Mean from a Frequency Table

Statement

The mean from a frequency table is calculated using:

\[ \bar{x} = \frac{\sum fx}{\sum f} \]

where \(f\) is the frequency of each value \(x\), and \(fx\) is the product of each value with its frequency.

Why it’s true

• The mean is the total of all values divided by how many values there are.
• When data is grouped in a frequency table, each value \(x\) occurs \(f\) times.
• Multiplying \(x\) by \(f\) gives the contribution to the total from that value.
• Adding up all \(fx\) gives the grand total, while \(\sum f\) is the total number of values.

Recipe (how to use it)

1. Make columns for \(x\), \(f\), and \(fx\).
2. Multiply each value by its frequency to get \(fx\).
3. Add all the \(fx\) values (\(\sum fx\)).
4. Add all the frequencies (\(\sum f\)).
5. Divide \(\sum fx\) by \(\sum f\).

Spotting it

You use this whenever data is given in a frequency table, instead of listing every number individually.

Common pairings

• Statistics questions about averages.
• Grouped data where exact values aren’t all listed.
• Real-world surveys and test scores.

Mini examples

1. Data: x=1,2,3; f=2,3,5. \(fx=2,6,15.\) \(\sum fx=23, \sum f=10.\) Mean=23/10=2.3.
2. Data: Marks: 5, 10, 15; f=4,6,10. \(fx=20,60,150.\) \(\sum fx=230, \sum f=20.\) Mean=230/20=11.5.

Pitfalls

• Forgetting to divide by total frequency: Don’t just use \(\sum fx\).
• Arithmetic mistakes: Double-check sums.
• Confusing grouped vs ungrouped data: With grouped data, use midpoints.

Exam strategy

• Always set out a clear table with columns for \(x\), \(f\), and \(fx\).
• Calculate carefully and show totals.
• Check: mean should lie between the smallest and largest values.

Summary

The mean from a frequency table is found by dividing the total of \(fx\) by the total frequency. Formula: \(\bar{x} = \tfrac{\sum fx}{\sum f}\).