Q1: In a dataset of 50 observations arranged in ascending order, what is the position of the 40th percentile? (a) 16th observation (b) 20th observation (c) 21st observation (d) 25th observation
Solution:
Ans: (b) Explanation: The position of the percentile is calculated using the formula \(L = \frac{P}{100} \times n\), where \(P = 40\) and \(n = 50\). Thus, \(L = \frac{40}{100} \times 50 = 20\). The 40th percentile is at the 20th observation.
Q2: The 75th percentile is also known as the: (a) First quartile (b) Second quartile (c) Third quartile (d) Median
Solution:
Ans: (c) Explanation: The 75th percentile divides the dataset so that 75% of the data falls below it. This is also called the third quartile (Q3). The first quartile is the 25th percentile, the second quartile is the 50th percentile (median), making option (c) correct.
Q3: If a student scores at the 85th percentile on a standardized test, this means: (a) The student answered 85% of the questions correctly (b) The student scored better than 85% of the test takers (c) 85% of students scored better than this student (d) The student's score is 85 out of 100
Solution:
Ans: (b) Explanation: A score at the 85th percentile means that the student performed better than 85% of all test takers. It does not refer to the percentage of correct answers or the actual numerical score.
Q4: In a grouped frequency distribution, which formula is used to calculate the kth percentile? (a) \(P_k = L + \left(\frac{\frac{kn}{100} - cf}{f}\right) \times h\) (b) \(P_k = U + \left(\frac{\frac{kn}{100} - cf}{f}\right) \times h\) (c) \(P_k = L - \left(\frac{\frac{kn}{100} - cf}{f}\right) \times h\) (d) \(P_k = L + \left(\frac{cf - \frac{kn}{100}}{f}\right) \times h\)
Solution:
Ans: (a) Explanation: For a grouped frequency distribution, the kth percentile is calculated using \(P_k = L + \left(\frac{\frac{kn}{100} - cf}{f}\right) \times h\), where \(L\) is the lower boundary of the percentile class, \(cf\) is the cumulative frequency before the percentile class, \(f\) is the frequency of the percentile class, and \(h\) is the class width.
Q5: What percentage of data values lie below the 30th percentile? (a) 70% (b) 30% (c) 50% (d) 100%
Solution:
Ans: (b) Explanation: By definition, the 30th percentile is the value below which 30% of the data falls. Therefore, 30% of data values lie below the 30th percentile.
Q6: In a dataset with 100 values, if the 60th percentile is 45, how many data values are less than or equal to 45? (a) 40 values (b) 45 values (c) 60 values (d) 100 values
Solution:
Ans: (c) Explanation: The 60th percentile indicates that 60% of the data values are less than or equal to 45. With 100 total values, \(0.60 \times 100 = 60\) values are less than or equal to 45.
Q7: The interquartile range (IQR) is calculated as: (a) \(Q_3 - Q_1\) (b) \(Q_3 + Q_1\) (c) \(Q_2 - Q_1\) (d) \(Q_3 - Q_2\)
Solution:
Ans: (a) Explanation: The interquartile range (IQR) measures the spread of the middle 50% of the data and is calculated as \(IQR = Q_3 - Q_1\), where \(Q_3\) is the 75th percentile and \(Q_1\) is the 25th percentile.
Q8: If the 50th percentile of a dataset is 72, what other statistical measure is also equal to 72? (a) Mode (b) Mean (c) Median (d) Range
Solution:
Ans: (c) Explanation: The 50th percentile is the value that divides the dataset into two equal halves, which is the definition of the median. Therefore, the 50th percentile and median are identical.
Section B: Fill in the Blanks
Q9:The value below which a specified percentage of observations in a dataset falls is called a __________.
Solution:
Ans: percentile Explanation: A percentile is a measure that indicates the value below which a given percentage of observations in a group of observations falls.
Q10:The 25th percentile is also called the __________ quartile.
Solution:
Ans: first (or lower) Explanation: The 25th percentile divides the lowest 25% of data from the rest and is known as the first quartile (Q1) or lower quartile.
Q11:In the percentile formula for grouped data, the symbol \(cf\) represents __________.
