--- title: "Central tendency measures; wage distribution" subject: "Probability and Statistics" chapter: 1 key: "semester_5/probability_and_statistics:central-tendency-wages-approx" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/central-tendency-wages-approx site: BCT Engineering --- # Central tendency measures; wage distribution Probability and Statistics · Chapter 1: Descriptive Statistics and Basic Probability - Asked in: 2074 Chaitra, 1+2+3 marks - Video: [Mean, Median, and Mode: Measures of Central Tendency: Crash Course Statistics #3](https://www.youtube.com/watch?v=kn83BA7cRNM) — CrashCourse - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/central-tendency-wages-approx - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1 ## Question What are measures of central tendencies? Write favorable points of each of them. Calculate approximate measures of central tendency from following data: | Wages in Rs/week | Less than 35 | 35-37 | 38-40 | 41-43 | Over 43 | |---|---|---|---|---|---| | No. of wage earned | 14 | 82 | 99 | 18 | 7 | ## Short answer Approximate measures: Median = Rs 37.92, Mode = Rs 38.02, Q₁ = Rs 36.00, Q₃ = Rs 39.59. Open‑ended classes prevent mean calculation; positional averages from continuous classes (34.5–37.5, etc.) with N=220. ## Answer #### Measures of central tendency A measure of central tendency is a single value that represents the centre of a data set. The main measures are the arithmetic mean, median, mode, geometric mean and harmonic mean. #### Favourable points - **Arithmetic mean:** easy to understand; uses every observation; suitable for further algebraic work. - **Median:** not affected by extreme values; can be found even with open-ended classes; good for skewed data. - **Mode:** the most typical value; easy to locate; usable for qualitative data. - **Geometric mean:** suitable for ratios, rates of growth and index numbers. - **Harmonic mean:** suitable for averaging rates such as speed. #### Calculation The first and last classes are open-ended, so the mean cannot be found. The **approximate (positional) averages**, namely the median, quartiles and mode, are used instead. They only need the middle classes. Make the classes continuous (35-37 becomes 34.5-37.5, and so on): | Wages (Rs/week) | f | cf | |---|---|---| | Less than 34.5 | 14 | 14 | | 34.5-37.5 | 82 | 96 | | 37.5-40.5 | 99 | 195 | | 40.5-43.5 | 18 | 213 | | Over 43.5 | 7 | 220 | N = 220, class width h = 3. **Median:** N/2 = 110, so the median class is 37.5-40.5. ```text Median = 37.5 + (110 − 96)/99 × 3 = 37.5 + 0.424 = Rs 37.92 ``` **Mode:** modal class 37.5-40.5 (f₁ = 99, f₀ = 82, f₂ = 18) ```text Mode = 37.5 + (99 − 82) / (2×99 − 82 − 18) × 3 = 37.5 + 17/98 × 3 = Rs 38.02 ``` **Quartiles:** ```text Q1 = 34.5 + (55 − 14)/82 × 3 = Rs 36.00 Q3 = 37.5 + (165 − 96)/99 × 3 = Rs 39.59 ``` **Answer:** median ≈ Rs 37.92, mode ≈ Rs 38.02, Q₁ = Rs 36.00 and Q₃ ≈ Rs 39.59 per week. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why can the mean not be found? A. First and last classes are open-ended B. Class widths are unequal C. Frequencies are missing D. Data are qualitative Answer: A. Midpoints of open classes are unknown. (Key point: Why not mean) 2. How are classes made continuous? A. 35-37 becomes 35-37 B. 35-37 becomes 34.5-37.5 C. 35-37 becomes 36-38 D. Add 3 to each limit Answer: B. Subtract 0.5 and add 0.5 to bridge the gaps. (Key point: Continuous classes) 3. What is the median? A. 37.5 + 14/99 = 37.64 B. 34.5 + 110/99 x 3 C. 37.5 + (110 - 96)/99 x 3 = 37.92 D. 40.5 Answer: C. N/2 = 110 lies in 37.5-40.5 with cf before = 96. (Key point: Median) 4. What is the mode formula used? A. L + (f1 + f0)/(2f1) x h B. L + N/2 x h C. L + (f2 - f0)/(2f1) x h D. L + (f1 - f0)/(2f1 - f0 - f2) x h = 38.02 Answer: D. Uses modal class frequency and neighbours. 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