Mode and Median

Measures of Central Tendency

In statistics, we often want to know where the "center" of our data lies. Mode and Median are two ways to see this center or tendency of the data.

Median: The Middle Value

The median is simply the data value that is exactly in the middle after all data has been sorted from smallest to largest.

How to Find the Median:

• If the Number of Data is Odd (n is odd):

  $$\text{Median} = \text{Value at position } \frac{n+1}{2}$$

  Example (Odd Data):

  Suppose the test scores of 5 students are: 7, 8, 6, 9, 7

  Sort: 6, 7, 7, 8, 9 (there are 5 data points, $n=5$)

  Middle position: $(5+1)/2 = 3$. The data at the 3rd position is 7.

  So, Median = 7.

• If the Number of Data is Even (n is even):

  $$\text{Median} = \frac{\text{Value at position }(n/2) + \text{Value at position }((n/2)+1)}{2}$$

  Example (Even Data):

  Suppose the test scores of 6 students are: 7, 8, 6, 9, 7, 10

  Sort: 6, 7, 7, 8, 9, 10 (there are 6 data points, $n=6$)

  Middle positions: Position $6/2 = 3$ and position $(6/2) + 1 = 4$.

  The two middle numbers are 7 and 8.

  Median = Average of 7 and 8 = $(7+8)/2 = 7.5$.

  So, Median = $7.5$.

The median is useful because it is not heavily affected by extremely large or small values (outliers) at the ends of the data.

Mode: The Most Frequent Value

The mode is even simpler. The mode is the data value that appears most often or has the highest frequency.

How to Find the Mode:

Example:

Test scores data: 7, 8, 6, 9, 7, 8, 7

• Score 6 appears 1 time
• Score 7 appears 3 times
• Score 8 appears 2 times
• Score 9 appears 1 time

The most frequent score is 7 (appears 3 times).

So, Mode = 7.

Important Notes about Mode:

• A dataset might have no mode (if all values appear with the same frequency).
• It can have one mode (unimodal).
• It can have two modes (bimodal), if there are two values with the same highest frequency.
• It can have more than two modes (multimodal).

Like the median, the mode is also not affected by extreme values (outliers).

A Quick Look at Range

Besides looking at the center of the data with mode and median, sometimes we also need to know how spread out our data is. One of the simplest ways is to look at the Range.

The range is the difference between the largest and smallest data values.

Example:

Data: 6, 7, 7, 8, 9, 10

Largest data = 10

Smallest data = 6

Range = $10 - 6 = 4$

The range gives us a quick idea of how far our data spreads from end to end.

So, Median and Mode focus on the "center" of the data, while Range focuses on the "spread" of the data.