10th Class Mathematics Unit 11 Statistics Numericals with Solutions

11.1.1.The following frequency distribution represents the achievement of a student's group upon a memory test: Scores 44-48, 49-53, 54-58, 59-63, 64-68, 69-73, 74-78 with No. of Students (f) = 3, 12, 15, 23, 12, 8, 7. (i) Plot cumulative frequency graph of group scores. (ii) Determine median, lower and upper quartiles, D3, P90 and interquartile range graphically.

Given

Frequency table / N N = sum f = 80

Less-than cumulative frequency table C.F. = 3, 15, 30, 53, 65, 73, 80

Formula

Q1 position Q₁ = N4 = 804 = 20 lies in class 54-58

Q1 Q₁ = 53.5 + 20-1515 × 5 = 55.17 ≈ 55

Median position Median = N2 = 40 lies in class 59-63

Median Median = 58.5 + 40-3023 × 5 = 60.67 ≈ 61

Q3 position Q₃ = 3N4 = 60 lies in class 64-68

Q3 Q₃ = 63.5 + 60-5312 × 5 = 66.42 ≈ 66

D3 position D₃ = 3N10 = 24 lies in class 54-58

D3 D₃ = 53.5 + 24-1515 × 5 = 56.50 ≈ 57

P90 position P₉₀ = 90N100 = 72 lies in class 69-73

P90 P₉₀ = 68.5 + 72-658 × 5 = 72.88 ≈ 73

The book prints the lower boundary as 63.5 in this line, but that value belongs to class 59-63, not the class 69-73 identified one line above; the class 69-73 boundary is 68.5. The final numeric answer (72.88≈73) is only reachable using 68.5, so this is a transcription slip that did not propagate to the printed result.

IQR IQR = Q₃ - Q₁ = 66.42 - 55.17 = 11.25 ≈ 11

Result

Summary Median ≈ 61, Q₁ ≈ 55, Q₃ ≈ 66, D₃ ≈ 57, P₉₀ ≈ 73, IQR ≈ 11

11.1.2.Construct an ogive for the following distribution of scores: Scores 69-79, 79-89, 89-99, 99-109, 109-119 with f = 4, 10, 13, 8, 5. Determine median and lower and upper quartiles, D6, P80 and interquartile range graphically.

Given

Frequency table / N N = sum f = 40

Less-than cumulative frequency table C.F. = 4, 14, 27, 35, 40

Formula

Q1 position Q₁ = N4 = 10 lies in class 79-89

Q1 Q₁ = 79.5 + 10-410 × 10 = 85.5 ≈ 86

Median position Median = N2 = 20 lies in class 89-99

Median Median = 89.5 + 20-1413 × 10 = 94.1 ≈ 94

Q3 position Q₃ = 3N4 = 30 lies in class 99-109

Q3 Q₃ = 99.5 + 30-278 × 10 = 103.25 ≈ 103

D6 position D₆ = 6N10 = 24 lies in class 89-99

D6 D₆ = 89.5 + 24-1413 × 10 = 97.2 ≈ 97

P80 position P₈₀ = 80N100 = 32 lies in class 99-109

P80 P₈₀ = 99.5 + 32-278 × 10 = 105.75 ≈ 106

IQR IQR = Q₃ - Q₁ = 103.25 - 85.5 = 17.75 ≈ 18

Result

Summary Median ≈ 94, Q₁ ≈ 86, Q₃ ≈ 103, D₆ ≈ 97, P₈₀ ≈ 106, IQR ≈ 18

11.1.3.Find Q1, Q3, median, range and IQR for the dataset given below: 3, 6, 8, 4, 7, 5, 10, 11, 13, 9, 14, 12, 15, 16, 17. Also draw a box-and-whisker plot.

Sorted data (n=15) 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17

Formula

Median position n+12 = 8^th ⇒ Median = 10

Q1 position n+14 = 4^th ⇒ Q₁ = 6

Q3 position 3(n+1)4 = 12^th ⇒ Q₃ = 14

Range Range = 17 - 3 = 14

IQR IQR = Q₃ - Q₁ = 14 - 6 = 8

Result

Summary Q₁ = 6, Q₃ = 14, Median = 10, Range = 14, IQR = 8

11.1.4.Find Q1, Q3, median, range, IQR and extreme values plot for the following data: 102, 98, 95, 100, 93, 110, 108, 104, 97, 96, 92, 101, 99, 105, 107. Also draw a box-and-whisker plot.

Sorted data (n=15) 92,93,95,96,97,98,99,100,101,102,104,105,107,108,110

Formula

Median position n+12 = 8^th ⇒ Median = 100

Q1 position n+14 = 4^th ⇒ Q₁ = 96

Q3 position 3(n+1)4 = 12^th ⇒ Q₃ = 105

Range Range = 110 - 92 = 18

IQR IQR = Q₃ - Q₁ = 105 - 96 = 9

Result

Summary Q₁ = 96, Q₃ = 105, Median = 100, Range = 18, IQR = 9

11.1.5.Find Q1, Q3, median, minimum and maximum values for the following box-and-whisker plot (scale 10 to 50).

Reading the plot Minimum = 15

Reading the plot Q₁ = 22

Reading the plot Median = 25

Reading the plot Q₃ = 35

Result

Reading the plot Maximum = 45

11.1.6.Draw the box-and-whisker plot, if min. value = 12, max. value = 26, median = 19, Q1 = 14 and Q3 = 24.

Given

Five-number summary Min=12, Q₁=14, Median=19, Q₃=24, Max=26

Result

Plot Box-and-whisker plot as required

11.1.7.Construct a scatter diagram and draw a line of best fit for the following quiz scores for 8 students in a class: 1st quiz (x): 14, 28, 34, 36, 43, 45, 46, 49; 2nd quiz (y): 16, 20, 33, 35, 38, 39, 42, 43.

11.1.8.The following table shows the test scores (one measuring speed and the other measuring strength) of football players out of 200: 1st Test (x): 20, 38, 70, 40, 100, 110, 50, 136, 150, 80; 2nd Test (y): 160, 142, 130, 120, 70, 50, 140, 44, 36, 100. (i) Construct a scatter diagram and draw a line of best fit. (ii) Describe correlation between the 1st and 2nd tests. (iii) Abdullah scores 60 in the 2nd test. Estimate his score in the 1st test.

11.1.9.The following table shows the number of daily social media hours and GPA of university students. Social Media Hours (x): 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9; GPA (y): 4.0, 3.9, 3.7, 3.5, 3.3, 3.1, 2.9, 2.8, 2.6, 2.5. (i) Construct a scatter diagram and draw a line of best fit. (ii) Identify the nature of correlation between social media use and GPA. (iii) Estimate the GPA for a student who spends 6.5 hours on social media.

Unit 11: Statistics — Exercise 11 1