Question 1.EX.4: (a) For the data in Exercise 2, calculate the inter-quartile...
(a) For the data in Exercise 2, calculate the inter-quartile range, the variance and the standard deviation.
(b) Calculate the coefficient of variation.
(c) Check if the relationship between the IQR and the standard deviation stated in the text is approximately true for this distribution.
(d) Approximately how much of the distribution lies within one standard deviation either side of the mean? How does this compare with the prediction from Chebyshev’s inequality?
(a) Q1 relates to observation 27.5 (= 110/4). This observation lies in the 11–30 range. There are 20 observations in the first class interval, so Q1 will relate to observation 7.5 in the second interval. Hence we need to go 7.5/40 of the way through the interval. This gives 11 + (7.5/40) × 19 = 14.6. Similarly, Q3 is 22.5/30 of the way through the third interval, yielding Q3 = 31 + 22.5/30 × 29 = 52.8. The IQR is therefore 38, approximately. For the variance we obtain Σfx = 3850 and Σfx² = 205 250. The variance is therefore σ² = 205 250/110 − 35² = 640.9 and the standard deviation 25.3.
(b) CV = 25.3/35 = 0.72.
(c) 1.3 × 25.3 = 32.9, not far from the IQR value of 38.
(d) 1 standard deviation either side of the mean takes us from 9.7 up to 60.3. This contains all 70 observations in the second and third intervals plus perhaps one from the first interval. Thus we obtain approximately 71 observations within this range. Chebyshev’s inequality does not help us here as it is not defined for k ≤ 1.