How to Calculate the Mean, Median, Mode, and Range

🌟 What is the Mean?

Mean, or Arithmetic Mean, tells you the average value of a set of numbers.

🌟 How Do You Calculate the Mean?

You calculate the mean by dividing the sum of all values by the total number of values:

$$ \text{Mean} = \frac{ \text{Sum of all values}}{\text{Total number of values}} $$

🌟 Example of Finding the Mean:

Imagine your scores in archery are: 5, 7, 6, 3, 5. To find your average score, you can calculate the mean:

• Step 1: Add up all the scores: $$5 + 7 + 6 + 3 + 5 = 26 $$
• Step 2: Divide by the total number of shots

$$ \text{Mean} = \frac{5 + 7 + 6 + 3 + 5}{5} = \frac{26}{5} = 5.2 $$

• This means, on average, your archery score is 5.2.

🌟 What is the Median?

The median is the value in the middle after sorting the numbers from smallest to largest.

🌟 How Do You Calculate the Median?

💡 If the number of values is odd, the middle number is the median.

• • Example: Scores after sorting = 3, 5, 5, 6, 7
  • Here, 5 is in the middle, so Median = 5

💡If the number of values is even, take the average of the two middle numbers.

• • Example: Scores after sorting = 3, 5, 5, 6, 7, 7
  • The middle numbers are 5 and 6. To find their average, add them and divide by 2:

$$ \text{Median} = \frac{5 + 6}{2} = \frac{11}{2} = 5.5 $$

To find the position of the median in an ordered data set, use the formula:

$$ \text{Median Position} = \frac{n + 1}{2} $$

Where $n$ is the total number of values.

This formula is especially useful when there are many numbers in a dataset.

🚨 Be careful: This gives you the position in the list, not the value itself!

🌟 Example of Finding the Median:

Data set (already sorted): 2, 2, 2, 3, 4, 6, 6, 7, 9, 9

• There are 10 numbers, so:

$$ \text{Median Position} = \frac{10 + 1}{2} = \frac{11}{2} = 5.5 $$

• This means the median is halfway between the 5th and 6th values.
• The values at those positions are 4 (5th) and 6 (6th).
• Now calculate the median:

$$ \text{Median} = \frac{4 + 6}{2} = \frac{10}{2} = 5 $$

The mode is the value that appears most often in a set of numbers.

• Example 1: Scores = 3, 5, 5, 6, 7
  Here, 5 appears twice, while other numbers only appear once. So, Mode = 5
  
  
• Example 2: Scores = 3, 5, 5, 6, 7, 7
  Here, 5 and 7 both appear twice, more than the others. So, Mode = 5 and 7

🌟 Exam tip: A data set can have:

• one mode (if one number appears most often),
• more than one mode (if multiple numbers appear with the same highest frequency),
• no mode at all (if no number repeats).

The range shows the spread of a data set. You find the range by subtracting the smallest value from the largest value:

$$ \text{Range} = \text{max value} \; – \; \text{min value} $$

• For example: 5, 7, 6, 3, 5
  • Max value = 7
  • Min value = 3
  • So the Range is 7−3=4
    
    

🌟 What the Range Tells You:

• A small range means the numbers are close together → The data is more consistent, and the results are steadier or more reliable.
• A large range means the numbers are spread out → The data is less consistent and shows more variation.