The Law of Large Numbers and the Mean

The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean [latex]\displaystyle\overline{{x}}[/latex] of the sample is very likely to get closer and closer to [latex]µ[/latex]. This is discussed in more detail later in the text.

Sampling Distributions and Statistic of a Sampling Distribution

You can think of a sampling distribution as a relative frequency distribution with a great many samples. Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below.

If you let the number of samples get very large (say, 300 million or more), the relative frequency table becomes a relative frequency distribution.

A statistic is a number calculated from a sample. Statistic examples include the mean, the median, and the mode as well as others. The sample mean [latex]\displaystyle\overline{{x}}[/latex] is an example of a statistic which estimates the population mean [latex]μ[/latex].

Calculating the Mean of Grouped Frequency Tables

When only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean:
[latex]\displaystyle\text{mean}=\frac{{\text{data sum}}}{{\text{number of data values}}}[/latex]. We simply need to modify the definition to fit within the restrictions of a frequency table.

Since we do not know the individual data values we can instead find the midpoint of each interval. The midpoint is [latex]\displaystyle\frac{{\text{lower boundary } + \text{ upper boundary}}}{{2}}[/latex] We can now modify the mean definition to be [latex]\displaystyle\text{Mean of Frequency Table} = \frac{\sum\nolimits{fm}}{\sum\nolimits{f}}[/latex] where [latex]f[/latex] = the frequency of the interval and [latex]m[/latex] = the midpoint of the interval.