The table shows the length of 50 rods. Find: The frequency of the modal class. The actual mean, variance and standard deviation of the lengths. Take assumed mean A = 18.

Question

Asked on April 13, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-13T16:45:38.052Z

a) To find the frequency of the modal class, we identify the class with the highest frequency. From the table: Length (cm) | No. of rods (frequency, f) ------------|----------------------------- 8-10 | 4 11-13 | 7 14-16 | 11 17-19 | 15 20-22 | 8 23-25 | 5 The highest frequency is 15, which corresponds to the class 17-19. The frequency of the modal class is 15. b) To calculate the actual mean, variance, and standard deviation using an assumed mean A = 18, we first find the midpoints (x) of each class and then calculate the deviation d = x - A. Step 1: Calculate midpoints (x) for each class. For 8-10, x = (8+10)/(2) = 9 For 11-13, x = (11+13)/(2) = 12 For 14-16, x = (14+16)/(2) = 15 For 17-19, x = (17+19)/(2) = 18 For 20-22, x = (20+22)/(2) = 21 For 23-25, x = (23+25)/(2) = 24 Step 2: Create a table to calculate f, fd, and fd^2. Let the assumed mean A = 18. | Class | x | f | d = x - 18 | fd | d^2 | fd^2 | |---------|-----|-----|--------------|----------------|-------|----------------| | 8-10 | 9 | 4 | 9 - 18 = -9| 4 × (-9) = -36 | (-9)^2 = 81 | 4 × 81 = 324 | | 11-13 | 12 | 7 | 12 - 18 = -6| 7 × (-6) = -42 | (-6)^2 = 36 | 7 × 36 = 252 | | 14-16 | 15 | 11 | 15 - 18 = -3| 11 × (-3) = -33 | (-3)^2 = 9 | 11 × 9 = 99 | | 17-19 | 18 | 15 | 18 - 18 = 0| 15 × 0 = 0 | 0^2 = 0 | 15 × 0 = 0 | | 20-22 | 21 | 8 | 21 - 18 = 3| 8 × 3 = 24 | 3^2 = 9 | 8 × 9 = 72 | | 23-25 | 24 | 5 | 24 - 18 = 6| 5 × 6 = 30 | 6^2 = 36 | 5 × 36 = 180 | | Total | | f = 50 | | fd = -57 | | fd^2 = 927 | i) Actual mean (x) The formula for the actual mean using an assumed mean is: x = A + ( fd)/( f) Substitute the values: x = 18 + (-57)/(50) x = 18 - 1.14 x = 16.86 The actual mean is 16.86 cm. ii) Variance (^2) The formula for variance using an assumed mean is: ^2 = ( fd^2)/( f) - (( fd)/( f))^2 Substitute the values: ^2 = (927)/(50) - ((-57)/(50))^2 ^2 = 18.54 - (-1.14)^2 ^2 = 18.54 - 1.2996 ^2 = 17.2404 The variance is 17.2404 cm^2. iii) Standard deviation () The formula for standard deviation is the square root of the variance: = sqrt(^2) Substitute the variance: = sqrt(17.2404) ≈ 4.152156 Rounding to three decimal places: ≈ 4.152 The standard deviation is 4.152 cm. Send me the next one 📸

Accepted Answer · 2026-04-13T16:45:38.052Z

a) To find the frequency of the modal class, we identify the class with the highest frequency. From the table: Length (cm) | No. of rods (frequency, f) ------------|----------------------------- 8-10 | 4 11-13 | 7 14-16 | 11 17-19 | 15 20-22 | 8 23-25 | 5 The highest frequency is 15, which corresponds to the class 17-19. The frequency of the modal class is 15. b) To calculate the actual mean, variance, and standard deviation using an assumed mean A = 18, we first find the midpoints (x) of each class and then calculate the deviation d = x - A. Step 1: Calculate midpoints (x) for each class. For 8-10, x = (8+10)/(2) = 9 For 11-13, x = (11+13)/(2) = 12 For 14-16, x = (14+16)/(2) = 15 For 17-19, x = (17+19)/(2) = 18 For 20-22, x = (20+22)/(2) = 21 For 23-25, x = (23+25)/(2) = 24 Step 2: Create a table to calculate f, fd, and fd^2. Let the assumed mean A = 18. | Class | x | f | d = x - 18 | fd | d^2 | fd^2 | |---------|-----|-----|--------------|----------------|-------|----------------| | 8-10 | 9 | 4 | 9 - 18 = -9| 4 × (-9) = -36 | (-9)^2 = 81 | 4 × 81 = 324 | | 11-13 | 12 | 7 | 12 - 18 = -6| 7 × (-6) = -42 | (-6)^2 = 36 | 7 × 36 = 252 | | 14-16 | 15 | 11 | 15 - 18 = -3| 11 × (-3) = -33 | (-3)^2 = 9 | 11 × 9 = 99 | | 17-19 | 18 | 15 | 18 - 18 = 0| 15 × 0 = 0 | 0^2 = 0 | 15 × 0 = 0 | | 20-22 | 21 | 8 | 21 - 18 = 3| 8 × 3 = 24 | 3^2 = 9 | 8 × 9 = 72 | | 23-25 | 24 | 5 | 24 - 18 = 6| 5 × 6 = 30 | 6^2 = 36 | 5 × 36 = 180 | | Total | | f = 50 | | fd = -57 | | fd^2 = 927 | i) Actual mean (x) The formula for the actual mean using an assumed mean is: x = A + ( fd)/( f) Substitute the values: x = 18 + (-57)/(50) x = 18 - 1.14 x = 16.86 The actual mean is 16.86 cm. ii) Variance (^2) The formula for variance using an assumed mean is: ^2 = ( fd^2)/( f) - (( fd)/( f))^2 Substitute the values: ^2 = (927)/(50) - ((-57)/(50))^2 ^2 = 18.54 - (-1.14)^2 ^2 = 18.54 - 1.2996 ^2 = 17.2404 The variance is 17.2404 cm^2. iii) Standard deviation () The formula for standard deviation is the square root of the variance: = sqrt(^2) Substitute the variance: = sqrt(17.2404) ≈ 4.152156 Rounding to three decimal places: ≈ 4.152 The standard deviation is 4.152 cm. Send me the next one 📸

The table shows the length of 50 rods. Find: The frequency of the modal class. The actual mean, variance and standard deviation of the lengths. Take assumed mean A = 18.

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The table shows the length of 50 rods. Find: The frequency of the modal class. The actual mean, variance and standard deviation of the lengths. Take assumed mean A = 18.

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More Mathematics Questions