Descriptive Stats - Chapter 2 - Calculations Flashcards
Given the data: Categories (A, B, C, D) with frequencies (10, 20, 15, 5), calculate the percentage each category represents for a pie chart and verify the total is 100%.
Total frequency = 10 + 20 + 15 + 5 = 50
Percentage for A = (10 / 50) × 100 = 20%
Percentage for B = (20 / 50) × 100 = 40%
Percentage for C = (15 / 50) × 100 = 30%
Percentage for D = (5 / 50) × 100 = 10%
Verification: 20% + 40% + 30% + 10% = 100%
Answer: A: 20%, B: 40%, C: 30%, D: 10%; Total = 100%
From the histogram example on Page 37 (Height in cm: 40-45, 45-50, 50-55, 55-60, 60-65; Number of Boys: 12, 18, 15, 9, 8), what is the total number of boys?
Total = 12 + 18 + 15 + 9 + 8
Calculation: 12 + 18 = 30, 30 + 15 = 45, 45 + 9 = 54, 54 + 8 = 62
Answer: 62 boys
Using the histogram data from Page 44 (Height: 40-45, 45-50, 50-55; Boys: 10, 15, 25), calculate the percentage of boys in the 50-55 cm height range.
Total boys = 10 + 15 + 25 = 50
Percentage for 50-55 = (25 / 50) × 100 = 50%
Answer: 50%
For the histogram on Page 38 (weekly wages), if 20 workers earn ₹60-70, estimate the number of workers earning ₹70-80 based on a similar bar height (assume equal intervals).
The histogram isn’t provided, but the question implies estimation. Since it says “similar bar height” and assumes equal intervals, we assume the frequency for ₹70-80 is also 20 (as no other data is given).
Answer: 20 workers (assumed based on “similar bar height”)
Construct a frequency distribution table from the histogram on Page 44 (Height: 40-45, 45-50, 50-55; Boys: 10, 15, 25).
Table:
Height (cm) Frequency
40-45 10
45-50 15
50-55 25
From the same data (Page 44), calculate the cumulative frequency for the class 45-50
Cumulative frequency is the running total up to that class.
40-45: 10
45-50: 10 + 15 = 25
Answer: 25
Using the histogram on Page 38, if the frequencies are (40-50: 10, 50-60: 15, 60-70: 20, 70-80: 25), what is the cumulative frequency for the class 50-60?
40-50: 10
50-60: 10 + 15 = 25
Answer: 25
For the data on Page 39 (assume frequencies 10-20: 5, 20-30: 10, 30-40: 15), calculate the cumulative frequency for 20-30.
10-20: 5
20-30: 5 + 10 = 15
Answer: 15
For the data on Page 40 (Class A: 85, 92, 78, 88, 76, 95, 80, 87, 90, 91), calculate Q1, Q2 (median), and Q3.
Sort: 76, 78, 80, 85, 87, 88, 90, 91, 92, 95
Q2 (median): (87 + 88) / 2 = 87.5 (5th and 6th values)
Q1 (lower half: 76, 78, 80, 85, 87): Median = 80 (3rd value)
Q3 (upper half: 88, 90, 91, 92, 95): Median = 91 (3rd value)
Answer: Q1 = 80, Q2 = 87.5, Q3 = 91
Using the same Class A data, compute the IQR and determine if there are any outliers.
IQR = Q3 - Q1 = 91 - 80 = 11
Lower bound = Q1 - 1.5 × IQR = 80 - 1.5 × 11 = 80 - 16.5 = 63.5
Upper bound = Q3 + 1.5 × IQR = 91 + 1.5 × 11 = 91 + 16.5 = 107.5
Range of data: 76 to 95, all within 63.5 to 107.5
Answer: IQR = 11, No outliers
For Class B (Page 40: 70, 65, 72, 68, 69, 75, 80, 82, 74, 77, 73, 76, 78, 79, 80), find Q1, Q2, and Q3.
Sort: 65, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 80, 82
Q2 (median): 75 (8th value)
Q1 (lower half: 65, 68, 69, 70, 72, 73, 74): Median = 70 (4th value)
Q3 (upper half: 76, 77, 78, 79, 80, 80, 82): Median = 79 (4th value)
Answer: Q1 = 70, Q2 = 75, Q3 = 79
Using Class B data, calculate the IQR and identify any outliers.
IQR = Q3 - Q1 = 79 - 70 = 9
Lower bound = 70 - 1.5 × 9 = 70 - 13.5 = 56.5
Upper bound = 79 + 1.5 × 9 = 79 + 13.5 = 92.5
Range: 65 to 82, all within 56.5 to 92.5
Answer: IQR = 9, No outliers
From Page 42 (weights: 55, 57, 60, 62, 63, 64, 65, 66, 67, 95), verify the outlier (95) using the IQR method.
