
=11.84
standard deviation=√11.84=3.12
Mean of the squares of the deviations from mean is called the:- a)Mode
- b)Standard deviation
- c)Variance
- d)Quartile deviation
Correct answer is option 'C'. Can you explain this answer?
Mean of the squares of the deviations from mean is called the:
a)
Mode
b)
Standard deviation
c)
Variance
d)
Quartile deviation
|
Jyoti Kapoor answered |
Squared deviations from the mean Squared deviations from the mean (SDM) are involved in various calculations. In probability theory and statistics, the definition of variance is either the expected value of the SDM (when considering a theoretical distribution) or its average value (for actual experimental data).
A batsman scores runs in 10 innings 38,70,48,34,42,55,63,46,54 and 44 , then the mean deviation is- a)6.4
- b)9.6
- c)10.6
- d)8.6
Correct answer is option 'D'. Can you explain this answer?
A batsman scores runs in 10 innings 38,70,48,34,42,55,63,46,54 and 44 , then the mean deviation is
a)
6.4
b)
9.6
c)
10.6
d)
8.6
|
|
Jyoti Sengupta answered |
Arranging the given data in ascending order,
we have 34, 38, 42, 44, 46, 48, 54, 55, 63, 70
Here, Median M = (46+48)/2
=47
(∵ n = 10, median is the mean of 5th and 6th items)
∴ Mean deviation = ∑|xi−M|/n
=∑|xi−47|/10
= (13+9+5+3+1+1+7+8+16+23)/10
=8.6
we have 34, 38, 42, 44, 46, 48, 54, 55, 63, 70
Here, Median M = (46+48)/2
=47
(∵ n = 10, median is the mean of 5th and 6th items)
∴ Mean deviation = ∑|xi−M|/n
=∑|xi−47|/10
= (13+9+5+3+1+1+7+8+16+23)/10
=8.6
The H.M. of 4,8,16 is- a)6.7
- b)7.8
- c)6.85
- d)6.4
Correct answer is option 'C'. Can you explain this answer?
The H.M. of 4,8,16 is
a)
6.7
b)
7.8
c)
6.85
d)
6.4
|
|
Vivek answered |
HM = 3*(1/4 + 1/8 + 1/16)-¹
= 6.85
KEY POINT →→ HM of n terms = n*[1/a + 1/b + (upto
n terms) ]-¹
If in moderately asymmetrical distribution mode and mean of the data are 6 μ and 9 μ respectively, then median is- a)8 μ
- b)6 μ
- c)5 μ
- d)7 μ
Correct answer is option 'A'. Can you explain this answer?
If in moderately asymmetrical distribution mode and mean of the data are 6 μ and 9 μ respectively, then median is
a)
8 μ
b)
6 μ
c)
5 μ
d)
7 μ
|
Ananya Das answered |
Median = [mode + 2(mean)]/3
= [6μ+2(9μ)]/3
= 24μ/3
= 8μ
= [6μ+2(9μ)]/3
= 24μ/3
= 8μ
The Mode of the following items is 0,1,6,7,2,3,7,6,6,2,6,0,5,6,0.- a)2
- b)5
- c)0
- d)6
Correct answer is option 'D'. Can you explain this answer?
The Mode of the following items is 0,1,6,7,2,3,7,6,6,2,6,0,5,6,0.
a)
2
b)
5
c)
0
d)
6
|
|
Manish Aggarwal answered |
To find the mode, we count how many times each number appears:
- 0 appears 3 times.
- 1 appears 1 time.
- 6 appears 5 times.
- 7 appears 2 times.
- 2 appears 2 times.
- 3 appears 1 time.
- 5 appears 1 time.
Since 6 occurs the most frequently (5 times), the mode is 6.
Thus, the correct answer is:
D: 6
If the median = (mode + 2 mean) μ, then μ is equal to- a)3
- b)1/3
- c)2/3
- d)2
Correct answer is option 'B'. Can you explain this answer?
If the median = (mode + 2 mean) μ, then μ is equal to
a)
3
b)
1/3
c)
2/3
d)
2
|
|
Muskaan Singh answered |
If the median equals the mode plus twice the mean, we can express this relationship mathematically as:
Median = Mode + 2 * Mean
Median = Mode + 2 * Mean
For a normal distribution, we have- a)mean = median = mode
- b)mean = mode
- c)median = mode
- d)mean = median
Correct answer is option 'A'. Can you explain this answer?
