Question 1178024
Let's solve this problem step-by-step.

**1. Sturge's Approximation Rule**

Sturge's rule helps determine the number of classes (k) for a frequency distribution:

* k = 1 + 3.322 * log10(n)

Where n is the number of data points (n = 80 in this case).

* k = 1 + 3.322 * log10(80)
* k = 1 + 3.322 * 1.9031
* k ≈ 1 + 6.322
* k ≈ 7.322

We round k to the nearest whole number, so k = 7.

**2. Range and Class Width**

* **Minimum Value:** 53
* **Maximum Value:** 97
* **Range:** 97 - 53 = 44

* **Class Width (w):** Range / k = 44 / 7 ≈ 6.286

We round the class width up to the nearest convenient whole number, so w = 7.

**3. Frequency Distribution Table**

| Class Interval | Class Midpoint (x) | Frequency (f) | fx | f(x-mean)^2 |
|----------------|--------------------|---------------|----|-------------|
| 53 - 59        | 56                 | 3             | 168 | 2755.07     |
| 60 - 66        | 63                 | 11            | 693 | 1968.64     |
| 67 - 73        | 70                 | 13            | 910 | 258.91      |
| 74 - 80        | 77                 | 22            | 1694 | 2.64        |
| 81 - 87        | 84                 | 8             | 672 | 1146.64     |
| 88 - 94        | 91                 | 12            | 1092 | 2673.91     |
| 95 - 101       | 98                 | 11            | 1078 | 4991.64     |
| **Total** |                    | **80** | **6307** | **13828.05** |

**4. Calculations**

* **Mean (x̄):** Σfx / n = 6307 / 80 ≈ 78.8375

* **Standard Deviation (s):**
    * s = √[Σf(x - x̄)² / (n - 1)]
    * s = √[13828.05 / 79]
    * s = √175.0386 ≈ 13.23

**5. Skewness and Kurtosis**

For this, we'll need to calculate the third and fourth moments.

* **Third Moment (m3):** Σf(x - x̄)³ / n
* **Fourth Moment (m4):** Σf(x - x̄)⁴ / n

Let's add those columns to our table.

| Class Interval | Class Midpoint (x) | Frequency (f) | fx | f(x-mean)^2 | f(x-mean)^3 | f(x-mean)^4 |
|----------------|--------------------|---------------|----|-------------|-------------|-------------|
| 53 - 59        | 56                 | 3             | 168 | 2755.07     | -13636.57    | 674482.16   |
| 60 - 66        | 63                 | 11            | 693 | 1968.64     | -6922.82     | 243288.58   |
| 67 - 73        | 70                 | 13            | 910 | 258.91      | -414.07      | 6625.16     |
| 74 - 80        | 77                 | 22            | 1694 | 2.64        | -0.82        | 0.25        |
| 81 - 87        | 84                 | 8             | 672 | 1146.64     | 3833.18      | 128362.43   |
| 88 - 94        | 91                 | 12            | 1092 | 2673.91     | 13783.50     | 710892.05   |
| 95 - 101       | 98                 | 11            | 1078 | 4991.64     | 33504.60     | 2252110.82  |
| **Total** |                    | **80** | **6307** | **13828.05** | **40567.00** | **3336361.45** |

* **m3:** 40567 / 80 ≈ 507.0875
* **m4:** 3336361.45 / 80 ≈ 41704.5181

* **Skewness (g1):** m3 / s³ = 507.0875 / 13.23³ ≈ 507.0875 / 2315.68 ≈ 0.219
* **Kurtosis (g2):** m4 / s⁴ - 3 = 41704.5181 / 13.23⁴ - 3 ≈ 41704.5181 / 30638.15 - 3 ≈ 1.361 - 3 ≈ -1.639

**Results**

1.  **Frequency Distribution Table:** As shown above.
2.  **Calculations:**
    * **Mean:** ≈ 78.84
    * **Standard Deviation:** ≈ 13.23
    * **Skewness:** ≈ 0.219 (slightly positive skew)
    * **Kurtosis:** ≈ -1.639 (platykurtic, flatter than normal)