Using two examples show how to calculate kurtosis of a distraction curve?

To calculate the kurtosis of a distribution curve, you need to have a dataset or a frequency distribution. Here are two examples:

Example 1: Dataset
Suppose you have the following dataset representing the heights (in inches) of a group of individuals: 65, 68, 70, 67, 69, 66, 68, 72, 71, 70. To calculate the kurtosis, follow these steps:

1. Calculate the mean (μ) and standard deviation (σ) of the dataset.
Mean (μ) = (65 + 68 + 70 + 67 + 69 + 66 + 68 + 72 + 71 + 70) / 10 = 68.6
Standard Deviation (σ) = √[((65-68.6)² + (68-68.6)² + ... + (70-68.6)²) / 10] = 2.302

2. Calculate the fourth central moment (M4) of the dataset.
M4 = [((65-68.6)⁴ + (68-68.6)⁴ + ... + (70-68.6)⁴) / 10] = 14.24

3. Calculate the kurtosis (K) using the formula:
K = M4 / (σ⁴) - 3 = 14.24 / (2.302⁴) - 3 = -0.34

The kurtosis of the height distribution is -0.34, indicating a platykurtic curve (less peaked and with lighter tails compared to a normal distribution).

Example 2: Frequency Distribution
Suppose you have the following frequency distribution table representing the number of hours spent by students on a particular activity:

| Hours Spent | Frequency |
|-------------|-----------|
| 0-2 | 5 |
| 2-4 | 10 |
| 4-6 | 15 |
| 6-8 | 20 |
| 8-10 | 10 |

To calculate the kurtosis, follow these steps:

1. Calculate the mean (μ) and standard deviation (σ) of the distribution.
Mean (μ) = [(1 * 5) + (3 * 10) + (5 * 15) + (7 * 20) + (9 * 10)] / 60 = 6.5
Standard Deviation (σ) = √[((1-6.5)² * 5 + (3-6.5)² * 10 + ... + (9-6.5)² * 10) / 60] = 2.872

2. Calculate the fourth central moment (M4) of the distribution.
M4 = [((1-6.5)⁴ * 5 + (3-6.5)⁴ * 10 + ... + (9-6.5)⁴ * 10) / 60] = 43.75

3. Calculate the kurtosis (K) using the formula:
K = M4 / (σ⁴) - 3 = 43.75 / (2.872⁴) - 3 = -0.34

The kurtosis of the distribution is -0.34, indicating a platykurtic curve (less peaked and with lighter tails compared to a normal distribution).