Solution:
Ans: cumulative frequency Explanation: In the percentile formula for grouped data, \(cf\) stands for the cumulative frequency of the class preceding the percentile class.
Q12:The difference between the 75th percentile and the 25th percentile is called the __________.
Solution:
Ans: interquartile range (or IQR) Explanation: The interquartile range (IQR) is calculated as \(Q_3 - Q_1\) and measures the spread of the middle 50% of the data.
Q13:To find the position of the kth percentile in ungrouped data with n observations, we use the formula \(L = \) __________.
Solution:
Ans: \(\frac{k}{100} \times n\) (or \(\frac{kn}{100}\)) Explanation: The position of the kth percentile in ungrouped data is found using the formula \(L = \frac{k}{100} \times n\), where \(k\) is the percentile rank and \(n\) is the total number of observations.
Q14:Percentiles divide a dataset into __________ equal parts.
Solution:
Ans: 100 Explanation:Percentiles divide a dataset into 100 equal parts, where each percentile represents 1% of the data.
Section C: Word Problems
Q15:The ages of 20 students in a class are as follows: 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19. Calculate the 60th percentile.
Solution:
Ans: Step 1: Calculate the position: \(L = \frac{60}{100} \times 20 = 12\) Step 2: Since the position is exactly 12, the 60th percentile is the 12th value in the ordered dataset. Step 3: The 12th value in the dataset is 17. Final Answer: 17 years
Q16:In a mathematics test taken by 80 students, a student scored 78 marks, which corresponds to the 70th percentile. How many students scored less than or equal to 78 marks?
Solution:
Ans: Step 1: The 70th percentile means 70% of students scored at or below this mark. Step 2: Calculate: \(0.70 \times 80 = 56\) Final Answer: 56 students
Q17:The following frequency distribution shows the weights (in kg) of 50 students. Calculate the 40th percentile.
Weight (kg)
Frequency
40-45
5
45-50
12
50-55
18
55-60
10
60-65
5
Solution:
Ans: Step 1: Total observations \(n = 50\). Position of 40th percentile: \(\frac{40 \times 50}{100} = 20\) Step 2: Create cumulative frequency table: 40-45: cf = 5 45-50: cf = 17 50-55: cf = 35 (percentile class) Step 3: For the percentile class 50-55: \(L = 50\), \(cf = 17\), \(f = 18\), \(h = 5\) Step 4: Apply formula: \(P_{40} = 50 + \left(\frac{20 - 17}{18}\right) \times 5 = 50 + \frac{3 \times 5}{18} = 50 + 0.833 = 50.83\) Final Answer: 50.83 kg
Q18:The heights (in cm) of 15 basketball players are: 178, 180, 182, 183, 185, 186, 188, 190, 191, 192, 194, 195, 196, 198, 200. Find the first quartile (Q1) and third quartile (Q3), then calculate the interquartile range.
Solution:
Ans: Step 1: Position of Q1 (25th percentile): \(L = \frac{25}{100} \times 15 = 3.75\), round up to 4th position. Q1 = 183 cm Step 2: Position of Q3 (75th percentile): \(L = \frac{75}{100} \times 15 = 11.25\), round up to 12th position. Q3 = 195 cm Step 3: IQR = Q3 - Q1 = 195 - 183 = 12 cm Final Answer: Q1 = 183 cm, Q3 = 195 cm, IQR = 12 cm
Q19:A survey recorded the daily screen time (in hours) of 25 teenagers: 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10. A teenager who spent 6 hours is at what percentile?
Solution:
Ans: Step 1: Count values less than 6: there are 14 values less than 6. Step 2: Count values equal to 6: there are 3 values equal to 6. Step 3: Take the average position for values equal to 6: values at positions 15, 16, 17. Middle position = 16. Step 4: Calculate percentile: \(\frac{16}{25} \times 100 = 64\%\) Final Answer: 64th percentile
Q20:The following data represents the scores of 30 students on a quiz: 12, 15, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66. Find the 90th percentile score.
Solution:
Ans: Step 1: Calculate position: \(L = \frac{90}{100} \times 30 = 27\) Step 2: Since the position is exactly 27, the 90th percentile is the 27th value in the ordered dataset. Step 3: The 27th value is 60. Final Answer: 60 points
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