For a normal distribution, we have
a)
mean = median = mode
b)
mean = mode
c)
median = mode
d)
mean = median
|
Nipuns Institute answered |
For a normal distribution, the following property holds:
Mean=Median=Mode
This is because a normal distribution is symmetric about its center, and the mean, median, and mode all lie at the same central point.
Mean=Median=Mode
This is because a normal distribution is symmetric about its center, and the mean, median, and mode all lie at the same central point.
If the mean of the first n odd natural numbers be n itself, then n is equal to- a)3
- b)any natural number
- c)2
- d)1
Correct answer is option 'B'. Can you explain this answer?
If the mean of the first n odd natural numbers be n itself, then n is equal to
a)
3
b)
any natural number
c)
2
d)
1
|
Tarun Kaushik answered |
The first n odd natural numbers are 1,3,5,…,(2n−1). The mean of these numbers is calculated as:
The sum of the first n odd natural numbers is:
Sum=n2
So, the mean becomes:
We are given that the mean is equal to n itself. This equality holds true for any natural number n.
The sum of the first n odd natural numbers is:
Sum=n2
So, the mean becomes:
We are given that the mean is equal to n itself. This equality holds true for any natural number n.
If the two lines of regression are at right angles, then ρ(X,Y) is equal to- a)- 1
- b)0
- c)1
- d)1 or - 1
Correct answer is option 'B'. Can you explain this answer?
If the two lines of regression are at right angles, then ρ(X,Y) is equal to
a)
- 1
b)
0
c)
1
d)
1 or - 1
|
|
Nabanita Bajaj answered |
Understanding Regression Lines and Correlation
When two lines of regression are at right angles, it provides critical information about the relationship between the variables X and Y.
Correlation Coefficient (ρ)
- The correlation coefficient (ρ) quantifies the degree of linear relationship between two variables.
- Its value ranges from -1 to +1:
- ρ = 1 indicates a perfect positive correlation.
- ρ = -1 indicates a perfect negative correlation.
- ρ = 0 indicates no correlation.
Right Angles and Independence
- Lines of regression are perpendicular (at right angles) when the product of their slopes equals -1.
- This condition signifies that changes in one variable do not predict changes in the other, indicating independence.
Conclusion: The Value of ρ
- When the lines of regression are at right angles, it implies that:
- There is no linear relationship between X and Y.
- Thus, ρ must equal 0.
This leads us to conclude that the correct answer to the question is option 'B', which states that ρ(X,Y) is equal to 0.
When two lines of regression are at right angles, it provides critical information about the relationship between the variables X and Y.
Correlation Coefficient (ρ)
- The correlation coefficient (ρ) quantifies the degree of linear relationship between two variables.
- Its value ranges from -1 to +1:
- ρ = 1 indicates a perfect positive correlation.
- ρ = -1 indicates a perfect negative correlation.
- ρ = 0 indicates no correlation.
Right Angles and Independence
- Lines of regression are perpendicular (at right angles) when the product of their slopes equals -1.
- This condition signifies that changes in one variable do not predict changes in the other, indicating independence.
Conclusion: The Value of ρ
- When the lines of regression are at right angles, it implies that:
- There is no linear relationship between X and Y.
- Thus, ρ must equal 0.
This leads us to conclude that the correct answer to the question is option 'B', which states that ρ(X,Y) is equal to 0.
The two lines of regression are x + 4y = 3 and 3x +y = 15. value of x for y = 3 is- a)- 4
- b)4
- c)- 9
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
The two lines of regression are x + 4y = 3 and 3x +y = 15. value of x for y = 3 is
a)
- 4
b)
4
c)
- 9
d)
none of these
|
Pragati Nair answered |
We have x + 4y = 3 and 3x + y = 15 as the lines of regression of y on x and x on y respectively.
Therefore, for y = 3 the required line is x on y
so put y= 3 in 3x + y = 15
we get, x = 4
Therefore, for y = 3 the required line is x on y
so put y= 3 in 3x + y = 15
we get, x = 4
The mean weight of a group of 10 items is 28 and that of another group of n items is 35.The mean of combined group of 10 + n items is found to be 30. The value of n is- a)12
- b)10
- c)4
- d)2
Correct answer is option 'C'. Can you explain this answer?
