The phrase what does mean mean in frequenct data refers to the meaning of the mean when the raw observations are summarized by counts (frequencies). The mean is still an average, but it must account for how many times each value occurs.
Meaning of the mean in frequency data
When values are listed with frequencies, the mean represents the average value per observation across the entire dataset. A value with a larger frequency influences the mean more because it appears more often.
How to calculate the mean from a frequency table
Suppose a value \(x_i\) occurs \(f_i\) times. The total number of observations is \(n=\sum f_i\). The mean is the weighted average:
Step-by-step procedure
- List each distinct value \(x_i\) and its frequency \(f_i\).
- Compute the products \(x_i f_i\) for each row.
- Sum the products \(\sum x_i f_i\) and sum the frequencies \(\sum f_i\).
- Divide: \(\bar{x}=(\sum x_i f_i)/(\sum f_i)\).
Worked example (frequency table)
A quiz score table is summarized as follows.
| Score \(x_i\) | Frequency \(f_i\) | Product \(x_i f_i\) |
|---|---|---|
| 60 | 2 | 120 |
| 70 | 5 | 350 |
| 80 | 8 | 640 |
| 90 | 3 | 270 |
| 100 | 2 | 200 |
| Totals | \( \sum f_i = 20 \) | \( \sum x_i f_i = 1580 \) |
Compute the mean:
Interpretation: across all 20 students, the average (mean) quiz score is 79 points.
Visualization: frequencies and the mean marker
Grouped classes (when the table uses intervals)
If the frequency table lists class intervals (such as 70–79, 80–89), the mean is estimated using the class midpoint \(m_i\) in place of \(x_i\):
This is an approximation because individual values inside each interval are unknown; the midpoint represents the class.
Key takeaway
In frequenct summaries, the mean is the average per observation after accounting for how many times each value occurs, computed as \( \bar{x}=(\sum x_i f_i)/(\sum f_i) \) (or using midpoints for grouped intervals).