Unit 2 — Advanced Measurements in Statistics · Statistics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 92 questions on this lesson, each with an explanation.
In the box plot of a set of data, the percentage of the data that lies between the lower quartile and the upper quartile = ……
Correct answer: (D) 50%
Explanation: Step 1: Q1 is preceded by 25% of the data and Q3 is preceded by 75% of the data. Step 2: Between them: 75% − 25% = 50% ∴ the box contains 50% of the data.
A box plot of the marks of a group of students shows: minimum value 15, lower quartile 32, median 40, upper quartile 49, maximum value 58. The semi-interquartile range = ……
Correct answer: (A) 8.5
Explanation: Step 1: Q1 = 32 and Q3 = 49 (the two ends of the box). Step 2: Q = (Q3 − Q1)/2 = (49 − 32)/2 = 17/2 ∴ Q = 8.5 (17 is the interquartile range, the length of the box; 21.5 = (58 − 15)/2 uses the maximum and minimum values; 4.5 = (49 − 40)/2 uses the median instead of Q1.)
The semi-interquartile range is better to use as a measure of dispersion when ……
Correct answer: (D) there are outlier values in the data
Explanation: Step 1: Advantages of the semi-interquartile range: it is better to use as a dispersion measure when there are outlier values, and it is easy and simple to calculate. Step 2: It depends only on Q1 and Q3, so the smallest and largest values do not affect it. ∴ it is preferred when the data contain outliers. (Taking all the values into account is a trap: not doing so is its disadvantage.)
In a stem-and-leaf display (Key 2|3 = 23): stem 2: 3, 6, 8 — stem 3: 0, 2, 5, 5, 9 — stem 4: 1, 4, 6, 8 — stem 5: 2, 7, 9. It was found that the value 23 was recorded by mistake, so it was deleted. The semi-interquartile range of the remaining data = ……
Correct answer: (C) 8.75
Explanation: Step 1: After deleting 23, n = 14 and n + 1 = 15 is not divisible by 4. Step 2: Order of Q1 = 15/4 = 3.75: Q1 = 30 + (32 − 30) × 0.75 = 31.5 Step 3: Order of Q3 = 45/4 = 11.25: Q3 = 48 + (52 − 48) × 0.25 = 49 ∴ Q = (49 − 31.5)/2 = 8.75 (9 is the value before deleting; 8 takes the 4th and 11th values without using the rule.)
Two classes have almost the same median. The semi-interquartile range of the marks is 4 for the first class and 9 for the second class. Which statement is true?
Correct answer: (B) The marks of the first class are more homogeneous than those of the second
Explanation: Step 1: The semi-interquartile range is a measure of dispersion: the smaller it is, the less dispersed (more homogeneous) the data. Step 2: 4 < 9 ∴ the marks of the first class are more homogeneous. (Equal medians do not mean equal dispersion, and the semi-interquartile range tells nothing about the mean.)
In an exam, 25% of the students scored less than 34 and 25% of the students scored more than 58. The semi-interquartile range of these marks = ……
Correct answer: (B) 12
Explanation: Step 1: The value preceded by 25% of the data is Q1 = 34; the value followed by 25% of the data (preceded by 75%) is Q3 = 58 Step 2: Q = (Q3 − Q1)/2 = (58 − 34)/2 ∴ Q = 12 (24 is the interquartile range; 46 = (58 + 34)/2 adds instead of subtracting.)
The ascending cumulative frequency table of the areas of 40 farms in hectares: less than 15: 0, less than 20: 4, less than 25: 12, less than 30: 28, less than 35: 36, less than 40: 40. Given that a hectare = 10000 square meters, the semi-interquartile range of the areas in square meters = ……
Correct answer: (D) 37500
Explanation: Step 1: Order of Q1 = 40/4 = 10, in the interval 20– (cumulative frequency before it 4, frequency 12 − 4 = 8): Q1 = 20 + (10 − 4)/8 × 5 = 23.75 Step 2: Order of Q3 = 3 × 40/4 = 30, in the interval 30– (cumulative frequency before it 28, frequency 8): Q3 = 30 + (30 − 28)/8 × 5 = 31.25 Step 3: Q = (31.25 − 23.75)/2 = 3.75 hectares ∴ Q = 3.75 × 10000 = 37500 square meters (3.75 is the answer in hectares before converting; 75000 is the interquartile range in square meters; 3750 multiplies by 1000 instead of 10000.)
The working hours of 36 workers: intervals 20–, 25–, 30–, 35–, 40– (each of width 5) with frequencies 2, 3, 10, 11, 10 respectively. 4 new workers join the factory, and the working hours of all of them lie in the interval 35–. The semi-interquartile range of the working hours after they join = ……
Correct answer: (C) 3.75
Explanation: Step 1: New frequencies: 2, 3, 10, 15, 10, n = 40; ascending cumulative frequencies: 2, 5, 15, 30, 40 Step 2: Order of Q1 = 10: Q1 = 30 + (10 − 5)/10 × 5 = 32.5; order of Q3 = 30: Q3 = 35 + (30 − 15)/15 × 5 = 40 ∴ Q = (40 − 32.5)/2 = 3.75 (4.25 is the value before they join; 3.5 is a trap: it keeps the old orders 9 and 27 with the new frequencies.)
The data (not in order): 30, 18, x, 25, 40, 12, 35. If the semi-interquartile range of these data = 10, then the possible values of x are ……
Correct answer: (A) 15 or 38
Explanation: Step 1: n = 7, so Q1 is the 2nd value and Q3 is the 6th value in ascending order. Step 2: If 12 ≤ x ≤ 18: Q1 = x and Q3 = 35, so (35 − x)/2 = 10 and x = 15 Step 3: If 35 ≤ x ≤ 40: Q1 = 18 and Q3 = x, so (x − 18)/2 = 10 and x = 38 ∴ x = 15 or x = 38 (any other position gives Q = 11.5, 8.5 or 11).
First factory: the wages of the workers in hundreds of pounds, intervals 20–, 25–, 30–, 35–, 40– (each of width 5) with frequencies 6, 8, 12, 8, 6 respectively. Second factory: a box plot of the wages (minimum value 20, lower quartile 29, median 34, upper quartile 37, maximum value 45). Using the semi-interquartile range, which statement is true?
Correct answer: (A) The wages in the second factory are more homogeneous (4 against 5)
Explanation: Step 1: First factory: n = 40, order of Q1 = 10: Q1 = 25 + (10 − 6)/8 × 5 = 27.5; order of Q3 = 30: Q3 = 35 + (30 − 26)/8 × 5 = 37.5; Q = (37.5 − 27.5)/2 = 5 Step 2: Second factory: Q = (37 − 29)/2 = 4 ∴ 4 < 5, so the wages in the second factory are less dispersed (more homogeneous). (12.5 is half the range of the second factory, 10 is the interquartile range of the first, and equal ranges do not mean equal semi-interquartile ranges.)
The first 10 questions in every subject are free, each with an explanation.