Chapter 15 – Statistics

Class 11 Mathematics · 34 questions · 0 with answers

Solved examples

example-1Short answer

Find the mean deviation about the mean of the following data: Size (x): 1 3 5 7 9 11 13 15 Frequency (f): 3 3 4 14 7 4 3 4 Solution Mean = x = ∑ fi xi = 3 + 9 + 20 + 98 + 63 + 44 + 39 + 60 = 336 = 8 ∑ fi 42 42 M.D. ( x ) = ∑ fi | xi − x | = 3(7) + 3(5) + 4(3) + 14(1) + 7(1) + 4(3) + 3(5) + 4(7) ∑ fi 42 STATISTICS 273

example-2Short answer

Find the variance and standard deviation for the following data: 57, 64, 43, 67, 49, 59, 44, 47, 61, 59 57 + 64 + 43 + 67 + 49 + 59 + 61 + 59 + 44 + 47 550 Solution Mean ( x ) = = = 55 10 10 Variance (σ ) = ∑ ( xi − x )

example-3Short answer

Show that the two formulae for the standard deviation of ungrouped data. σ= ∑( xi − x )2 and σ′ = ∑ xi2 − x 2 n n are equivalent. Solution We have ∑ ( xi − x )2 = ∑ ( xi2 − 2 x xi + x 2 ) = ∑ xi2 + ∑ − 2 x xi + ∑ x 2 = ∑xi2 − 2 x ∑ xi + ( x ) ∑ 1 = ∑xi2 − 2 x (n x ) + n x 2 = ∑ xi2 − n x 2 Dividing both sides by n and taking their square root, we get σ = σ′.

example-4Short answer

Calculate variance of the following data : Class interval Frequency 4-8 3 8 - 12 6 12 - 16 4 16 - 20 7 Mean ( x ) = ∑ fi xi = 3 × 6 + 6 × 10 + 4 × 14 + 7 × 18 = 13 ∑ fi 20 Solution Variance (σ2) = ∑ fi ( xi − x )2 = 3( −7)2 + 6( −3)2 + 4(1)2 + 7(5)2 ∑ fi 20 147 + 54 + 4 + 175 = = 19

example-5Short answer

Calculate mean, variation and standard deviation of the following frequency distribution: Classes Frequency 1 - 10 11 10 - 20 29 20 - 30 18

example-6Long answer

Life of bulbs produced by two factories A and B are given below: Length of life Factory A Factory B (in hours) (Number of bulbs) (Number of bulbs) 550 - 650 10 8 650 - 750 22 60 750 - 850 52 24 850 - 950 20 16 950 - 1050 16 12 120 120 The bulbs of which factory are more consistent from the point of view of length of life? Solution Here h = 100, let A (assumed mean) = 800. xi – A Length of life Mid values(xi) yi = Factory A Factory B (in hour) fi fi yi f i yi2 fi fi yi f i yi2 550 - 650 600 –2 10 –20 40 8 –16 32 650 - 750 700 –1 22 –22 22 60 – 60 60 750 - 850 800 0 52 0 0 24 0 0 850 - 950 900 1 20 20 20 16 16 16 950 - 1050 1000 2 16 32 64 12 24 48 120 10 146 120 –36 156 For factory A Mean ( x ) = 800 + × 100 = 816.67 hours S.D. = 120 (146) − 100 = 109.98 STATISTICS 277 S.D. 109.98 Therefore, Coefficient of variation (C.V.) = × 100 = × 100 = 13.47 x 816.67 For factory B −36 Mean = 800 + 100 = 770 S.D. = 120 (156) − ( −36)2 = 110 S.D. 110 Therefore, Coefficient of variation = × 100 = × 100 = 14.29 Mean 770 Since C.V. of factory B > C.V. of factory A ⇒ Factory B has more variability which means bulbs of factory A are more consistent.

example-7Multiple choice

The mean deviation of the data 2, 9, 9, 3, 6, 9, 4 from the mean is

  • (A)2.23
  • (B)2.57
  • (C)3.23
  • (D)3.57 Solution (B) is the correct answer ∑ | xi − x | = 4 + 3 + 3 + 3 + 0 + 3 + 2 = 2.57 M.D. ( x ) = n 7
example-8Multiple choice

Variance of the data 2, 4, 5, 6, 8, 17 is 23.33. Then variance of 4, 8, 10, 12, 16, 34 will be

  • (A)23.23
  • (B)25.33
  • (C)46.66
  • (D)48.66 Solution (C) is the correct answer. When each observation is multiplied by 2, then variance is also multiplied by 2.
example-9Multiple choice

A set of n values x1, x2, ..., xn has standard deviation 6. The standard deviation of n values x1 + k, x2 + k, ..., xn + k will be

  • (A)σ
  • (B)σ + k
  • (C)σ – k
  • (D)k σ Solution (A) is correct answer. If each observation is increased by a constant k, then standard deviation is unchanged.

