For the variable x, taking the values 0,1,2, …, k, the cumulative frequencies of more-than type

step 1 Okay, so we are giving a sample size of 20, and then we have a couple observed frequencies that

For the variable x, taking the values 0,1,2, …, k, the...

For the variable $x$, taking the values $0,1,2, \ldots, k$, the cumulative frequencies of more-than type are $F_{0}, F_{1}, F_{2}, \ldots, F_{k}$. Show that $\bar{x}=\sum_{i=1}^{k} F_{i} \mid n$, where $n$ is the total frequency.

Key Concepts

Frequency Distribution

A frequency distribution is a summary of how many times each value of a variable occurs within a dataset. It is critical in statistics as it organizes the data in a structured way, allowing for the computation of various statistical measures such as the mean, median, and mode. Understanding frequency distributions lays the groundwork for exploring more complex concepts like cumulative frequencies and weighted sums.

Cumulative Frequency

Cumulative frequency refers to the accumulation of frequencies up to a certain point in the dataset. In the more-than type cumulative frequency, each cumulative value represents the total frequency of observations greater than or equal to a certain value of the variable. This concept helps in visualizing and calculating percentiles and medians, and in the problem context, it is used to reframe the calculation of the sample mean.

Sample Mean and Weighted Sum Representation

The sample mean is the average value of a variable, calculated by summing the products of each value and its corresponding frequency, and then dividing by the total number of observations. In the problem, the sample mean is expressed as a weighted sum where the weights are the cumulative 'more-than' frequencies. This alternative representation leverages the cumulative frequencies to simplify the mean's computation, particularly in distributions where data is provided in a cumulative manner.

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