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A frequency table is equivalent to the full ordered list of data, just written more compactly. Because of this, any statistic that depends only on the values and how often they occur can still be found exactly from the table.
The mean is a weighted average because values with larger frequencies should influence the answer more strongly. The formula
works because represents the total contribution of each value repeated times.
The median is a positional measure, so you do not need to total products or use arithmetic weighting. Instead, you identify the middle position in the ordered data and use cumulative frequency to see which row contains that position.
Cumulative frequency means adding frequencies as you move down the table. It works for locating the median because it tells you how many data points have been accounted for up to each value.
The range depends only on the extreme data values, not on how many times they occur. This is why even a very large frequency in the middle of the table does not affect the range at all.
which gives the average value per observation.
This measures the total spread of the data values.
| Quantity | Meaning | Example role in calculation |
|---|---|---|
| actual data value | candidate for mode, part of range | |
| number of occurrences | used for weighting and positions | |
| total contribution of a row | used in the mean |
| Question | Correct output |
|---|---|
| What appears most often? | the data value |
| How many times does it appear? | the frequency |
| Idea | What it tells you |
|---|---|
| Median position | where to look |
| Median value | what the answer is |
| Feature | Mean | Median |
|---|---|---|
| Based on | all values and frequencies | middle position |
| Uses ? | yes | no |
| Affected by extreme values? | more | less |
Always label the columns mentally before calculating so you know which column contains values and which contains frequencies. This reduces errors such as giving a frequency as the mode or subtracting frequencies to find the range.
For the mean, set out an column clearly even if the table is short. This is the most reliable method because it shows the weighted structure of the calculation and makes checking totals easier.
Key formula to remember:
For the median, use cumulative frequency rather than expanding the full list unless the table is tiny. This is faster, more organised, and less likely to cause counting mistakes under time pressure.
Check whether the total frequency is odd or even before deciding how to locate the median. If it is even, the middle lies between two central positions, so you must be careful not to treat one position as the whole answer automatically.
Use a reasonableness check after finding the mean by seeing whether the answer lies between the smallest and largest data values. If it falls outside that interval, a multiplication, total, or division error has almost certainly occurred.
Read the wording of the question carefully for rounding or units. The arithmetic may be correct, but marks are often lost by omitting units, rounding too early, or giving too many or too few significant figures.
Averages from tables connect directly to ungrouped averages because the table is simply a compressed list of raw data. If all frequencies were expanded, the mode, median, mean, and range would match exactly.
This topic is a foundation for grouped data, where the exact values are no longer known and the mean becomes an estimate rather than an exact answer. Understanding exact frequency tables first helps students see why grouped tables require midpoints later.
Frequency tables also link to cumulative frequency graphs and statistical summaries because cumulative totals are a general tool for locating positions such as medians and quartiles. The same positional reasoning extends beyond tables into wider data handling.
In real applications, frequency tables are useful when many repeated values occur, such as test scores, survey counts, or size categories. They make the distribution easier to inspect while still allowing exact calculations for several summary measures.