The mean weight of a group of 10 items is 28 and that of another group of n items is 35.The mean of combined group of 10 + n items is found to be 30. The value of n is
a)
12
b)
10
c)
4
d)
2
|
Bhavana Shah answered |
Given:
- Mean weight of 10 items = 28
- Mean weight of n items = 35
- Mean weight of combined group of 10+n items = 30
To find: Value of n
Solution:
Let's start by using the formula for the mean of a group of items:
Mean = (Sum of all items) / (Number of items)
We can use this formula for each of the three groups mentioned in the question:
Group 1 (10 items):
Mean = 28
Number of items = 10
Sum of all items = 28 * 10 = 280
Group 2 (n items):
Mean = 35
Number of items = n
Sum of all items = 35n
Combined group (10+n items):
Mean = 30
Number of items = 10 + n
Sum of all items = 30 * (10 + n) = 300 + 30n
Now we can use the fact that the combined group consists of items from Group 1 and Group 2:
Sum of all items in combined group = Sum of all items in Group 1 + Sum of all items in Group 2
300 + 30n = 280 + 35n
Solving for n:
30n - 35n = 280 - 300
-5n = -20
n = 4
Therefore, the value of n is 4, which is option (c).
- Mean weight of 10 items = 28
- Mean weight of n items = 35
- Mean weight of combined group of 10+n items = 30
To find: Value of n
Solution:
Let's start by using the formula for the mean of a group of items:
Mean = (Sum of all items) / (Number of items)
We can use this formula for each of the three groups mentioned in the question:
Group 1 (10 items):
Mean = 28
Number of items = 10
Sum of all items = 28 * 10 = 280
Group 2 (n items):
Mean = 35
Number of items = n
Sum of all items = 35n
Combined group (10+n items):
Mean = 30
Number of items = 10 + n
Sum of all items = 30 * (10 + n) = 300 + 30n
Now we can use the fact that the combined group consists of items from Group 1 and Group 2:
Sum of all items in combined group = Sum of all items in Group 1 + Sum of all items in Group 2
300 + 30n = 280 + 35n
Solving for n:
30n - 35n = 280 - 300
-5n = -20
n = 4
Therefore, the value of n is 4, which is option (c).
For a moderately skewed distribution, quartile deviation and the standard deviation are related by- a)S.D. = 3/2 Q.D.
- b)S.D = 3/4 Q.D.
- c)S.D. = 4/3 Q.D.
- d)S.D. = 2/3 Q.D.
Correct answer is option 'A'. Can you explain this answer?
For a moderately skewed distribution, quartile deviation and the standard deviation are related by
a)
S.D. = 3/2 Q.D.
b)
S.D = 3/4 Q.D.
c)
S.D. = 4/3 Q.D.
d)
S.D. = 2/3 Q.D.
|
|
Sushant Chaudhary answered |
The relationship between the mean, quartile and the standard deviation are as follows:
Mean Deviation is the mean of all the absolute deviations of a set of data.
Quartile deviation is the difference between “first and third quartiles” in any distribution.
Standard deviation measures the “dispersion of the data set” that is relative to its mean.
Mean Deviation = 4/5 × Quartile deviation
Standard Deviation = 3/2 × Quartile deviation
Mean Deviation is the mean of all the absolute deviations of a set of data.
Quartile deviation is the difference between “first and third quartiles” in any distribution.
Standard deviation measures the “dispersion of the data set” that is relative to its mean.
Mean Deviation = 4/5 × Quartile deviation
Standard Deviation = 3/2 × Quartile deviation
For a symmetrical distribution Q1 = 20 and Q3. = 40. The median of the data is- a)10
- b)40
- c)20
- d)30
Correct answer is option 'D'. Can you explain this answer?
For a symmetrical distribution Q1 = 20 and Q3. = 40. The median of the data is
a)
10
b)
40
c)
20
d)
30
|
|
Tarun Kaushik answered |
In a symmetrical distribution, the median lies exactly halfway between the first quartile (Q1) and the third quartile (Q3). The formula for the median in such cases is:
Given:
Q1=20andQ3=40
Given:
Q1=20andQ3=40
Mean deviation from the mean for the observations – 1, 0 ,4 is- a)2/3
- b)3/2
- c)2
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
Mean deviation from the mean for the observations – 1, 0 ,4 is
a)
2/3
b)
3/2
c)
2
d)
none of these
|
|
Bhargavi Bose answered |
To calculate the mean deviation from the mean for a set of observations, follow these steps:
1. Calculate the mean of the observations by adding all the values together and dividing by the total number of observations.
