For the following distribution, calculate first four central moments using recurrence
relations. Also find the coefficients of skewness and kurtosis.
| Marks: | 2.7- 7.5 | 7.5- 12.5 | 12.5- 17.5 | 17.5- 22.5 | 22.5- 27.5 | 27.5- 32.5 | 32.5- 37.5 |
| Frequency: | 06 | 13 | 23 | 39 | 19 | 15 | 05 |
To calculate the first four central moments using recurrence relations, you can follow these steps:
Step 1: Calculate the mean (µ) and the total number of data points (N).
Mean (µ) = Σ [Midpoint × Frequency] / N
N = Σ Frequency
Given the data:
Midpoints: 5.1, 10, 15, 20, 25, 30, 35
Frequencies: 6, 13, 23, 39, 19, 15, 5
Calculate N:
N = 6 + 13 + 23 + 39 + 19 + 15 + 5 = 120
Now, calculate the mean (µ):
µ = (5.1*6 + 10*13 + 15*23 + 20*39 + 25*19 + 30*15 + 35*5) / 120
µ ≈ 18.575
Step 2: Calculate the central moments.
Central Moment of order 1 (µ1) = Σ [(Midpoint - µ)^1 * Frequency] /
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