## NCERT Exemplar Problems Class 11 Mathematics Chapter 15 Statistics

**Short Answer Type Questions**

**Q1. Find the mean deviation about the mean of the distribution:
**

Size | 20 | 21 | 22 | 23 | 24 |

Frequency | 6 | 4 | 5 | 1 | 4 |

**Q2. Find the mean deviation about the median of the following distribution:
**

Marks obtained | 10 | 11 | 12 | 14 | 15 |

Number of students | 2 | 3 | 8 | 3 | 4 |

**Q3. Calculate the mean deviation about the mean of the set of first n natural numbers when n is an odd number.**

**Sol:** Consider first natural number when n is an odd i.e., 1, 2, 3,4,… , n [odd].

**Q4. Calculate the mean deviation about the mean of the set of first n natural numbers when n is an even number.**

**Sol:** Consider first **n** natural number, when **n** is even i.e., 1, 2, 3,4..n.

**Q5. Find the standard deviation of the first n natural numbers.**

**Q6. 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 s.**

**Q8. 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.
**

**Q9. The frequency distribution:**

X | A | 2A | 3 A | 4A | 5 A | 6A |

f | 2 | 1 | 1 | 1 | 1 | 1 |

where A is a positive integer, has a variance of 160. Determine the value of A.

**Q10. For the frequency distribution:
**

X | 2 | 3 | 4 | 5 | 6 | 7 |

f | 4 | 9 | 16 | 14 | 11 | 6 |

Find the standard deviation.

**Q11. 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 | i | 2 | 3 | 4 | 5 |

Frequency | x – 2 | X | x^{2} |
(x+1)^{2} |
2x | x + 1 |

**where x is a positive integer. Determine the mean and standard deviation of the marks.
**

**Q12. 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.
**

**Q13. Mean and standard deviation of 100 items are 50 and 4, respectively. Then find the sum of all the item and the sum of the squares of the items.
**

**Q14. If for a distribution Σ (x -5)= 3,Σ (x -5) ^{2}= 43 and the total number of item is 18, find the mean and standard deviation.**

**Sol:**Given, n = 18, Σ (x – 5) = 3 and Σ (x – 5)

^{2}= 43

**Q15. Find the mean and variance of the frequency distribution given below:
**

**Long Answer Type Questions**

**Q16. 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 |

**Q17. 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 |

**Q18. 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 |

Q19. The weights of coffee in 70 jars are shown in the following table:

Weight (in grams) | Frequency |

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.
**

**Q20. Determine mean and standard deviation of first n terms of an A.P. whose first term is a and common difference is d.
**

Q21. Following are the marks obtained, out of 100, by two students Ravi and Hashinain 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?

**Q22. 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.
**

**Q23. 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.
**

**Objective Type Questions**

**Q24. 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**

**Sol:** (b) Given, observations are 3, 10, 10, 4, 7, 10 and 5.

**Q26. 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
**

**Q27. 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**

**Sol:** (c) Since, marks obtained by 9 students in Mathematics are 50,69,20,33,53, 39,40, 65 and 59.

Rewrite the given data in ascending order.

20, 33, 39,40, 50, 53, 59, 65, 69,

Here, n = 9 [odd]

**Q30. 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
**

**Q31. Let a, b, c, d, e be the observations with mean m and standard deviation V. The standard deviation of the observations a + k,b + k,c + k,d+k,e + k is
**

**Q34. Standard deviations for first 10 natural numbers is**

**(a) 5.5 (b) 3.87 (c) 2.97 (d) 2.87**

**Q35. 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**

**(a) 6.5 (b) 2.87 (c) 3.87 (d) 8.25**

**Sol:** (d) Given numbers are 1, 2, 3,4, 5, 6, 7, 8, 9 and 10

If 1 is added to each number, then observations will be 2, 3,4, 5, 6,7, 8, 9, 10 and 11.

**Q36. 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
Sol:**(a) Since, the first 10 positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.

On multiplying each number by -1, we get

-1, -2, -3, -4, -5, -6, -7, -8, -9, -10 On adding 1 in each number, we get

0, -1, -2, -3, -4, -5, -6, -7, -8, -9

**Q38. 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
**

**Q39. The standard deviation of some temperature data in °C is 5. If the data were converted into °F, the variance would be**

**(a) 81 (b) 57 (c) 36 (d) 25
**

**Fill in the Blanks
**

**Q43. The standard deviation of a data is _____ of any change in origin, but is ________ on the change of scale.**

Sol: The standard deviation of a data is independent of any change in origin but is dependent of charge of scale.

**Q44. The sum of the squares of the deviations of the values of the variable is ________ when taken about their arithmetic mean.**

**Sol:** The sum of the squares of the deviations of the values of the variable is minimum when taken about their arithmetic mean.

**Q45. The mean deviation of the data is ________ when measured from the median.**

**Sol:** The mean deviation of the data is least when measured from the median.

**Q46. The standard deviation is________ to the mean deviation taken from the arithmetic mean.**

**Sol:** The standard deviation is greater than or equal to the mean deviation taken from the arithmetic mean.