Statistics · Class 10 Mathematics: 174 MCQs with answers

Q1. The mean of grouped data must be estimated because

Q2. The class mark of a class is

Q3. The class mark of 20–30 is

Q4. In the direct method, the mean is

Q5. In the assumed-mean method, $d_i$ is

Q6. The three methods of this topic give

Q7. For classes 0–10, 10–20, 20–30, 30–40, 40–50 the class marks are

Q8. With frequencies 2, 3, 9, 10, 6 on those classes, $\sum f_i$ is

Q9. $\sum f_i$ is the same as

Q10. Which column belongs in a direct-method table?

Q11. A class 45–60 has width

Q12. Define the class mark and find it for the classes 0–10 and 45–60.

Q13. The mode of a data set is

Q14. The modal class of a grouped distribution is

Q15. The formula for the mode of grouped data is

Q16. In that formula $l$ is

Q17. In that formula $f_0$ is

Q18. In that formula $f_2$ is

Q19. For frequencies 2, 3, 9, 10, 6 on classes of width 10 from 0, the modal class is

Q20. For frequencies 5, 12, 8, 3 on classes of width 10 from 0, the modal class is

Q21. A shoe shop deciding which size to stock most heavily should use

Q22. In a histogram, the modal class is

Q23. The mode of the list 3, 7, 7, 9, 7, 4 is

Q24. The answer to a mode question should be given as

Q25. Define the mode and the modal class, and explain the difference between them.

Q26. State the formula for the mode of grouped data and say what each symbol means.

Q27. The median of a data set is

Q28. The cumulative frequency of a class is

Q29. The last entry of a cumulative frequency column should equal

Q30. The median formula for grouped data is

Q31. For frequencies 2, 3, 9, 10, 6, the cumulative frequencies are

Q32. With $n = 30$, the value of $\frac{n}{2}$ is

Q33. The first step in any median question should be to

Q34. State the median formula for grouped data and explain each symbol.

Q35. The class mark of 45–60 is

Q36. The divisor in every mean formula of this topic is

Q37. Dividing $\sum f_i x_i$ by the number of CLASSES instead of $\sum f_i$ gives

Q38. The assumed-mean formula is

Q39. A middle class mark is usually chosen for $a$ because

Q40. In the step-deviation method, $u_i$ is

Q41. The step-deviation formula is

Q42. Step deviation can be used only when

Q43. For that distribution, $\sum f_i x_i$ is

Q44. Assertion (A): The assumed-mean and direct methods always give the same answer. Reason (R): The $a$ subtracted in each deviation is cancelled by dividing at the end.

Q45. For that distribution the mean is

Q46. Taking $a = 25$ for that distribution, the deviations $d_i$ are

Q47. The corresponding $u_i$, with $h = 10$, are

Q48. Which method best suits small class marks and few classes?

Q49. If a question says 'by the step-deviation method', showing only the direct method

Q50. If $n$ is given and one frequency is unknown, you can find it because

Q51. The mean of grouped data is best described as

Q52. Which columns belong in a step-deviation table?

Q53. Explain why the mean of grouped data is only an estimate, and what assumption is made in computing it.

Q54. Find the mean of the distribution 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 2, 3, 9, 10, 6, by the direct method.

Q55. Repeat that calculation by the assumed-mean method, taking $a = 25$.

Q56. State when the step-deviation method may be used and what $u_i$ measures.

Q57. Describe the two commonest errors in this topic and how to avoid each.

Q58. Explain why choosing a poor value of $a$ costs nothing but arithmetic.

Q59. The modal class and the mode are

Q60. The denominator of the fraction is

Q61. For that distribution, $f_0$ is

Q62. For that distribution, $f_2$ is

Q63. A computed mode that falls OUTSIDE the modal class means

Q64. If the modal class is the FIRST class, then

Q65. If the modal class is the LAST class, then

Q66. Assertion (A): The modal class may be quoted as the mode. Reason (R): The modal class is the class with the largest frequency.

Q67. A distribution with two classes tied for the largest frequency is called

Q68. Which average can be used on data with no numerical order at all?

Q69. Can a data set have more than one mode?

Q70. The commonest error in the mode formula is

Q71. How many numbers from the table does the mode formula need?

Q72. For classes 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 2, 3, 9, 10, 6, find the mode.

Q73. What is a bimodal distribution, and why does the syllabus avoid one?

Q74. State the five-step method for finding a mode, and say where marks are usually lost.

Q75. Explain why the mode is the only average usable on non-numerical data.