Questions

Q21Short answer

+ 15 + 12 + 14 + 7 + 12 + 15 + 28 62 = = = 2.95 42 21

Q22Short answer

+ 92 + 122 + 122 + 62 + 42 + 62 + 42 + 112 + 82 = = = 66.2 Standard deviation (σ) = σ 2 = 66.2 = 8.13

Q30Long answer

- 40 4 40 - 50 5 50 - 60 3 STATISTICS 275 Solution Let A, the assumed mean, be 25.5. Here h = 10 xi – 25.5 Classes xi yi = fi fi y i f i yi2 1 - 10 5.5 –2 11 –22 44 10 - 20 15.5 –1 29 –29 29 20 - 30 25.5 0 18 0 0 30 - 40 35.5 1 4 4 4 40 - 50 45.5 2 5 10 20 50 - 60 55.5 3 3 9 27 70 –28 124 ∑ fi yi = −28 = − 0.4 x′ = ∑ fi 70 Mean = x = 25.5 + (–10) (0.4) = 21.5 h 2 Variance 2 (σ ) =  N∑ fi yi2 − (∑ f y )  N 10 × 10 = [70(124) – (–28)2] 70 × 70 70 (124) 28 × 28 1240 = − = − 16 = 161 7×7 7×7 7 S.D. (σ) = 161 = 12.7

Q1Short answer

Find the mean deviation about the mean of the distribution: Size 20 21 22 23 24 Frequency 6 4 5 1 4

Q2Short answer

Find the mean deviation about the median of the following distribution: Marks obtained 10 11 12 14 15 No. of students 2 3 8 3 4

Q3Short answer

Calculate the mean deviation about the mean of the set of first n natural numbers when n is an odd number.

Q4Short answer

Calculate the mean deviation about the mean of the set of first n natural numbers when n is an even number.

Q5Short answer

Find the standard deviation of the first n natural numbers.

Q6Short answer

The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results: Number of observations = 25, mean = 18.2 seconds, standard deviation = 3.25 seconds. Further, another set of 15 observations x1, x2, ..., x15, also in seconds, is now 15 15 available and we have ∑ xi = 279 and ∑ xi = 5524 . Calculate the standard i =1 i =1 derivation based on all 40 observations.

Q7Short answer

The mean and standard deviation of a set of n1 observations are x1 and s1, respectively while the mean and standard deviation of another set of n2 observations are x2 and s2, respectively. Show that the standard deviation of the combined set of (n1 + n2) observations is given by n1 ( s1 ) 2 + n2 ( s2 ) 2 n1n2 ( x1 − x2 ) 2 S.D. = + n1 + n2 (n1 + n2 ) 2 STATISTICS 279

Q8Short answer

Two sets each of 20 observations, have the same standard derivation 5. The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.

Q9Short answer

The frequency distribution: x A 2A 3A 4A 5A 6A f 2 1 1 1 1 1 where A is a positive integer, has a variance of 160. Determine the value of A.

Q10Short answer

For the frequency distribution: x 2 3 4 5 6 7 f 4 9 16 14 11 6 Find the standard distribution.

Q11Short answer

There are 60 students in a class. The following is the frequency distribution of the marks obtained by the students in a test: Marks 0 1 2 3 4 5 Frequency x–2 x x2 (x + 1)2 2x x+1 where x is a positive integer. Determine the mean and standard deviation of the marks.

Q12Short answer

The mean life of a sample of 60 bulbs was 650 hours and the standard deviation was 8 hours. A second sample of 80 bulbs has a mean life of 660 hours and standard deviation 7 hours. Find the overall standard deviation.

Q13Short answer

Mean and standard deviation of 100 items are 50 and 4, respectively. Find the sum of all the item and the sum of the squares of the items.

Q14Short answer

If for a distribution ∑( x − 5) = 3, ∑ ( x − 5)2 = 43 and the total number of item is 18, find the mean and standard deviation.

Q15Short answer

Find the mean and variance of the frequency distribution given below: x 1≤x<3 3≤x<5 5≤x<7 7 ≤ x < 10 f 6 4 5 1

Q16Long answer

Calculate the mean deviation about the mean for the following frequency distribution: Class interval 0-4 4-8 8 - 12 12 - 16 16 - 20 Frequency 4 6 8 5 2

Q17Long answer

Calculate the mean deviation from the median of the following data: Class interval 0-6 6 - 12 12 - 18 18 - 24 24 - 30 Frequency 4 5 3 6 2

Q18Long answer

Determine the mean and standard deviation for the following distribution: Marks 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Frequency 1 6 6 8 8 2 2 3 0 2 1 0 0 0 1

Q19Long answer

The weights of coffee in 70 jars is shown in the following table: Weight Frequency (in grams) 200 - 201 13 201 - 202 27 202 - 203 18 203 - 204 10 204 - 205 1 205 - 206 1 Determine variance and standard deviation of the above distribution.

