Class 11 Economics Chapter 5 Question Answer | Measures of Central Tendency | English Medium | ASSEB
Welcome, learners! This article on HSLC Guru presents complete English-medium notes for Class 11 Economics (Statistics for Economics) Chapter 5 — Measures of Central Tendency, prepared strictly as per the ASSEB (Assam State School Education Board) syllabus. The chapter explains the meaning of an average, properties of a good average, computation of the Arithmetic Mean, Median and Mode under individual, discrete and continuous series, the empirical relationship among them, and positional measures such as Quartiles, Deciles and Percentiles. Numerical illustrations are supplied wherever the syllabus expects calculations.
Chapter Summary
Meaning of Central Tendency. A measure of central tendency is a single value that represents the entire data set by indicating the centre around which the observations cluster. It condenses a mass of figures into one representative value, makes comparison between two or more series possible, and forms the foundation for further statistical analysis like dispersion, correlation and regression. The three most widely used measures are the Arithmetic Mean (mathematical average), Median (positional average) and Mode (frequency-based average).
Properties of a Good Average as laid down by Prof. G. U. Yule include: (i) it should be rigidly defined so that the same value is obtained by different investigators, (ii) it should be based on all items of the series, (iii) it should be easy to understand and simple to calculate, (iv) it should be capable of further algebraic treatment, (v) it should not be affected unduly by extreme values, and (vi) it should have sampling stability — that is, results from different samples should not vary widely.
Arithmetic Mean (AM) is obtained by dividing the sum of observations by their number. Simple AM assumes equal importance for every item, while Weighted AM assigns weights according to relative importance. Formulae: Individual series — Direct: X̄ = ΣX/N; Short-cut: X̄ = A + Σd/N where d = X − A. Discrete series — Direct: X̄ = ΣfX/Σf; Short-cut: X̄ = A + Σfd/Σf. Continuous series — Direct: X̄ = Σfm/Σf where m is the mid-value; Step-deviation: X̄ = A + (Σfd′/Σf) × c, where d′ = (m − A)/c and c is the class-width. Weighted Mean: X̄w = ΣwX/Σw.
Median is the middle-most value when items are arranged in ascending or descending order. Individual series: M = size of ((N+1)/2)th item. Discrete series: locate ((N+1)/2)th item in the cumulative frequency column. Continuous series: M = L + ((N/2 − cf)/f) × h, where L is the lower limit of the median class, cf is the cumulative frequency preceding it, f is the frequency of the median class and h is its width. Median can also be located graphically as the X-coordinate of the point of intersection of the less-than and more-than ogives.
Mode is the value that occurs most frequently. In an individual or discrete series it can be located by inspection (the item with the highest frequency) or, when frequencies are irregular, by the grouping method followed by an analysis table. In a continuous series, Mode = L + ((f1 − f0)/(2f1 − f0 − f2)) × h, where L is the lower limit of the modal class, f1 its frequency, f0 the frequency of the preceding class and f2 of the succeeding class. Empirical relationship for a moderately asymmetrical distribution: Mode = 3 Median − 2 Mean. Quartiles divide the distribution into four equal parts (Q1, Q2 = Median, Q3); Deciles into ten parts (D1…D9); Percentiles into hundred parts (P1…P99). For continuous series, Qj = L + ((jN/4 − cf)/f) × h; similar formulae apply with N/10 and N/100 for deciles and percentiles.
Textbook Questions and Answers
A. Very Short Answer Questions (1 Mark)
Q1. What is meant by an average?
Answer: An average is a single value that represents a group of values and lies somewhere within the range of the data, indicating its central tendency.
Q2. Name the three most commonly used measures of central tendency.
Answer: Arithmetic Mean, Median and Mode.
Q3. Which average is most affected by extreme values?
Answer: The Arithmetic Mean.
Q4. Define Median.
Answer: Median is the value of the middle item when observations are arranged in order of magnitude; it divides the series into two equal parts.
Q5. Define Mode.
Answer: Mode is the value that occurs most frequently in a series.
Q6. Write the empirical relationship between Mean, Median and Mode.
Answer: Mode = 3 Median − 2 Mean.
Q7. What does Q2 represent?
Answer: Q2 represents the second quartile, which is the same as the Median.