Q76. The median is preferred to the mean for incomes because

Q77. The median class is

Q78. For grouped data, the position used is

Q79. In that formula, $cf$ is

Q80. The letter $f$ in the median formula stands for

Q81. For that distribution, the median class is

Q82. If the median class is the FIRST class, then

Q83. The empirical relationship between the three averages is

Q84. Which average uses every observation?

Q85. Which average is least disturbed by a few extreme values?

Q86. Is the cumulative frequency GRAPH part of the current syllabus?

Q87. Which average would a government use to report typical income?

Q88. A median that comes out outside its class means

Q89. Assertion (A): The median can be found without computing the mean. Reason (R): The mean and the median of a grouped distribution are always equal anyway.

Q90. Define the median and explain why it is preferred to the mean for reporting incomes.

Q91. Explain what a cumulative frequency column is and how the median class is identified from it.

Q92. State the empirical relationship between the three averages, and explain in what sense it holds.

Q93. A distribution has mean 30 and median 31. Estimate its mode, and rearrange the relation to give the mean in terms of the other two.

Q94. The value of $a$ in the assumed-mean method may be

Q95. Forgetting the $h$ in the step-deviation formula makes the answer

Q96. What does $u_i$ measure?

Q97. Assertion (A): The step-deviation method may be used on classes of unequal width. Reason (R): The method divides every deviation by a single class width $h$.

Q98. Multiplying each $u_i$ by its frequency and adding gives, for that distribution,

Q99. Hence the step-deviation calculation reads

Q100. Taking $d_i = a - x_i$ instead of $x_i - a$

Q101. Which method best suits classes of unequal width?

Q102. Classes of widths 10, 10, 20, 10 mean that

Q103. If every observation were increased by 5, the mean would

Q104. Taking $a$ equal to the actual mean would make $\sum f_i d_i$

Q105. Step deviation is best described as

Q106. Assertion (A): The divisor in the direct method is the number of classes. Reason (R): Each class contributes one term to the sum $\sum f_i x_i$.

Q107. Assertion (A): Choosing a poor value of $a$ makes the computed mean wrong. Reason (R): A poor choice of $a$ makes the deviations larger and the arithmetic heavier.

Q108. For classes 10–20, 20–30, 30–40 with frequencies 5, 8, 7, the mean is

Q109. Repeat it once more by the step-deviation method, and say why the arithmetic is lighter.

Q110. Prove that the assumed-mean method gives the same answer as the direct method, whatever $a$ is chosen.

Q111. A distribution has classes of widths 10, 10, 20 and 10. Which method should be used and why?

Q112. (a) Explain why grouped data need class marks, and define the class mark. (b) Find the mean of 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 2, 3, 9, 10, 6 by the DIRECT method, setting out a table. (c) Verify your answer by the step-deviation method with $a = 25$.

Q113. (a) State all three methods for the mean of grouped data. (b) Prove algebraically that the assumed-mean method gives the direct-method answer for every choice of $a$. (c) Explain what $u_i$ measures and why the $h$ in the step-deviation formula cannot be omitted.

Q114. (a) Give a rule for choosing between the three methods. (b) Find the mean of 10–20, 20–30, 30–40 with frequencies 5, 8, 7, showing your table. (c) Explain what would change if the second class were 20–40 with the same frequency.

Q115. (a) Explain the role of the total frequency in these formulas, and the error that arises from dividing by the number of classes. (b) A distribution of 30 observations has frequencies 2, 3, 9, $f$, 6 over five classes. Find $f$. (c) Explain why the mean computed here should be described as an estimate.

Q116. For that distribution, the denominator $2f_1 - f_0 - f_2$ is

Q117. Hence the mode of that distribution is

Q118. That mode lies near the LOWER end of its class because

Q119. Why does the mode always lie inside the modal class?

Q120. If both neighbours of the modal class are equally tall, the mode sits

Q121. For classes 0–10, 10–20, 20–30 with frequencies 12, 8, 5, the mode is

Q122. Assertion (A): The mode of grouped data always lies within the modal class. Reason (R): The fraction in the mode formula lies strictly between 0 and 1 whenever the modal frequency is the largest.

Q123. The syllabus avoids bimodal distributions because

Q124. Classes written as 0–9, 10–19, 20–29 must be adjusted because

Q125. After adjustment, the class 10–19 becomes

Q126. Why does that adjustment matter more for the mode than for the mean?

Q127. For that distribution, the mode is

Q128. Which of these is NOT needed to find the mode of grouped data?

