Let's explore how to calculate the mean (average), median (middle value), and mode (most frequent value) of a given data set. These are fundamental measures of central tendency in statistics.
\( A = \{1, 2, 2, 2, 5, 8, 8, 9\} \)
\[ \bar{x} = \frac{1 + 2 + 2 + 2 + 5 + 8 + 8 + 9}{8} = \frac{37}{8} = 4.625 \]
The mean is the sum of all values divided by the total number of values.
Since the set has 8 numbers (even), the median is the average of the 4th and 5th values after sorting:
\[ x_{\text{median}} = \frac{x_4 + x_5}{2} = \frac{2 + 5}{2} = 3.5 \]
Count the frequencies of each number:
\[ \text{Frequencies: } F(2) = 3,\quad F(8) = 2,\quad F(1) = F(5) = F(9) = 1 \Rightarrow x_{\text{mode}} = 2 \]
The mode is the value that appears most frequently — here, it's 2.