2. Subtract the mean from each individual observation to find the deviation from the mean.
3. Take the absolute value of each deviation to ensure that negative values do not cancel out positive values.
4. Add up all the absolute deviations.
5. Divide the sum of the absolute deviations by the total number of observations to find the mean deviation from the mean.
Here is an example:
Observations: 5, 7, 9, 12, 15
1. Mean = (5 + 7 + 9 + 12 + 15) / 5 = 9.6
2. Deviations from the mean: -4.6, -2.6, -0.6, 2.4, 5.4
3. Absolute deviations: 4.6, 2.6, 0.6, 2.4, 5.4
4. Sum of absolute deviations: 15.6
5. Mean deviation from the mean = 15.6 / 5 = 3.12
Therefore, the mean deviation from the mean for these observations is 3.12.
1. Calculate the mean of the observations by adding all the values together and dividing by the total number of observations.
2. Subtract the mean from each individual observation to find the deviation from the mean.
3. Take the absolute value of each deviation to ensure that negative values do not cancel out positive values.
4. Add up all the absolute deviations.
5. Divide the sum of the absolute deviations by the total number of observations to find the mean deviation from the mean.
Here is an example:
Observations: 5, 7, 9, 12, 15
1. Mean = (5 + 7 + 9 + 12 + 15) / 5 = 9.6
2. Deviations from the mean: -4.6, -2.6, -0.6, 2.4, 5.4
3. Absolute deviations: 4.6, 2.6, 0.6, 2.4, 5.4
4. Sum of absolute deviations: 15.6
5. Mean deviation from the mean = 15.6 / 5 = 3.12
Therefore, the mean deviation from the mean for these observations is 3.12.
The arithmetic mean of the numerical values of the deviations of items from some average value is called the- a)Standard deviation
- b)Range
- c)Quartile deviation
- d)Mean deviation
Correct answer is option 'D'. Can you explain this answer?
The arithmetic mean of the numerical values of the deviations of items from some average value is called the
a)
Standard deviation
b)
Range
c)
Quartile deviation
d)
Mean deviation
|
|
Maulik Majumdar answered |
Mean deviation of a data set is the average of the absolute deviations from a central point (Average value).
S.D. of a data is 6. When each observation is increased by 1, then the S.D. of new data is- a)8
- b)6
- c)5
- d)7
Correct answer is option 'B'. Can you explain this answer?
S.D. of a data is 6. When each observation is increased by 1, then the S.D. of new data is
a)
8
b)
6
c)
5
d)
7
|
EduRev JEE answered |
Answer: B: 6
Explanation: When a constant is added to every observation in a data set, the spread or dispersion of the data does not change. Therefore, adding 1 to each observation leaves the standard deviation unchanged.
If the mean of 3,4,x,7,10, is 6, then the value of x is- a)7
- b)5
- c)4
- d)6
Correct answer is option 'D'. Can you explain this answer?
If the mean of 3,4,x,7,10, is 6, then the value of x is
a)
7
b)
5
c)
4
d)
6
|
|
Tarun Kaushik answered |
To find the value of x, we use the formula for the mean:
Step 1: Write the equation for the mean
Given:
Step 1: Write the equation for the mean
Given:
- Observations: 3,4,x,7,10
- Mean: 6
Substitute into the formula:
Step 2: Simplify the equation
Multiply through by 5:
30=24+x
Solve for x:
x=30−24=6
Step 2: Simplify the equation
Multiply through by 5:
30=24+x
Solve for x:
x=30−24=6
If the two lines of regression are y = 3x – 5 and y = 2x – 4 , then ρ(X,Y) is equal t- a)
- b)
- c)
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
If the two lines of regression are y = 3x – 5 and y = 2x – 4 , then ρ(X,Y) is equal t
a)
b)
c)
d)
none of these
|
|
Tarun Kaushik answered |
To determine the correlation coefficient ρ(X,Y) from the two regression lines, we use the property that the product of the slopes of the two regression lines is equal to ρ 2 .
Given regression lines:
y=3x−5
Slope (m1 ) = 3
Given regression lines:
y=3x−5
Slope (m1 ) = 3
y=2x−4
Slope ( m2) = 2
Step 1: Check the product of slopes
The product of the slopes of the regression lines is:
m 1 ⋅m 2 =3⋅2=6
However, for the two regression lines, the product of their slopes must satisfy:
m1 ⋅m2 =ρ 2
Thus: ρ2 =6
Step 2: Conclusion
Since ρ2 cannot exceed 1, this configuration of regression lines is not possible. Hence, ρ(X,Y) is undefined for the given regression lines.
