--- title: "Central tendency; middle 70% income, median" subject: "Probability and Statistics" chapter: 1 key: "semester_5/probability_and_statistics:central-tendency-middle-70-income" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/central-tendency-middle-70-income site: BCT Engineering --- # Central tendency; middle 70% income, median Probability and Statistics · Chapter 1: Descriptive Statistics and Basic Probability - Asked in: 2076 Asoj, 6 marks - Video: [Median of Grouped Data │ Statistics](https://www.youtube.com/watch?v=wrstK3HbR3Y) — GoTutor Math - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/central-tendency-middle-70-income - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1 ## Question Describe the strong and weak points of various measures of Central tendency. From the following frequency distribution find the range of income of middle 70% of the employees and the median income. Also find mean deviation from mean. | Income in Rs. | 500-600 | 600-700 | 700-800 | 800-900 | 900-1000 | |---|---|---|---|---|---| | No. of employees | 150 | 300 | 500 | 200 | 50 | ## Short answer Range of income of middle 70% is Rs 610 to Rs 835 (P₁₅ = 610, P₈₅ = 835). Median = Rs 730. Mean = Rs 725; mean deviation from mean = Rs 81.25. ## Answer #### Measures of central tendency: strong and weak points | Measure | Strong points | Weak points | |---|---|---| | Mean | Uses all values; easy; good for further algebra | Affected by extreme values; not usable for open-ended classes | | Median | Not affected by extremes; works with open-ended classes; good for skewed data | Ignores the size of most values; needs ordering; weak algebra | | Mode | Easy; shows the most typical value; works for qualitative data | May be absent or not unique; unstable | | Geometric mean | Best for ratios and growth rates | Not defined for zero or negative values; harder to compute | | Harmonic mean | Good for rates such as speed | Affected by very small values; not defined with zeros | #### Numerical part N = 1200. Cumulative frequencies (cf): 150, 450, 950, 1150, 1200. The middle 70% of employees lie between the 15th and 85th percentiles (P₁₅ and P₈₅). **P₁₅:** position 0.15 × 1200 = 180, class 600-700 (cf before = 150) ```text P15 = 600 + (180 − 150)/300 × 100 = 610 ``` **P₈₅:** position 0.85 × 1200 = 1020, class 800-900 (cf before = 950) ```text P85 = 800 + (1020 − 950)/200 × 100 = 835 ``` **Median:** position 600, class 700-800 (cf before = 450) ```text Median = 700 + (600 − 450)/500 × 100 = 730 ``` **Mean deviation from mean:** | Class | Mid x | f | fx | Deviation from 725 | f × deviation | |---|---|---|---|---|---| | 500-600 | 550 | 150 | 82,500 | 175 | 26,250 | | 600-700 | 650 | 300 | 195,000 | 75 | 22,500 | | 700-800 | 750 | 500 | 375,000 | 25 | 12,500 | | 800-900 | 850 | 200 | 170,000 | 125 | 25,000 | | 900-1000 | 950 | 50 | 47,500 | 225 | 11,250 | | Total | | 1200 | 870,000 | | 97,500 | ```text Mean = 870,000 / 1200 = Rs 725 MD about mean = 97,500 / 1200 = Rs 81.25 ``` **Answer:** the middle 70% of employees earn between Rs 610 and Rs 835; median income = Rs 730; mean deviation from the mean = Rs 81.25. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which percentiles bound the middle 70% of employees? A. P30 and P70 B. P15 and P85 C. P0 and P70 D. P10 and P80 Answer: B. 15% lie below and 15% above, leaving 70% between. (Key point: Middle 70%) 2. Why does the median lie in 700-800? A. The class has the highest mid-value B. It is the last class C. N/2 = 600 falls where cf rises from 450 to 950 D. It has the least frequency Answer: C. The first class with cf >= 600 is the median class. (Key point: Median class) 3. How is mean deviation about the mean found? A. Σf(x - mean)/Σf B. Σf(x - mean)^2/Σf C. Σf|x - median|/N D. Σf|x - mean|/Σf = 97,500/1200 Answer: D. Absolute deviations from the mean (725) are weighted by f. (Key point: Mean deviation) 4. What is P15? A. 610 B. 600 C. 625 D. 580 Answer: A. 600 + (180 - 150)/300 x 100 = 610. 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