Q8. Into how many equal parts do percentiles divide a distribution?
Answer: Hundred (100) equal parts.
Q9. Which average can be calculated graphically using ogives?
Answer: The Median.
Q10. Write the formula for the Arithmetic Mean of a discrete series (direct method).
Answer: X̄ = ΣfX / Σf.
B. Short Answer Questions (2-3 Marks)
Q1. State any three properties of a good average.
Answer: (i) It should be rigidly defined so that all investigators arrive at the same figure. (ii) It should be based on all items of the series so that it is representative. (iii) It should be easy to compute and understand, and capable of further algebraic treatment.
Q2. Distinguish between Simple and Weighted Arithmetic Mean.
Answer: Simple AM treats every item as equally important and is computed as ΣX/N. Weighted AM assigns weights to items according to their relative importance and is computed as ΣwX/Σw. Weighted Mean is preferable when items differ in importance, e.g., calculating average marks where subjects carry different credit weights.
Q3. Calculate the Arithmetic Mean: 25, 30, 35, 40, 45, 50, 55.
Answer: ΣX = 25 + 30 + 35 + 40 + 45 + 50 + 55 = 280; N = 7. X̄ = 280/7 = 40.
Q4. Find the Median: 12, 15, 18, 22, 25, 28, 32.
Answer: N = 7; Median = size of ((7+1)/2) = 4th item = 22.
Q5. Find the Mode by inspection: 5, 7, 8, 8, 9, 10, 8, 11, 12.
Answer: The value 8 occurs most frequently (3 times). Hence, Mode = 8.
Q6. If Mean = 50 and Median = 48, find Mode using the empirical formula.
Answer: Mode = 3 Median − 2 Mean = 3(48) − 2(50) = 144 − 100 = 44.
C. Long Answer Questions (5-6 Marks)
Q1. Calculate the Arithmetic Mean of the following continuous series by the direct method.
| Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
| No. of students (f) | 5 | 10 | 20 | 10 | 5 |
Answer: Mid-values (m): 5, 15, 25, 35, 45. fm: 25, 150, 500, 350, 225. Σf = 50; Σfm = 1250. X̄ = Σfm/Σf = 1250/50 = 25 marks.
Q2. Compute the Median for the following distribution.
| Wages (Rs.) | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
| No. of workers | 4 | 6 | 10 | 5 | 5 |
Answer: Cumulative frequencies: 4, 10, 20, 25, 30. N = 30; N/2 = 15. The class containing the 15th item is 20-30 (cf jumps from 10 to 20). L = 20, cf = 10, f = 10, h = 10. M = L + ((N/2 − cf)/f) × h = 20 + ((15 − 10)/10) × 10 = 20 + 5 = Rs. 25.
Q3. Calculate the Mode for the following data.
| Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
| Frequency | 5 | 8 | 15 | 9 | 3 |
Answer: Modal class = 20-30 (highest frequency 15). L = 20, f1 = 15, f0 = 8, f2 = 9, h = 10. Mode = L + ((f1 − f0)/(2f1 − f0 − f2)) × h = 20 + ((15 − 8)/(30 − 8 − 9)) × 10 = 20 + (7/13) × 10 = 20 + 5.38 = 25.38.
Q4. The mean weight of 50 students is 60 kg. Of these, the mean weight of 30 boys is 65 kg. Find the mean weight of the 20 girls.
Answer: Total weight of all students = 50 × 60 = 3000 kg. Total weight of boys = 30 × 65 = 1950 kg. Total weight of girls = 3000 − 1950 = 1050 kg. Mean weight of girls = 1050/20 = 52.5 kg.
Q5. Calculate Q1 and Q3 from the following continuous series.
| Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
| Frequency | 4 | 8 | 12 | 10 | 6 |
Answer: Cumulative frequencies: 4, 12, 24, 34, 40. N = 40. Q1: N/4 = 10; class = 10-20 (cf passes 10 here). L = 10, cf = 4, f = 8, h = 10. Q1 = 10 + ((10 − 4)/8) × 10 = 10 + 7.5 = 17.5. Q3: 3N/4 = 30; class = 30-40 (cf passes 30 here). L = 30, cf = 24, f = 10, h = 10. Q3 = 30 + ((30 − 24)/10) × 10 = 30 + 6 = 36.
