Turn a Frequency Table Into a Descriptive Summary

A student hands you a frequency table: value 2 appears 3 times, value 5 appears 2 times, value 7 appears 1 time. Six observations total. The question is whether you can turn a frequency table into a descriptive summary that gives the same counts, proportions, and center measures you would get from the original list. The answer is yes for counts and proportions, yes for mean and median with specific formulas, and yes for mode only in the sense of a modal class. The limits come from the fact that grouped data hides the exact values inside each bin, which matters for variance but not for the measures discussed here.

From any frequency table you can recover the raw counts. Each row's frequency f is already a count. The total n is the sum of the f column. The proportions are the relative frequencies f/n, which you compute by dividing each count by n. For the example table with values 2, 5, and 7, the counts are 3, 2, and 1. The total n is 6. The proportions are 3/6 = 0.5, 2/6 ≈ 0.333, and 1/6 ≈ 0.167. The relative frequencies sum to 1.0 exactly if you do not round intermediate steps.

Work the example from the fact sheet. The frequency table is value 2 (freq 3), value 5 (freq 2), value 7 (freq 1). Total n = 6.

Counts And Proportions

Counts are the frequencies themselves: 3, 2, 1. Proportions (relative frequencies) are those counts divided by 6: 0.5, 0.333, 0.167. The sum of proportions is 0.5 + 0.333 + 0.167 = 1.0. If you rounded each proportion to two decimal places you would get 0.50 + 0.33 + 0.17 = 1.00, which works here but can produce 0.99 or 1.01 in other datasets. The safe practice is to report the unrounded sum or note the rounding error.

Mean From Grouped Data

The formula for the mean from grouped data is Σ(midpoint × frequency) / total frequency. For a table with single values, the midpoint is the value itself. So the mean is (2×3 + 5×2 + 7×1) / 6 = (6 + 10 + 7) / 6 = 23/6 ≈ 3.833. Use a statistics calculator to verify this step and avoid arithmetic slip.

Median From Grouped Data

To find the median, reconstruct the ordered list: 2,2,2,5,5,7. The median of six observations is the average of the third and fourth values: (2+5)/2 = 3.5. The grouped-data median formula gives the same result when the data are already sorted by value. The formula is L + ((n/2 - cf) / f) × w. Here the median class is the interval containing the 3rd and 4th values, which is the value 2 and the value 5. For single values, L is the lower boundary of the median class (2 for the first class), cf is the cumulative frequency before the median class (0 for the first class), f is the frequency of that class (3), and w is the class width (1 for a single value). Then 2 + ((3 - 0)/3) × 1 = 2 + 1 = 3. But that gives the median of the lower half only, which is why the standard method uses the average of the two middle values when n is even. The grouped-data median formula is designed for continuous binned data, not for discrete values with integer frequencies. For this exact example, the correct median is 3.5, which you get from the ordered list, not from a single application of the formula.

Mode

The mode is the value with the highest frequency: 2 appears 3 times, which is more than any other value, so the mode is 2. For grouped intervals, you identify the modal class (the interval with the highest frequency) and report its midpoint as the approximate mode.

The formulas for mean and median from grouped data are defined in OpenStax Introductory Statistics 2e, Chapter 2. The mean formula Σ(midpoint × f) / n is an approximation when the data are binned into intervals because it assumes all values in a bin fall at the midpoint. If the true data are skewed within a bin, the approximation drifts. The median formula L + ((n/2 - cf) / f) × w works exactly only when the data are uniformly distributed within the median class. For nominal or ordinal data, the mean is meaningless. Only the mode and proportions are recoverable for categorical variables. The median is recoverable only if the categories have a natural order.

Consider a frequency table with two intervals: [0-10] with frequency 5, and [10-20] with frequency 5. The total n is 10. The mean is approximated as (5×5 + 15×5) / 10 = (25 + 75) / 10 = 10. The true mean is also 10 if the actual data are 0,0,0,0,0,20,20,20,20,20. In that extreme case, the mean is still 10, but the variance is completely lost. The variance from the grouped data formula would be small (assuming midpoints), while the true variance is 100. The mean is correct, but the spread of the data is invisible. This is the fundamental limit: a frequency table preserves counts, proportions, and center measures, but discards the distribution shape within each bin. The median formula also fails for this dataset because the median class contains the boundary value 10, which could belong to either bin depending on the bin definition. If the bins are defined as [0,10) and [10,20), then the value 10 falls in the second bin, shifting the median calculation.

This method suits middle-school students constructing basic frequency tables, high-school students computing joint and marginal frequencies, introductory college students in non-math majors checking hand calculations, teachers building answer keys, and survey analysts converting raw tallies into percentages. Skip it if you are an advanced statistician needing inferential methods, confidence intervals, or hypothesis tests. Also skip it if you have not yet learned what a fraction or a percentage is. The most common mistake is treating the relative frequency from the table as a true probability for the next draw, when it only describes the data you collected.