Tenth Grade Mathematics: Empirical Relationship Between Mean, Median, and Mode

The three primary measures of central tendency—the mean, the median, and the mode—each capture a different aspect of a data distribution. In a perfectly symmetric distribution (such as the classical bell-shaped normal curve), all three measures coincide at the exact same numerical value. However, real-world data is almost invariably skewed, with frequencies trailing off more gradually on one side than the other. In moderately skewed unimodal distributions, a consistent structural relationship binds the three measures together, known as Karl Pearson's Empirical Relationship.

Symmetry and Skewness in Distributions

A continuous frequency distribution can exhibit three fundamental structural profiles:

1. Perfectly Symmetric Distribution (Zero Skewness):
   The frequency curve is an identical mirror image about its central vertical axis.
   The balance point (mean), the central observation (median), and the highest peak
   (mode) all coincide at the exact same location:
                         Mean = Median = Mode

2. Positively Skewed Distribution (Right-Skewed):
   The distribution has a long tail extending toward high positive values.
   Extreme high values pull the mean to the right. The mode remains at the peak:
                         Mode < Median < Mean

3. Negatively Skewed Distribution (Left-Skewed):
   The distribution has a long tail extending toward low negative values.
   Extreme low values pull the mean to the left:
                         Mean < Median < Mode

Visualizing Skewness and Relative Positions:

   Symmetric                Positively Skewed               Negatively Skewed
      /\                            /\                             /     /  \                          /  \                           /      /    \                        /    \                         /       /      \                      /      \____                   /        +---+----+                    +---+---+----+--               +---+----+----+
  Mean=Med=Mode                 Mode Med Mean                  Mean Med Mode

Notice that in both skewed distributions, the median ALWAYS lies between the mode and the mean!

Karl Pearson's Empirical Formula

For moderately skewed distributions, the great English statistician Karl Pearson observed that the median divides the distance between the mode and the mean approximately in a 2 : 1 ratio:

Geometric Alignment along the Number Line:
                Mode                       Median               Mean
                 +---------------------------+-------------------+
                 <--------- 2 units --------><----- 1 unit ------>
                 <-------------------- 3 units ------------------>

Distance from Mode to Mean   = 3 * (Distance from Median to Mean)
Mean - Mode                  = 3 * (Mean - Median)

Expanding and rearranging this equation:

Mean - Mode = 3*Mean - 3*Median
-Mode       = 3*Mean - Mean - 3*Median
-Mode       = 2*Mean - 3*Median
Multiply by -1:

Theorem (Karl Pearson's Empirical Relationship):
                         Mode = 3 * Median - 2 * Mean

Equivalent Transpositions:
1. 3 * Median = Mode + 2 * Mean
2. Median     = (Mode + 2 * Mean) / 3
3. Mean       = (3 * Median - Mode) / 2

This empirical relationship allows any one of the three central tendencies to be estimated if the other two are known.

Determining When to Use Each Measure of Central Tendency

+-------------------+-----------------------------------+------------------------------------+
| Measure           | Optimal Real-World Use Case       | Primary Limitation                 |
+-------------------+-----------------------------------+------------------------------------+
| Mean (x̄)          | Symmetric data, physical          | Highly distorted by extreme        |
|                   | measurements, engineering tests   | outliers and skewed data           |
+-------------------+-----------------------------------+------------------------------------+
| Median            | Skewed economic data, salaries,   | Does not use exact numerical       |
|                   | real estate, examination ranks    | values of non-central observations |
+-------------------+-----------------------------------+------------------------------------+
| Mode              | Retail inventory, shoe sizes,     | May not exist, or multiple modes   |
|                   | popular voting, peak usage hours  | may exist (bimodal/multimodal)     |
+-------------------+-----------------------------------+------------------------------------+

Chapter Exercises

Exercise 1: In a moderately skewed frequency distribution, the values of the mean and median are 28.5 and 27.9 respectively. Use the empirical relationship to calculate the mode of the distribution.

Exercise 2: The mode of a dataset is 24 and its mean is 60. Find the median using Pearson's empirical formula.

Exercise 3: If for a moderately skewed distribution, the difference between the mode and the mean is 12, find the difference between the median and the mean.

Solutions and Explanations

Solution 1: We are given Mean = 28.5 and Median = 27.9. Karl Pearson's empirical formula states: Mode = 3 * Median - 2 * Mean. Substituting the values: Mode = 3 * (27.9) - 2 * (28.5) = 83.7 - 57.0 = 26.7. Thus, the estimated mode of the distribution is 26.7.

Solution 2: We are given Mode = 24 and Mean = 60. Using the empirical relationship: 3 * Median = Mode + 2 * Mean. Substituting the given numbers: 3 * Median = 24 + 2 * (60) = 24 + 120 = 144. Solving for the median: Median = 144 / 3 = 48. Thus, the median of the dataset is 48.

Solution 3: We start from the structural form of the empirical relationship: (Mean - Mode) = 3 * (Mean - Median). We are given that the difference between the mode and the mean is 12, so |Mean - Mode| = 12. If Mean - Mode = 12: 12 = 3 * (Mean - Median). Dividing by 3 gives: Mean - Median = 12 / 3 = 4. (If Mode - Mean = 12, then Median - Mean = 4). Therefore, the difference between the median and the mean is 4.