Mode and Median

Measures of Central Tendency

Measures of central tendency describe where a data set is centered. The median and mode describe that center in different ways. The median splits the ordered values in half, while the mode reports the value that occurs most often.

The Median as the Middle Value

The median is the middle value of an ordered data set.

How to find the median:

1. Sort the data from smallest to largest.
2. Find the middle position or positions.

• If the number of data points is odd ($n$ is odd):

  The median is the value at position $\frac{n+1}{2}$ in the sorted data.

  Example with an odd number of values:

  Suppose the scores of $5 \text{ students}$ 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 position $3$ is $7$.

  So, $\text{Median} = 7$.

• If the number of data points is even ($n$ is even):

  Write $x_{(i)}$ for the value at position $i$ after sorting the data from smallest to largest. The median is the mean of the values at positions $n/2$ and $n/2+1$:

  $$\text{Median}=\frac{x_{(n/2)}+x_{(n/2+1)}}{2}$$

  Example with an even number of values:

  Suppose the scores of $6 \text{ students}$ 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 is the average of $7$ and $8$: $(7+8)/2 = 7.5$.

  So, the median is $7.5$.

The median depends on the middle position in the ordered data. Extremely large or small values at the ends therefore have little effect on it.

The Mode as the Most Frequent Value

The mode is the value that appears most often, meaning the value with the highest frequency.

How to find the mode:

1. Count how often each value appears.
2. Identify the value or values with the highest frequency.

Example:

Test scores:

• Score $6$ appears $1 \text{ time}$
• Score $7$ appears $3 \text{ times}$
• Score $8$ appears $2 \text{ times}$
• Score $9$ appears $1 \text{ time}$

The most frequent score is $7$ (appears $3 \text{ times}$).

So, the mode is $7$.

Important notes about the mode:

• If every value has the same frequency, no value is uniquely most frequent. Under the convention used here, the dataset is reported as having no mode.
• A dataset can have one mode (unimodal).
• A dataset can have two modes (bimodal), if two values share the highest frequency.
• A dataset can have more than two modes (multimodal).

A single isolated extreme value usually does not change the mode because its frequency is only one.

Range Shows Data Spread

Besides locating the center with the median and mode, we may also want to know how spread out the data is. The range provides one quick measure.

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 shows the distance between the two endpoints, but it does not describe how values are distributed between them.

The median and mode describe the center of a data set, while the range describes its overall width.