Slope ( m2) = 2
Step 1: Check the product of slopes
The product of the slopes of the regression lines is:
m 1 ⋅m 2 =3⋅2=6
However, for the two regression lines, the product of their slopes must satisfy:
m1 ⋅m2 =ρ 2
Thus: ρ2 =6
Step 2: Conclusion
Since ρ2 cannot exceed 1, this configuration of regression lines is not possible. Hence, ρ(X,Y) is undefined for the given regression lines.
Which of the following, in case of a discrete data, is not equal to the median?- a)5th decile
- b)2nd quartile
- c)lower quartile
- d)50th percentile
Correct answer is option 'C'. Can you explain this answer?
Which of the following, in case of a discrete data, is not equal to the median?
a)
5th decile
b)
2nd quartile
c)
lower quartile
d)
50th percentile
|
KP Classes answered |
The median is the middle value of a dataset when arranged in ascending order. It is also represented as:
- The 5th decile (because the median divides the data into two equal parts, which is the 5th decile in the decile system).
- The 2nd quartile (as quartiles divide the data into four equal parts, and the median is at the 2nd quartile).
- The 50th percentile (since percentiles divide the data into 100 equal parts, and the median corresponds to the 50th percentile).
The lower quartile (Q1) is the first quartile, representing the 25th percentile, and is not equal to the median.
If mean = (3 median – mode) x , then the value of x is- a)1
- b)1/2
- c)3/2
- d)2
Correct answer is option 'B'. Can you explain this answer?
If mean = (3 median – mode) x , then the value of x is
a)
1
b)
1/2
c)
3/2
d)
2
|
Nipun Tuteja answered |
We are given the relationship:
Mean=(3⋅Median−Mode)⋅x
Step 1: Recall the empirical relationship
The empirical relationship between the mean, median, and mode is:
Mode=3⋅Median−2⋅Mean
Step 2: Compare the given formula with the empirical relationship
Rewriting the empirical formula for the mean:
Compare this with the given formula:
Mean=(3⋅Median−Mode)⋅x
Equating the two expressions for the mean:
Step 3: Solve for x
Cancel
3⋅Median−Mode (non-zero):
Mean=(3⋅Median−Mode)⋅x
Step 1: Recall the empirical relationship
The empirical relationship between the mean, median, and mode is:
Mode=3⋅Median−2⋅Mean
Step 2: Compare the given formula with the empirical relationship
Rewriting the empirical formula for the mean:
Compare this with the given formula:
Mean=(3⋅Median−Mode)⋅x
Equating the two expressions for the mean:
Step 3: Solve for x
Cancel
3⋅Median−Mode (non-zero):
The mean and S.D. of 1,2,3,4,5,6 is- a)3,3
- b)7/2,√3
- c)7/2,
- d)3, 35/12
Correct answer is option 'C'. Can you explain this answer?
The mean and S.D. of 1,2,3,4,5,6 is
a)
3,3
b)
7/2,√3
c)
7/2,
d)
3, 35/12
|
|
Learners Habitat answered |
We are given the data: 1, 2, 3, 4, 5, 6
Step 1: Calculate the Mean
The formula for the mean is:
Mean = (Sum of all values) / (Number of values)
Substituting the values:
Mean = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 7/2
Step 2: Calculate the Variance
The formula for variance is:
Variance = Σ(xi - Mean)² / n
1. Subtract the mean (7/2) from each value and square the result:
- (1 - 7/2)² = (-5/2)² = 25/4
- (2 - 7/2)² = (-3/2)² = 9/4
- (3 - 7/2)² = (-1/2)² = 1/4
- (4 - 7/2)² = (1/2)² = 1/4
- (5 - 7/2)² = (3/2)² = 9/4
- (6 - 7/2)² = (5/2)² = 25/4
2. Sum the squared values:
(25/4) + (9/4) + (1/4) + (1/4) + (9/4) + (25/4) = 70/4 = 35/2
3. Divide by n (6):
Variance = (35/2) / 6 = 35/12
Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
Standard Deviation = √(35/12)
Final Answer
The mean is 7/2, and the standard deviation is √(35/12).
Correct option: (c) 7/2, √(35/12)
If the variance of the data is V, then its S.D. is- a)
- b)V2
- c)±
- d)-
Correct answer is option 'A'. Can you explain this answer?
