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Question
The monthly incomes of 8 families in rupees in a certain locality are given below. Calculate the mean, the geometric mean and the harmonic mean and confirm that the relations among them holds true. Verify their relationships among averages.
| Family: | A | B | C | D | E | F | G | H |
| Income (Rs.): | 70 | 10 | 50 | 75 | 8 | 25 | 8 | 42 |
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Solution
| Family | Income (₹) (x) |
log x | `1/"x"` |
| A | 70 | 1.8451 | 0.0143 |
| B | 10 | 1.000 | 0.1 |
| C | 50 | 1.6990 | 0.02 |
| D | 75 | 1.875 | 0.0133 |
| E | 8 | 0.9031 | 0.125 |
| F | 25 | 1.3979 | 0.04 |
| G | 8 | 0.9031 | 0.125 |
| H | 42 | 1.6232 | 0.0239 |
| 288 | 11.2465 | 0.4615 |
(i) Arithimetric Mean (AM) = `(sum "x")/"n" = 288/8` = 36
(ii) Geometric Mean (GM) = Antilog `((sum "log x")/"n")`
= Antilog `((11.2465)/8)`
= Antilog (1.4058)
= 25.46
(iii) Harmonic Mean (HM) = `"n"/(sum (1/"x"))`
= `8/0.4615`
= 17.33
Thus, 36 > 25.46 > 17.33
∴ AM > GM > HM
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