Question 1175464: Consider the following grouped frequency distribution, which portrays the duration (in minutes) of 50 lectures
delivered by a Chemistry professor.
Duration (Minutes) Frequency
30.00 < 40.00 3
40.00 < 50.00 7
50.00 < 60.00 18
60.00 < 70.00 13
70.00 < 80.00 8
80.00 < 90.00 1
Using computational formulae, calculate the:
(i) estimated mean.
(ii) mode.
(iii) standard deviation.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's calculate the estimated mean, mode, and standard deviation for this grouped frequency distribution.
**1. Estimated Mean**
* **Midpoints (x_m):** Find the midpoint of each class interval.
* 30-40: 35
* 40-50: 45
* 50-60: 55
* 60-70: 65
* 70-80: 75
* 80-90: 85
* **Multiply Midpoints by Frequencies (f * x_m):**
* 35 * 3 = 105
* 45 * 7 = 315
* 55 * 18 = 990
* 65 * 13 = 845
* 75 * 8 = 600
* 85 * 1 = 85
* **Sum of (f * x_m):** 105 + 315 + 990 + 845 + 600 + 85 = 2940
* **Sum of Frequencies (n):** 3 + 7 + 18 + 13 + 8 + 1 = 50
* **Estimated Mean (x̄):** (Σ f * x_m) / n = 2940 / 50 = 58.8
**(i) Estimated Mean = 58.8 minutes**
**2. Mode**
* **Modal Class:** The class with the highest frequency is 50-60 (frequency = 18).
* **Lower Boundary (L):** 50
* **Class Width (w):** 10
* **Frequency of Modal Class (f_m):** 18
* **Frequency of Class Before Modal Class (f_1):** 7
* **Frequency of Class After Modal Class (f_2):** 13
* **Mode Formula:** L + [(f_m - f_1) / ((f_m - f_1) + (f_m - f_2))] * w
* Mode = 50 + [(18 - 7) / ((18 - 7) + (18 - 13))] * 10
* Mode = 50 + [11 / (11 + 5)] * 10
* Mode = 50 + (11 / 16) * 10
* Mode = 50 + (0.6875) * 10
* Mode = 50 + 6.875
* Mode = 56.875
**(ii) Mode = 56.875 minutes**
**3. Standard Deviation**
* **Calculate (x_m - x̄)²:**
* (35 - 58.8)² = (-23.8)² = 566.44
* (45 - 58.8)² = (-13.8)² = 190.44
* (55 - 58.8)² = (-3.8)² = 14.44
* (65 - 58.8)² = (6.2)² = 38.44
* (75 - 58.8)² = (16.2)² = 262.44
* (85 - 58.8)² = (26.2)² = 686.44
* **Multiply by Frequencies (f * (x_m - x̄)²):**
* 3 * 566.44 = 1699.32
* 7 * 190.44 = 1333.08
* 18 * 14.44 = 259.92
* 13 * 38.44 = 499.72
* 8 * 262.44 = 2099.52
* 1 * 686.44 = 686.44
* **Sum of (f * (x_m - x̄)²):** 1699.32 + 1333.08 + 259.92 + 499.72 + 2099.52 + 686.44 = 6578.00
* **Variance (s²):** (Σ f * (x_m - x̄)²) / (n - 1) = 6578.00 / (50 - 1) = 6578 / 49 = 134.2449
* **Standard Deviation (s):** √Variance = √134.2449 ≈ 11.5864
**(iii) Standard Deviation ≈ 11.59 minutes**
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