Statistics & Probability: Mean (Arithmetic Average)

The arithmetic mean (commonly called the average) represents the central balance point of a dataset. It is calculated by adding all the values together and dividing by the total number of items:

Think of data points as physical weights placed along a seesaw number line. The mean is the exact location of the fulcrum (pivot point) where the beam balances perfectly! The sum of distances to the left of the mean equals the sum of distances to the right.

To find the mean of any dataset:
1. Add all numbers together to find the total sum.
2. Count how many numbers there are.
3. Divide the sum by the count: $\text{Mean} = \frac{\text{Total Sum}}{\text{Number of Values}}$.

When you know several values (like test scores) and need to find a missing score $x$ to reach a specific target mean, work backwards using algebra:

1. Count Grows by One: You already know your current count of values. Since you are adding one new value ($x$), the new total count becomes $\text{Total Count} = \text{Known Count} + 1$.
2. The Total Sum Principle: Multiply the new total count by the target mean to find the grand total points needed:

3. Sum Known Values: Add up the scores you already have to find your current sum.
4. Isolate the Missing Value ($x$): Subtract the known sum from the required total sum to find the missing score $x$:

When data is organized in a frequency table, each value appears multiple times (given by its frequency / count). Follow these calculation steps:

1. Multiply each value by its count: $\text{Value} \times \text{Count}$ to find the subtotal for that value.
2. Sum all products together to get the grand total sum.
3. Sum all frequencies together to find the total count of items.
4. Divide the grand sum by the total count to find the mean.