Ninth Grade Mathematics: Measures of Central Tendency: Median & Mode

While the arithmetic mean is the most common measure of central tendency, it suffers from a major weakness: it is highly sensitive to extreme outliers. The median and mode provide robust alternative perspectives on central clustering. In this chapter, you will master the calculation of the median (the spatial middle value) and the mode (the most frequent value), and explore Pearson's empirical relationship connecting all three measures.

The Median: The Positional Middle Value

The median of a dataset is the value of the middle observation when all data points are arranged in ascending or descending order of magnitude. It divides the distribution into two equal halves: exactly 50% of observations lie below the median, and 50% lie above it.

The Median Algorithm for Raw Data:
Step 1: Arrange the raw data into an ordered array (ascending order).
Step 2: Count the total number of observations, n.

Case 1: If n is ODD:
The median is the unique single middle observation:
               / n + 1      Median = | ------- | -th observation
               \   2   /

Case 2: If n is EVEN:
There are two middle observations at positions (n/2) and (n/2 + 1).
The median is the arithmetic mean of these two middle values:
                [ (n/2)-th value ] + [ (n/2 + 1)-th value ]
     Median =  ---------------------------------------------
                                     2

Example 1 (Odd n = 7):
Data: 3, 5, 7, 8, 12, 14, 19
Position: (7 + 1) / 2 = 4th observation.
Median = 8.

Example 2 (Even n = 8):
Data: 4, 6, 9, 11, 15, 18, 22, 30
Middle positions: 8/2 = 4th (value 11) and 8/2 + 1 = 5th (value 15).
Median = (11 + 15) / 2 = 26 / 2 = 13.

Notice in Example 2 that the median (13) is not even a member of the original dataset; it is the numerical midpoint between the two central values.

The Mode: The Most Frequent Observation

The mode of a dataset is the value that occurs with the greatest frequency (the most popular value).

Classifications of Mode:
- Unimodal:  A distribution with exactly one mode (e.g., 2, 4, 4, 4, 7, 9  ===> Mode = 4).
- Bimodal:   A distribution with two distinct modes having equal top frequency.
- Multimodal:A distribution with three or more peaks of equal maximum frequency.
- No Mode:   When every value in the dataset occurs with the exact same frequency
             (e.g., 3, 5, 8, 11, 14 has NO mode).

The mode is the only measure of central tendency that can be used for qualitative categorical data (for example, finding the most popular shoe size or political candidate).

Comparing Mean, Median, and Mode: Sensitivity to Outliers

An outlier is an extreme observation that differs substantially from the rest of the data. Consider the weekly salaries of five employees: $600, $650, $700, $750, $7,300 (CEO salary).

Statistical Comparison:
Mean:   (600 + 650 + 700 + 750 + 7300) / 5 = 10,000 / 5 = $2,000
Median: Middle value = $700
Mode:   No mode

The arithmetic mean ($2,000) paints a deceptive picture: four out of five workers earn less than half the mean! The outlier ($7,300) heavily dragged the mean upward. The median ($700), by contrast, remains completely unaffected by the extreme value. For skewed distributions (such as household income or real estate prices), the median is vastly superior to the mean as a measure of typical value.

Pearson's Empirical Relationship

For moderately skewed, unimodal distributions, Karl Pearson discovered an empirical relationship connecting the three measures:

Pearson's Empirical Relationship:

Mode  =  3 * Median  -  2 * Mean

Equivalently:
Mean - Mode = 3 * (Mean - Median)

If any two of these measures are known, this empirical formula allows you to estimate the third with remarkable accuracy.

Chapter Exercises

Exercise 1: Find the median of the following set of 10 test scores: 45, 62, 88, 54, 71, 95, 62, 78, 83, 67. Also identify the mode of the dataset.

Exercise 2: The following observations are arranged in ascending order: 11, 13, 15, 19, (x + 2), (x + 4), 30, 35, 39, 43. If the median of the data is 24, find the value of x.

Exercise 3: In a moderately skewed distribution, the mean is 42.5 and the median is 40.0. Use Pearson's empirical formula to estimate the mode of the distribution.

Solutions and Explanations

Solution 1: Step 1: Arrange the 10 scores in ascending order: 45, 54, 62, 62, 67, 71, 78, 83, 88, 95. Step 2: The number of observations is n = 10 (even). The two middle positions are n/2 = 5th observation (67) and n/2 + 1 = 6th observation (71). The median is the average of these two values: Median = (67 + 71) / 2 = 138 / 2 = 69. Step 3: To find the mode, identify the value that occurs with greatest frequency. The score 62 appears twice, while all other scores appear only once. Therefore, the mode is 62.

Solution 2: The dataset contains n = 10 observations already arranged in ascending order. Since n is even, the median is the average of the 5th and 6th observations. Looking at the ordered list: the 5th observation is (x + 2), and the 6th observation is (x + 4). Set up the median equation: [ (x + 2) + (x + 4) ] / 2 = 24. Simplify the numerator: (2*x + 6) / 2 = 24, which gives x + 3 = 24. Subtract 3 from both sides: x = 21. Substituting x = 21 back into the array gives 5th value = 23 and 6th value = 25, whose average is (23 + 25)/2 = 24.

Solution 3: We are given Mean = 42.5 and Median = 40.0. Apply Pearson's empirical relationship: Mode = 3 * Median - 2 * Mean. Substitute the given values: Mode = 3 * (40.0) - 2 * (42.5) = 120.0 - 85.0 = 35.0. The estimated mode of the distribution is 35.0.