Sort: 55, 57, 60, 62, 63, 64, 65, 66, 67, 95
Q2 = (63 + 64) / 2 = 63.5
Q1 (55, 57, 60, 62, 63) = 60
Q3 (64, 65, 66, 67, 95) = 66
IQR = 66 - 60 = 6
Lower bound = 60 - 1.5 × 6 = 60 - 9 = 51
Upper bound = 66 + 1.5 × 6 = 66 + 9 = 75
95 > 75, so it’s an outlier
Answer: 95 is an outlier
For the data on Page 45 (15, 20, 9, 32, 27, 19, 16, 28, 31, 7, 11, 19), calculate Q1, Q2, Q3, and IQR.
Sort: 7, 9, 11, 15, 16, 19, 19, 20, 27, 28, 31, 32
Q2 = (19 + 19) / 2 = 19
Q1 (7, 9, 11, 15, 16, 19) = (11 + 15) / 2 = 13
Q3 (19, 20, 27, 28, 31, 32) = (27 + 28) / 2 = 27.5
IQR = 27.5 - 13 = 14.5
Answer: Q1 = 13, Q2 = 19, Q3 = 27.5, IQR = 14.5
For Page 47 (ages: 22, 24, 25, 26, 28, 30, 31, 32, 33, 34, 35, 60), calculate Q1, Q3, and IQR, and identify the outlier.
Sort: 22, 24, 25, 26, 28, 30, 31, 32, 33, 34, 35, 60
Q2 = (30 + 31) / 2 = 30.5
Q1 (22, 24, 25, 26, 28, 30) = (25 + 26) / 2 = 25.5
Q3 (31, 32, 33, 34, 35, 60) = (34 + 35) / 2 = 34.5
IQR = 34.5 - 25.5 = 9
Lower bound = 25.5 - 1.5 × 9 = 25.5 - 13.5 = 12
Upper bound = 34.5 + 1.5 × 9 = 34.5 + 13.5 = 48
60 > 48, so it’s an outlier
Answer: Q1 = 25.5, Q3 = 34.5, IQR = 9, Outlier = 60
Using the same data, determine the lower and upper bounds for outliers and identify any if present.
Lower bound = 13 - 1.5 × 14.5 = 13 - 21.75 = -8.75
Upper bound = 27.5 + 1.5 × 14.5 = 27.5 + 21.75 = 49.25
Range: 7 to 32, all within -8.75 to 49.25
Answer: No outliers
For Group A scores on Page 45 (65, 70, 75, 80, 85, 90, 92, 87, 88, 95), calculate Q1, Q2, and Q3.
Sort: 65, 70, 75, 80, 85, 87, 88, 90, 92, 95
Q2 = (85 + 87) / 2 = 86
Q1 (65, 70, 75, 80, 85) = 75
Q3 (87, 88, 90, 92, 95) = 90
Answer: Q1 = 75, Q2 = 86, Q3 = 90
For Group B scores (55, 60, 65, 70, 72, 75, 78, 80, 85, 90, 95, 100, 105), compute Q1, Q2, Q3, and check for outliers.
Sort: 55, 60, 65, 70, 72, 75, 78, 80, 85, 90, 95, 100, 105
Q2 = 78 (7th value)
Q1 (55, 60, 65, 70, 72, 75) = (65 + 70) / 2 = 67.5
Q3 (80, 85, 90, 95, 100, 105) = (90 + 95) / 2 = 92.5
IQR = 92.5 - 67.5 = 25
Lower bound = 67.5 - 1.5 × 25 = 67.5 - 37.5 = 30
Upper bound = 92.5 + 1.5 × 25 = 92.5 + 37.5 = 130
Range: 55 to 105, all within 30 to 130
Answer: Q1 = 67.5, Q2 = 78, Q3 = 92.5, No outliers
From Page 35 (1, 5, 6, 7, 10, 15, 25, 100), recalculate the IQR and confirm 100 as an outlier
Sort: 1, 5, 6, 7, 10, 15, 25, 100
Q2 = (7 + 10) / 2 = 8.5
Q1 (1, 5, 6, 7) = (5 + 6) / 2 = 5.5
Q3 (10, 15, 25, 100) = (15 + 25) / 2 = 20
IQR = 20 - 5.5 = 14.5
Lower bound = 5.5 - 1.5 × 14.5 = 5.5 - 21.75 = -16.25
Upper bound = 20 + 1.5 × 14.5 = 20 + 21.75 = 41.75
100 > 41.75, so it’s an outlier
Answer: IQR = 14.5, 100 is an outlier
Given a dataset (10, 12, 15, 18, 20, 22, 25, 50), calculate Q1, Q3, IQR, and determine if 50 is an outlier.
Sort: 10, 12, 15, 18, 20, 22, 25, 50
Q2 = (18 + 20) / 2 = 19
Q1 (10, 12, 15, 18) = (12 + 15) / 2 = 13.5
Q3 (20, 22, 25, 50) = (22 + 25) / 2 = 23.5
IQR = 23.5 - 13.5 = 10
Lower bound = 13.5 - 1.5 × 10 = 13.5 - 15 = -1.5
Upper bound = 23.5 + 1.5 × 10 = 23.5 + 15 = 38.5
50 > 38.5, so it’s an outlier
Answer: Q1 = 13.5, Q3 = 23.5, IQR = 10, Outlier = 50
