Given the frequency distribution of ages, calculate the 1st quartile, 2nd quartile, 3rd quartile, 4th decile, and 5th decile.

To compute the quartiles (Q_1, Q_2, Q_3) and deciles (D_4, D_5), we first need to determine the class boundaries and cumulative frequencies. The given data is: | Age (Class Interval) | Frequency (f) | |:---------------------|:--------------| | 0-9 | 19 | | 10-19 | 47 | | 20-29 | 105 | | 30-39 | 93 | | 40-49 | 16 | The class width (h) is 10 - 0 = 10 (or 19.5 - 9.5 = 10). The total frequency (N) is 19 + 47 + 105 + 93 + 16 = 280. First, we construct the cumulative frequency table: | Class Interval | Class Boundaries | Frequency (f) | Cumulative Frequency (cf) | |:---------------|:-----------------|:--------------|:--------------------------| | 0-9 | -0.5 - 9.5 | 19 | 19 | | 10-19 | 9.5 - 19.5 | 47 | 19 + 47 = 66 | | 20-29 | 19.5 - 29.5 | 105 | 66 + 105 = 171 | | 30-39 | 29.5 - 39.5 | 93 | 171 + 93 = 264 | | 40-49 | 39.5 - 49.5 | 16 | 264 + 16 = 280 | The general formula for quartiles (Q_j) and deciles (D_j) for grouped data is: P_k = L + ((kN)/(X) - cf_bf_p)h where: • P_k is the k-th percentile (or quartile/decile). • L is the lower class boundary of the class containing P_k. • N is the total frequency. • X is 4 for quartiles, 10 for deciles. • cf_b is the cumulative frequency of the class before the class containing P_k. • f_p is the frequency of the class containing P_k. • h is the class width. Step 1: Calculate Q_1 (First Quartile) The position of Q_1 is (1N)/(4) = (1 × 280)/(4) = 70. The 70th value falls in the 20-29 class (since cf=66 for 10-19 and cf=171 for 20-29). For Q_1: L = 19.5, cf_b = 66, f_Q = 105, h = 10. Q_1 = 19.5 + ((70 - 66)/(105))10 Q_1 = 19.5 + ((4)/(105))10 Q_1 = 19.5 + (40)/(105) Q_1 = 19.5 + 0.38095 Q_1 = 19.881 Step 2: Calculate Q_2 (Second Quartile / Median) The position of Q_2 is (2N)/(4) = (280)/(2) = 140. The 140th value falls in the 20-29 class. For Q_2: L = 19.5, cf_b = 66, f_Q = 105, h = 10. Q_2 = 19.5 + ((140 - 66)/(105))10 Q_2 = 19.5 + ((74)/(105))10 Q_2 = 19.5 + (740)/(105) Q_2 = 19.5 + 7.04762 Q_2 = 26.548 Step 3: Calculate Q_3 (Third Quartile) The position of Q_3 is (3N)/(4) = (3 × 280)/(4) = 210. The 210th value falls in the 30-39 class (since cf=171 for 20-29 and cf=264 for 30-39). For Q_3: L = 29.5, cf_b = 171, f_Q = 93, h = 10. Q_3 = 29.5 + ((210 - 171)/(93))10 Q_3 = 29.5 + ((39)/(93))10 Q_3 = 29.5 + (390)/(93) Q_3 = 29.5 + 4.19355 Q_3 = 33.694 Step 4: Calculate D_4 (Fourth Decile) The position of D_4 is (4N)/(10) = (4 × 280)/(10) = 112. The 112th value falls in the 20-29 class. For D_4: L = 19.5, cf_b = 66, f_D = 105, h = 10. D_4 = 19.5 + ((112 - 66)/(105))10 D_4 = 19.5 + ((46)/(105))10 D_4 = 19.5 + (460)/(105) D_4 = 19.5 + 4.38095 D_4 = 23.881 Step 5: Calculate D_5 (Fifth Decile) The position of D_5 is (5N)/(10) = (5 × 280)/(10) = 140. This is the same position as Q_2. The 140th value falls in the 20-29 class. For D_5: L = 19.5, cf_b = 66, f_D = 105, h = 10. D_5 = 19.5 + (140 - 661