Variance of 1,3,5,7,9

The Correct Answer and Explanation is :

To calculate the variance of the dataset (1, 3, 5, 7, 9), follow these steps:

Step 1: Calculate the Mean

The mean (average) of the dataset is calculated as follows:

[
\text{Mean} = \frac{\sum x_i}{n} = \frac{1 + 3 + 5 + 7 + 9}{5} = \frac{25}{5} = 5
]

Step 2: Calculate Each Deviation from the Mean

Next, we find the deviation of each number from the mean:

For 1: (1 – 5 = -4)
For 3: (3 – 5 = -2)
For 5: (5 – 5 = 0)
For 7: (7 – 5 = 2)
For 9: (9 – 5 = 4)

Step 3: Square Each Deviation

Now, square each deviation:

For 1: ((-4)^2 = 16)
For 3: ((-2)^2 = 4)
For 5: (0^2 = 0)
For 7: (2^2 = 4)
For 9: (4^2 = 16)

Step 4: Calculate the Variance

The variance is the average of these squared deviations.

[
\text{Variance} = \frac{\sum (\text{deviation}^2)}{n} = \frac{16 + 4 + 0 + 4 + 16}{5} = \frac{40}{5} = 8
]

Conclusion

Thus, the variance of the dataset (1, 3, 5, 7, 9) is (8).

Explanation of Variance

Variance measures the degree of spread in a set of data points. It quantifies how much the numbers in a dataset deviate from the mean. A higher variance indicates that the data points are more spread out, while a lower variance signifies that they are closer to the mean.

In our example, the dataset consists of odd integers centered around the mean of 5. The variance of (8) indicates that the values tend to vary, but since they are evenly distributed around the mean, this value reflects a moderate spread. Understanding variance is crucial in statistics, as it informs us about the consistency and reliability of the data, particularly in fields such as finance, research, and quality control. By calculating variance, we gain insights into the behavior of datasets, aiding in decision-making and predictions.