Summarizing Data Numerically: Measures of Center

The Median

• Definition: The median is identified as the $50\text{th}$ percentile of a data set.

• Calculation Method:

  • Arrange observations from smallest to largest.

  • If the number of observations ($n$) is odd, the median is the middle observation.

  • If the number of observations ($n$) is even, the median is the average of the two middle observations.

• Notation:

  • Sample Median: $\tilde{x}$

  • Population Median: $M$

• Properties: The median is resistant to outliers. In an example of personal savings, an outlier of $914$ thousand dollars did not change the median from $8$ thousand dollars.

Summation Notation and The Mean

• Sigma Notation ($\sum$): Used to denote the sum of observations. For example, $\sum x_i$ represents the sum of all values in a data set.

• Sample Mean (Arithmetic Mean): Calculated by dividing the sum of all observations by the total number of observations in the sample.

  • Formula: $\bar{x} = \frac{\sum x_i}{n}$

• Population Mean ($\mu$): Calculated by dividing the sum of all observations by the population size ($N$).

  • Formula: $\mu = \frac{\sum x_i}{N}$

• Properties: The mean is sensitive to outliers. In the savings example, an outlier of $914$ thousand dollars caused the mean to increase from $8.2$ thousand to $188.2$ thousand dollars.

Comparing Measures of Center by Distribution Shape

• Symmetric Distribution: The mean is approximately equal to the median ($\bar{x} \approx \tilde{x}$). Both are considered reasonable measures of center and represent "typical" values.

• Skewed Left: The mean is usually less than the median (\bar{x} < \tilde{x}). The median is generally the better measure of center.

• Skewed Right: The mean is usually greater than the median (\bar{x} > \tilde{x}). The median is generally the better measure of center.

• Bimodal Distribution: Neither the mean nor the median may adequately describe a "typical" value, as seen in the distribution of days served by United States presidents. Stratifying the data into distinct groups (one-term vs. two-term) may provide more useful analysis.

The Mode

• Definition: The observation that occurs with the greatest frequency.

• Variations:

  • Data sets can have more than one mode.

  • If all observations occur only once (frequency of $1$), there is no mode.

• Applications: The mode can be used for categorical data, such as identifying the United States as the most frequent winner of the women's Olympic $100\text{-meter}$ run from $1984$ to $2016$.