Mean, Median, and Mode: Definitions
Mean:
The mean (average) of a set of numbers is the sum of all the values divided by the number of values.
$$ \text{Mean} = \frac{x_1 + x_2 + \cdots + x_n}{n} $$
Median:
The median is the middle value when the data are arranged in order. If there is an even number of values, the median is the average of the two middle numbers.
Mode:
The mode is the value that appears most frequently in the data set.
Worked Example
Suppose you have the following data set:
$$ {3,, 7,, 7,, 2,, 9} $$
Step 1: Arrange the data in order
$$ 2,, 3,, 7,, 7,, 9 $$
Step 2: Find the Mean
$$ \text{Mean} = \frac{2 + 3 + 7 + 7 + 9}{5} = \frac{28}{5} = 5.6 $$
Step 3: Find the Median
There are 5 numbers (odd), so the median is the middle value:
$$ \text{Median} = 7 $$
Step 4: Find the Mode
The number 7 appears twice, more than any other value.
$$ \text{Mode} = 7 $$
Takeaways
- Mean is the arithmetic average: add all values, divide by the count.
- Median is the middle value when data are ordered.
- Mode is the most frequently occurring value in the set.