Q129. If $f_1 - f_0$ is large, the mode lies

Q130. If $f_1 - f_0$ is small, the mode lies

Q131. The mode formula is a form of

Q132. For frequencies 4, 9, 9, 2, the mode formula

Q133. Assertion (A): The mode of grouped data cannot be found unless the total number of observations is known. Reason (R): The formula uses only the modal class's boundary and width and three frequencies.

Q134. Assertion (A): $l$ in the mode formula is the class mark of the modal class. Reason (R): The formula adds a fraction of the class width to $l$.

Q135. Explain why the mode of grouped data always lies inside the modal class.

Q136. Explain how the neighbouring frequencies decide where in the modal class the mode falls.

Q137. For classes 0–10, 10–20, 20–30 with frequencies 12, 8, 5, find the mode.

Q138. Explain what must be done with classes written 0–9, 10–19, 20–29 before the mode formula is used.

Q139. (a) Define the modal class and state the mode formula, explaining every symbol. (b) For classes 0–10 up to 40–50 with frequencies 2, 3, 9, 10, 6, find the mode. (c) Explain where in the modal class your answer sits, and why.

Q140. (a) Prove that the mode found by the formula always lies inside the modal class. (b) Show what happens when both neighbouring frequencies are equal. (c) For classes 0–10, 10–20, 20–30 with frequencies 12, 8, 5, find the mode, explaining the special feature of this case.

Q141. (a) List the errors most often made in this topic. (b) Explain what must be done with classes written 0–9, 10–19, 20–29, and why the mean is less affected. (c) For those classes with frequencies 4, 11, 6, find the mode.

Q142. (a) Explain why the mode is the only average that works on non-numerical data. (b) Explain what a bimodal distribution is and why the formula fails on one. (c) For classes of width 10 from 0 with frequencies 5, 12, 8, 3, find the mode and check it lies inside the modal class.

Q143. Using the median class's OWN cumulative frequency as $cf$ makes the fraction

Q144. For that distribution, $cf$ is

Q145. For that distribution, $f$ is

Q146. Hence the median of that distribution is

Q147. The median lies just inside the LOWER end of its class there because

Q148. Why does the median always lie inside the median class?

Q149. Assertion (A): The median of grouped data always lies inside the median class. Reason (R): The median class is chosen so that $cf < \frac{n}{2} \le cf + f$.

Q150. Assertion (A): The empirical relationship between the three averages is an exact identity. Reason (R): It was obtained by observing many real distributions rather than by proof.

Q151. A distribution has mean 30 and median 31. The empirical estimate of its mode is

Q152. Rearranged for the median, the empirical relation reads

Q153. Rearranged for the mean, it reads

Q154. The empirical relation is exact when the distribution is

Q155. For a distribution with $n = 40$ and cumulative frequencies 6, 15, 27, 40, the median class is the

Q156. For that distribution, $cf$ would be

Q157. For that distribution, $f$ would be

Q158. The median is sometimes called

Q159. Multiplying the largest observation by ten would

Q160. Which average would a total wage bill, shared equally, require?

Q161. Which quantity appears in the MEDIAN formula but not in the mode formula?

Q162. In a perfectly symmetric distribution, the mean, median and mode are

Q163. For frequencies 5, 8, 12, 10, 5 the median class, with $n = 40$, is

Q164. Assertion (A): $f$ in the median formula is the cumulative frequency of the median class. Reason (R): The formula divides by the number of observations inside the median class.

Q165. The purpose of the fraction in the median formula is to

Q166. For classes 0–10 up to 40–50 with frequencies 2, 3, 9, 10, 6, find the median.

Q167. Explain why the median always lies inside the median class.

Q168. Explain the median formula as an instruction rather than a rule to be memorised.

Q169. Explain when the empirical relationship becomes exact, and what makes it rougher.

Q170. Describe the two errors most often made in the median formula and how to avoid them.

Q171. (a) Explain how the median class is found, and state the median formula with every symbol explained. (b) For classes 0–10 up to 40–50 with frequencies 2, 3, 9, 10, 6, build the cumulative frequency column and find the median. (c) Check that your answer lies inside the median class and explain why it must.

Q172. (a) State the empirical relationship and explain what 'empirical' means here. (b) For the distribution with mean 30, median 31 and true mode 32, test the relation and comment. (c) Give the two rearranged forms of the relation and say when each is used.

Q173. (a) Compare the mean, median and mode as descriptions of a data set. (b) Explain, with an example, why incomes are reported as medians. (c) For frequencies 5, 8, 12, 10, 5 over classes of width 10 from 0, find the median.

Q174. (a) Read the median formula as an instruction and explain what each part is doing. (b) Describe the two commonest errors in applying it. (c) Explain what happens to the formula when the median class is the first class, and why.