Calculating the Mean from a Frequency Distribution Table

Traditionally, the mean (arithmetic average) is calculated by summing every individual score in a dataset and dividing that total sum by the count of how many scores are present.
This method is efficient for small datasets but becomes cumbersome as the volume of data increases.
Alternative methods for finding the mean include using a calculator or utilizing a frequency distribution table.
A frequency distribution table serves as an organized shortcut to handle larger amounts of data without manually typing every individual number into a calculator.

A standard frequency distribution table consists of several columns used to organize data: - Score Column ( $x$ ): This column lists the distinct values appearing in the dataset. - Tally Column: This column is used during the initial data entry phase to count occurrences of each score. - Frequency Column ( $f$ ): This column represents the final count of how many times each specific score occurs in the set.
To calculate the mean specifically from this table, an additional column is required: - $fx$ Column: This is also referred to as the "frequency times score" column.

In algebraic notation, when two letters are placed side-by-side (such as $f$ and $x$ ), there is an invisible multiplication symbol between them.
Therefore, the $fx$ column is calculated by multiplying the value in the score column ( $x$ ) by its corresponding frequency in the ( $f$ ) column.
The logic behind the $fx$ column is that it calculates the total contribution of a specific score to the overall sum: - Example: If the score is $2$ and it occurs $3$ times, the calculation is $2 \times 3 = 6$ , adding $6$ to the total sum. - Example: if the score is $7$ and it occurs $3$ times, the calculation is $7 \times 3 = 21$ , adding $21$ to the total sum.
Consequently, the $f$ column represents total count ( $n$ ), and the $fx$ column represents the partial totals that will eventually make up the grand total of all scores.

The following dataset (previously utilized in a frequency distribution table video) is used to illustrate the process: - Score ( $x$ ) of $2$ with a frequency ( $f$ ) of $3$ results in $fx = 6$ . - Score ( $x$ ) of $3$ with a frequency ( $f$ ) of $7$ results in $fx = 21$ . - Score ( $x$ ) of $4$ with a frequency ( $f$ ) of $2$ results in $fx = 8$ . - Score ( $x$ ) of $5$ with a frequency ( $f$ ) of $6$ results in $fx = 30$ (six scores of five). - Score ( $x$ ) of $6$ with a frequency ( $f$ ) of $5$ results in $fx = 30$ (five scores of six). - Score ( $x$ ) of $7$ with a frequency ( $f$ ) of $2$ results in $fx = 14$ .
Summation of Columns: - The sum of the frequency column ( $f$ ) is found by adding $3 + 7 + 2 + 6 + 5 + 2 = 25$ . There are $25$ total scores. - The sum of the $fx$ column is found by adding $6 + 21 + 8 + 30 + 30 + 14 = 109$ . The total sum of all scores is $109$ .

The mean is calculated by dividing the total sum of items by the number of items.
Formula Arrangement: $\text{Mean} = \frac{\text{Sum of the } fx \text{ column}}{\text{Sum of the } f \text{ column}}$ .
For this specific dataset: - Numerator: $109$ - Denominator: $25$ - Calculation: $\frac{109}{25} = 4.36$
The resulting mean for this frequency distribution table is $4.36$ .

Many textbooks employ a specific mathematical symbol to represent the phrase "sum of."
The symbol is the Greek letter Sigma ( $\Sigma$ ), which often looks like a "funny looking e symbol."
Using this symbol, the formula for the mean can be written as: - $\frac{\sum fx}{\sum f}$
This notation is functionally identical to the verbal instruction to divide the total of the $fx$ column by the total of the $f$ column.

It serves as a shortcut for complex datasets.
It prevents the need to type out long lists of numbers into a calculator.
If a frequency distribution table has already been constructed, finding the mean requires only one additional column ( $fx$ ) and two column totals.
This presentation was provided by Peter at blakemath.com.