Calculating Standard Deviation for Frequency Data

To find the standard deviation for the given frequency distribution, we follow these steps:

1. Calculate the mean ($\bar{x}$).
2. Calculate the variance ($\sigma^2$).
3. Calculate the standard deviation ($\sigma$).

Step 1: Calculate the Mean ($\bar{x}$)

The formula for the mean of a frequency distribution is:

$ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} $

First, we create a table to calculate $\sum f_i$ and $\sum f_i x_i$:

Now, substitute the sums into the formula:

$ \bar{x} = \frac{614}{48} \approx 12.7917 $

Step 2: Calculate the Variance ($\sigma^2$)

We use the formula:

$ \sigma^2 = \frac{\sum f_i x_i^2}{\sum f_i} - (\bar{x})^2 $

Calculate $f_i x_i^2$:

Substitute the values:

$ \sigma^2 = \frac{9652}{48} - (12.7917)^2 $

$ \sigma^2 \approx 201.0833 - 163.6285 $

$ \sigma^2 \approx 37.4548 $

Step 3: Calculate the Standard Deviation ($\sigma$)

The standard deviation is the square root of the variance:

$ \sigma = \sqrt{\sigma^2} $

$ \sigma = \sqrt{37.4548} \approx 6.12 $

Therefore, the standard deviation for the given data is approximately 6.12.