Additional Multiple Choice Questions (MCQs)
Q1. Which of the following is a positional average? (a) Mean (b) Median (c) Geometric Mean (d) Harmonic Mean
Answer: (b) Median.
Q2. The sum of deviations of items from the Arithmetic Mean is always: (a) Maximum (b) Minimum (c) Zero (d) One
Answer: (c) Zero.
Q3. Mode of the data 2, 3, 5, 5, 6, 7, 5, 8 is: (a) 2 (b) 5 (c) 6 (d) 8
Answer: (b) 5.
Q4. Median of 4, 6, 8, 10, 12 is: (a) 6 (b) 8 (c) 10 (d) 12
Answer: (b) 8.
Q5. Quartiles divide a distribution into how many equal parts? (a) Two (b) Three (c) Four (d) Ten
Answer: (c) Four.
Q6. Which average is best for open-end classes? (a) Mean (b) Median (c) Mode (d) GM
Answer: (b) Median.
Q7. If Mean = 30 and Mode = 24, then Median is: (a) 26 (b) 27 (c) 28 (d) 29
Answer: (c) 28. (Using Mode = 3M − 2X̄ → 24 = 3M − 60 → M = 28.)
Q8. The 50th percentile is equal to: (a) Q1 (b) Median (c) Q3 (d) D9
Answer: (b) Median.
Q9. Which graph is used to locate the Median? (a) Histogram (b) Bar diagram (c) Ogive (d) Pie chart
Answer: (c) Ogive.
Q10. The formula X̄ = A + (Σfd′/Σf) × c is for: (a) Direct method (b) Step-deviation method (c) Median (d) Mode
Answer: (b) Step-deviation method.
Fill in the Blanks
Q1. The arithmetic mean of first 5 natural numbers is _______.
Answer: 3.
Q2. Median is the _______ value when items are arranged in order.
Answer: middle.
Q3. Mode = 3 Median − _______ Mean.
Answer: 2.
Q4. _______ divide a distribution into ten equal parts.
Answer: Deciles.
Q5. The Mean is most affected by _______ values.
Answer: extreme.
True or False
Q1. Median can be determined graphically. — True.
Q2. Mode is always unique in any data set. — False (a series may be bimodal or multimodal).
Q3. The sum of deviations of observations from the Mean is zero. — True.
Q4. The Median is influenced by every observation in the series. — False (only the position matters).
Q5. Q3 is also called the upper quartile. — True.
Glossary
| Term | Meaning |
|---|---|
| Central Tendency | Tendency of data to cluster around a central value. |
| Arithmetic Mean | Sum of items divided by their number. |
| Weighted Mean | Mean computed after assigning weights to items. |
| Median | Middle value of an ordered series. |
| Mode | Most frequently occurring value. |
| Ogive | Cumulative frequency curve used to locate Median, Quartiles etc. |
| Quartile | Value that divides data into four equal parts. |
| Decile | Value that divides data into ten equal parts. |
| Percentile | Value that divides data into hundred equal parts. |
| Modal Class | Class interval with the highest frequency. |
Formula Table
| Measure | Series | Formula |
|---|---|---|
| Arithmetic Mean (Direct) | Individual | X̄ = ΣX / N |
| Arithmetic Mean (Short-cut) | Individual | X̄ = A + Σd / N |
| Arithmetic Mean (Direct) | Discrete | X̄ = ΣfX / Σf |
| Arithmetic Mean (Direct) | Continuous | X̄ = Σfm / Σf |
| Arithmetic Mean (Step-deviation) | Continuous | X̄ = A + (Σfd′ / Σf) × c |
| Weighted Mean | — | X̄w = ΣwX / Σw |
| Median | Individual | M = size of ((N+1)/2)th item |
| Median | Continuous | M = L + ((N/2 − cf) / f) × h |
| Mode | Continuous | Z = L + ((f1 − f0) / (2f1 − f0 − f2)) × h |
| Empirical Relation | — | Mode = 3 Median − 2 Mean |
| Quartile | Continuous | Qj = L + ((jN/4 − cf) / f) × h |
| Decile | Continuous | Dj = L + ((jN/10 − cf) / f) × h |
| Percentile | Continuous | Pj = L + ((jN/100 − cf) / f) × h |
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