Question

Which of the following is correct for standard deviation, where \(\bar{X}\) = mean 

• A. \[ \sigma = \sqrt{\frac{ \sum_{i=1}^n (X_i - \bar{X})^2 }{n}} \]
• B. \[ \sigma = \sqrt{\frac{\sum_{i=1}^n (X_i - \bar{X})^2 }{n}} \]
• C. \[ \sigma = \sqrt{ \sum_{i=1}^n \left(\frac{(X_i - \bar{X})^2}{n}\right) } \]
• D. \[ \sigma = \sqrt{ \frac{\sum_{i=1}^n X_i - \bar{X}}{n} } \]

Answer 1

The Correct Option is B

To determine the correct expression for the standard deviation, let us first understand the formula for standard deviation itself. The standard deviation \(\sigma\) is defined as the square root of the variance. The variance is the average of the squared differences between each data point \(X_i\) and the mean \(\bar{X}\) of the dataset. The steps are as follows:

1. Compute the mean of the data set: \(\bar{X} = \frac{\sum_{i=1}^n X_i}{n}\).
2. Calculate each squared difference from the mean: \((X_i - \bar{X})^2\).
3. Sum all the squared differences: \(\sum_{i=1}^n (X_i - \bar{X})^2\).
4. Divide the sum by the number of data points to find the variance: \(\frac{\sum_{i=1}^n (X_i - \bar{X})^2}{n}\).
5. The standard deviation is the square root of the variance: \(\sigma = \sqrt{\frac{\sum_{i=1}^n (X_i - \bar{X})^2}{n}}\).

Hence, the correct option for the standard deviation is: \(\sigma = \sqrt{\frac{\sum_{i=1}^n (X_i - \bar{X})^2}{n}}\).