Finding the Mean
This section focuses on calculating the mean of grouped data, which is vital for summarizing large datasets. The mean is defined as the total sum of observations divided by the number of observations. For grouped data, finding the mean requires accounting for frequencies. This section elaborates on three main methods for determining the mean:
- Direct Method: This method sums the product of each observation and its frequency, then divides by the total frequency. The formula is given as:
$$x = \frac{\Sigma f x}{\Sigma f}$$
- Assumed Mean Method: This technique involves selecting an assumed mean 'a', calculating deviations from 'a', and adjusting the result based on these deviations using the formula:
$$x = a + \frac{\Sigma fd}{\Sigma f}$$
- Step-Deviation Method: A simplification of the assumed mean method, this method uses a common divisor 'h' (the class width) to recalculate the deviations. The formula is:
$$u = \frac{d}{h}$$ where $$d = x - a$$.
And the mean is then found using:
$$x = a + h\frac{\Sigma fu}{\Sigma f}$$.
These methods are illustrated with practical examples, making it clear how to navigate between them based on the data characteristics. The differences in results from the direct method and the assumed mean method are discussed, stressing the importance of precision in statistical analysis.