Statistics - Measures of Central Tendency (Mean and Median of ungrouped data)

Central Tendency Overview: It is a single value that represents the center or typical value of a data set. On a horizontal number line where data points are plotted, central tendency identifies the 'balancing point' or the area where the density of points is highest.

The Arithmetic Mean: The mean is the sum of all values divided by the total number of observations. Visually, if you represent each data point as a weight on a beam, the mean is the exact fulcrum point where the beam would stay perfectly level.

Mean of Discrete Frequency Distribution: When data is presented in a table with frequencies (ungrouped), the mean is calculated by multiplying each value ($x$) by its frequency ($f$), summing these products, and dividing by the total frequency. This accounts for 'heavier' points that appear more often.

The Median: The median is the middle-most value when data is arranged in order of magnitude. If you visualize the data as a set of bars of increasing height, the median is the height of the bar that splits the set into two equal halves.

Ordering for Median: Unlike the mean, calculating the median requires the data to be sorted in ascending or descending order. This creates a 'gradient' of values where the median sits at the center of the rank.

Impact of Outliers: The mean is sensitive to extreme values (outliers), whereas the median is robust. In a dot plot, a single point shifted far to the right will pull the mean toward it, but the median value will likely remain at the same position or move only slightly.

Effect of Constant Changes: If every value in a data set is increased or decreased by the same constant $k$, the mean also increases or decreases by $k$. Visually, this is equivalent to sliding the entire cluster of data points along the x-axis without changing their relative spacing.

Median Position for Even vs. Odd: For an odd number of data points, the median is a specific physical data point in the center. For an even number, there is a 'gap' in the center, and the median is the numerical midpoint between the two middle-most values.