Mean, median, MAD, IQR Knowledge-Based Study

Study notes

1. Understanding the Center of Data

   When analyzing a numerical dataset, a measure of center gives a single representative value that describes the middle or typical value of the distribution. The two primary measures of center are the mean and the median.

   1. The Mean

   The mean (often called the arithmetic average) is found by summing all values in the dataset and dividing by the total number of values.

   For a dataset of $n$ values $x_1, x_2, \dots, x_n$:

   $$\text{Mean} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + \dots + x_n}{n}$$

   Example: Find the mean of the scores: 4, 7, 8, 9, 12.

   $$\text{Mean} = \frac{4 + 7 + 8 + 9 + 12}{5} = \frac{40}{5} = 8$$

   2. The Median

   The median is the middle value when the data points are arranged in ascending order. It splits the dataset into two equal halves.

   • Odd number of observations ($n$): The median is the unique value at position $\frac{n+1}{2}$.
   • Even number of observations ($n$): The median is the average of the two middle values at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$.

   Example with even $n$: Find the median of 3, 5, 8, 10, 14, 18.

   Here, $n = 6$. The two central values are 8 and 10. Thus, $\text{Median} = \frac{8 + 10}{2} = 9$.

   Quantifying Spread: The Interquartile Range

   While measures of center indicate where values cluster, measures of variability (or spread) indicate how spread out the values are. One robust measure of spread is the interquartile range (IQR).

   Quartiles

   Quartiles divide an ordered dataset into four equal quarters:

   • First Quartile ($Q_1$): The median of the lower half of the data (representing the 25th percentile).
   • Second Quartile ($Q_2$): The median of the entire dataset (50th percentile).
   • Third Quartile ($Q_3$): The median of the upper half of the data (representing the 75th percentile).

   Note: When the total number of values $n$ is odd, the overall median ($Q_2$) is traditionally excluded when finding the lower and upper halves.

   Calculating the IQR

   The interquartile range measures the spread of the middle 50% of the data:

   $$\text{IQR} = Q_3 - Q_1$$

   Step-by-step Example:

   Consider the ordered dataset: 2, 4, 5, 7, 9, 11, 15.

   1. Find $Q_2$ (median): The middle value is 7.
   2. Lower half: 2, 4, 5. The median of this half is $Q_1 = 4$.
   3. Upper half: 9, 11, 15. The median of this half is $Q_3 = 11$.
   4. Compute IQR: $\text{IQR} = 11 - 4 = 7$.

   Because the IQR relies only on the middle 50% of data, it is not distorted by extreme values (outliers).

   Mean Absolute Deviation (MAD)

   The Mean Absolute Deviation (MAD) is another fundamental measure of variability. It represents the average distance between each data point and the dataset's mean.

   Formula for MAD

   For a dataset with values $x_1, x_2, \dots, x_n$ and mean ar{x}:

   \text{MAD} = \frac{\sum |x_i - ar{x}|}{n} = \frac{|x_1 - ar{x}| + |x_2 - ar{x}| + \dots + |x_n - ar{x}|}{n}

   Calculation Procedure

   1. Calculate the mean (ar{x}) of the dataset.
   2. Find the distance of each value from the mean by taking the absolute value of the difference: |x_i - ar{x}|.
   3. Find the mean of these absolute differences by adding them together and dividing by $n$.

   Example: Find the MAD for 3, 6, 6, 7, 8, 12.

   1. Calculate the mean: ar{x} = \frac{3 + 6 + 6 + 7 + 8 + 12}{6} = \frac{42}{6} = 7
   2. Find absolute deviations from 7:
      |3 - 7| = 4
      |6 - 7| = 1
      |6 - 7| = 1
      |7 - 7| = 0
      |8 - 7| = 1
      |12 - 7| = 5
   3. Average the deviations: $$\text{MAD} = \frac{4 + 1 + 1 + 0 + 1 + 5}{6} = \frac{12}{6} = 2$$

   A MAD of 2 means that, on average, a data value lies 2 units away from the mean.

   Comparing Center and Spread

   Choosing whether to use the mean and MAD or the median and IQR depends on the shape of the data distribution and the presence of extreme values (outliers).

   Symmetric vs. Skewed Distributions

   • Symmetric Distributions (without outliers):
     The mean and median are roughly equal. In this scenario, the mean is the preferred measure of center because it utilizes every value in the dataset, and the MAD is paired with it as the measure of variability.
   • Skewed Distributions or Datasets with Outliers:
     The mean is sensitive to extreme values, pulling the center toward the tail of the distribution. The median, however, remains unchanged by extreme values. Therefore, the median is the preferred measure of center, paired with the IQR as the measure of spread.

   Summary Table

   • Symmetric data: Report Mean and MAD.
   • Skewed data / Outliers: Report Median and IQR.