If the variance of the data is V, then its S.D. is
a)
b)
V2
c)
±
d)
-
|
Foothill Academy answered |
If the variance of the data is denoted by V, then the standard deviation (S.D.) is the square root of the variance.
Mathematically,
S.D. = √V
The correct answer is: a) √V
Mathematically,
S.D. = √V
The correct answer is: a) √V
The range of the following set of observations 2,3,5,9,8,7,6,5,7,4,3 is- a)5.5
- b)7
- c)6
- d)11
Correct answer is option 'B'. Can you explain this answer?
The range of the following set of observations 2,3,5,9,8,7,6,5,7,4,3 is
a)
5.5
b)
7
c)
6
d)
11
|
|
Nipun Tuteja answered |
Step 1: Definition of Range
The range of a dataset is the difference between the maximum and minimum values.
Step 2: Identify Maximum and Minimum Values
From the given dataset:
2,3,5,9,8,7,6,5,7,4,3
The range of a dataset is the difference between the maximum and minimum values.
Step 2: Identify Maximum and Minimum Values
From the given dataset:
2,3,5,9,8,7,6,5,7,4,3
- Maximum value: 9
- Minimum value: 2
Step 3: Calculate the Range
Range=Maximum−Minimum=9−2=7
Range=Maximum−Minimum=9−2=7
The median of the data 13,14,16,18,20,22 is- a)16
- b)19
- c)18
- d)17
Correct answer is option 'D'. Can you explain this answer?
The median of the data 13,14,16,18,20,22 is
a)
16
b)
19
c)
18
d)
17
|
|
Nipun Tuteja answered |
To find the median, we first arrange the data in ascending order (already done here):
13,14,16,18,20,22
Step 1: Count the number of observations
The number of data points is �� = 6 n=6 (even).
Step 2: Median for even n
For an even number of observations, the median is the average of the two middle values. The two middle values here are 16 and 18.
13,14,16,18,20,22
Step 1: Count the number of observations
The number of data points is �� = 6 n=6 (even).
Step 2: Median for even n
For an even number of observations, the median is the average of the two middle values. The two middle values here are 16 and 18.
The mean of 50 observations is 36, if two observations are 30 and 42 are deleted , then the mean of the remaining observations is- a)36
- b)48
- c)38
- d)46
Correct answer is option 'A'. Can you explain this answer?
The mean of 50 observations is 36, if two observations are 30 and 42 are deleted , then the mean of the remaining observations is
a)
36
b)
48
c)
38
d)
46
|
|
Learners Habitat answered |
Step 1: Calculate the total sum of the observations
The mean of 50 observations is 36. Therefore, the total sum of the 50 observations is: Total Sum = Mean × Number of Observations
Total Sum=Mean×Number of Observations
Total Sum = 36 × 50 = 1800 Total Sum=36×50=1800
Step 2: Subtract the deleted observations
Two observations, 30 and 42, are deleted. The sum of these two observations is: Sum of deleted observations = 30 + 42 = 72
Sum of deleted observations=30+42=72
The new total sum of the remaining observations is:
New Total Sum = 1800 − 72 = 1728 New Total Sum=1800−72=1728
Step 3: Calculate the mean of the remaining observations
The number of remaining observations is:
50 − 2 = 48 50−2=48
The new mean is:
50−2=48
The new mean is:
The mean of 50 observations is 36. Therefore, the total sum of the 50 observations is: Total Sum = Mean × Number of Observations
Total Sum=Mean×Number of Observations
Total Sum = 36 × 50 = 1800 Total Sum=36×50=1800
Step 2: Subtract the deleted observations
Two observations, 30 and 42, are deleted. The sum of these two observations is: Sum of deleted observations = 30 + 42 = 72
Sum of deleted observations=30+42=72
The new total sum of the remaining observations is:
New Total Sum = 1800 − 72 = 1728 New Total Sum=1800−72=1728
Step 3: Calculate the mean of the remaining observations
The number of remaining observations is:
50 − 2 = 48 50−2=48
The new mean is:
50−2=48
The new mean is:
Chapter doubts & questions for Statistics - Mathematics for A Level 2025 is part of A Level exam preparation. The chapters have been prepared according to the A Level exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for A Level 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.
Chapter doubts & questions of Statistics - Mathematics for A Level in English & Hindi are available as part of A Level exam.
Download more important topics, notes, lectures and mock test series for A Level Exam by signing up for free.
Mathematics for A Level
124 videos|166 docs|207 tests
|