Q20Long answer

Determine mean and standard deviation of first n terms of an A.P. whose first term is a and common difference is d. STATISTICS 281

Q21Long answer

Following are the marks obtained, out of 100, by two students Ravi and Hashina in 10 tests. Ravi 25 50 45 30 70 42 36 48 35 60 Hashina 10 70 50 20 95 55 42 60 48 80 Who is more intelligent and who is more consistent?

Q22Long answer

Mean and standard deviation of 100 observations were found to be 40 and 10, respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.

Q23Long answer

While calculating the mean and variance of 10 readings, a student wrongly used the reading 52 for the correct reading 25. He obtained the mean and variance as 45 and 16 respectively. Find the correct mean and the variance.

Q39Multiple choice

(M.C.Q.). 24. The mean deviation of the data 3, 10, 10, 4, 7, 10, 5 from the mean is

  • (A)2
  • (B)2.57
  • (C)3
  • (D)3.75 25. Mean deviation for n observations x1, x2, ..., xn from their mean x is given by 1 n (A) ∑ ( xi − x ) (B) ∑ xi − x n i =1 i =1 2 1 n 2 (C) ∑ ( xi − x ) (D) ∑ ( xi − x ) i =1 n i =1 26. When tested, the lives (in hours) of 5 bulbs were noted as follows: 1357, 1090, 1666, 1494, 1623 The mean deviations (in hours) from their mean is (A) 178 (B) 179 (C) 220 (D) 356 27. Following are the marks obtained by 9 students in a mathematics test: 50, 69, 20, 33, 53, 39, 40, 65, 59 The mean deviation from the median is: (A) 9 (B) 10.5 (C) 12.67 (D) 14.76 28. The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is 52 52 (A) (B) (C) 6 (D) 6 7 7 29. Let x1, x2, ..., xn be n observations and x be their arithmetic mean. The formula for the standard deviation is given by 2 ∑ ( xi − x )2 (A) ∑ ( xi − x ) (B) (C) ∑( xi − x ) (D) ∑ xi2 + x 2 n n 30. The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations is (A) 50000 (B) 250000 (C) 252500 (D) 255000 31. Let a, b, c, d, e be the observations with mean m and standard deviation s. The standard deviation of the observations a + k, b + k, c + k, d + k, e + k is (A) s (B) ks (C) s+k (D) 32. Let x1, x2, x3, x4, x5 be the observations with mean m and standard deviation s. The standard deviation of the observations kx1, kx2, kx3, kx4, kx5 is (A) k + s (B) (C) ks (D) s 33. Let x1, x2, ... xn be n observations. Let wi = lxi + k for i = 1, 2, ...n, where l and k are constants. If the mean of xi’s is 48 and their standard deviation is 12, the mean of wi’s is 55 and standard deviation of wi’s is 15, the values of l and k should be (A) l = 1.25, k = – 5 (B) l = – 1.25, k = 5 (C) l = 2.5, k = – 5 (D) l = 2.5, k = 5 34. Standard deviations for first 10 natural numbers is (A) 5.5 (B) 3.87 (C) 2.97 (D) 2.87 35. Consider the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If 1 is added to each number, the variance of the numbers so obtained is STATISTICS 283 (A) 6.5 (B) 2.87 (C) 3.87 (D) 8.25 36. Consider the first 10 positive integers. If we multiply each number by –1 and then add 1 to each number, the variance of the numbers so obtained is (A) 8.25 (B) 6.5 (C) 3.87 (D) 2.87 37. The following information relates to a sample of size 60: ∑ x 2 = 18000, ∑ x = 960 The variance is (A) 6.63 (B) 16 (C) 22 (D) 44 38. Coefficient of variation of two distributions are 50 and 60, and their arithmetic means are 30 and 25 respectively. Difference of their standard deviation is (A) 0 (B) 1 (C) 1.5 (D) 2.5 39. The standard deviation of some temperature data in °C is 5. If the data were converted into ºF, the variance would be (A) 81 (B) 57 (C) 36 (D) 25 Fill in the blanks in Exercises from 40 to 46. ...
Q40Multiple choice

Coefficient of variation = ×100 Mean

Q41Fill in the blanks

If x is the mean of n values of x, then ∑ ( xi − x ) is always equal to _______. i =1 If a has any value other than x , then ∑ ( xi − x ) is _________ than i =1 ∑ ( xi − a)2

Q42Fill in the blanks

If the variance of a data is 121, then the standard deviation of the data is _______.

Q43Fill in the blanks

The standard deviation of a data is ___________ of any change in orgin, but is _____ on the change of scale.

Q44Fill in the blanks

The sum of the squares of the deviations of the values of the variable is _______ when taken about their arithmetic mean.

Q45Fill in the blanks

The mean deviation of the data is _______ when measured from the median.

Q46Fill in the blanks

The standard deviation is _______ to the mean deviation taken from the arithmetic mean.