- 2071 Shrawan · 6 marks
Central tendency vs dispersion; mode 58
Short answer
Central tendency gives a typical value; dispersion shows spread. Both are needed to describe data. Missing frequencies: 18 (20-40) and 26 (60-80). Mean = 5200/100 = 52 marks.
Write difference between measure of central tendency and measure of dispersion and their importance. The following table represents the marks of 100 students.
| Marks | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |
|---|---|---|---|---|---|
| No. of Students | 14 | ? | 27 | ? | 15 |
| If the mode value is 58, find the missing frequencies and the mean of all 100 students. |
Difference and importance
| Basis | Central tendency | Dispersion |
|---|---|---|
| Meaning | A typical or central value of the data | The spread of the values about the centre |
| Examples | Mean, median, mode | Range, SD, variance, mean deviation |
| Importance | Summarises the whole data by one value; allows comparison of groups | Shows reliability of the average, consistency and risk |
| Limitation | Hides how scattered the data is | Does not tell the location of data |
Both are needed: two sets can have the same mean but different spread.
Missing frequencies
Let the frequency of 20-40 be x and of 60-80 be y. The total is 100, so
14 + x + 27 + y + 15 = 100 → x + y = 44
Mode = 58 lies in the class 40-60 (L = 40, h = 20, f₁ = 27, f₀ = x, f₂ = y).
Mode = L + (f1 − f0) / (2f1 − f0 − f2) × h
58 = 40 + (27 − x) / (54 − x − y) × 20
Since x + y = 44, the denominator is 54 − 44 = 10:
18 = (27 − x)/10 × 20 = 2(27 − x)
27 − x = 9 → x = 18, y = 44 − 18 = 26
Mean
| Marks | Mid x | f | fx |
|---|---|---|---|
| 0-20 | 10 | 14 | 140 |
| 20-40 | 30 | 18 | 540 |
| 40-60 | 50 | 27 | 1350 |
| 60-80 | 70 | 26 | 1820 |
| 80-100 | 90 | 15 | 1350 |
| Total | 100 | 5200 |
Mean = 5200 / 100 = 52
Answer: the missing frequencies are 18 (for 20-40) and 26 (for 60-80); the mean mark is 52.
✓ Answer checked by Claude Sonnet 5.5
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