- 2076 Asoj · 6 marks
Central tendency; middle 70% income, median
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.
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 |
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)
P15 = 600 + (180 − 150)/300 × 100 = 610
P₈₅: position 0.85 × 1200 = 1020, class 800-900 (cf before = 950)
P85 = 800 + (1020 − 950)/200 × 100 = 835
Median: position 600, class 700-800 (cf before = 450)
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 |
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.
✓ Answer checked by Claude Sonnet 5.5
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