Frequency Distribution Calculator
Generate complete frequency distribution tables with relative frequency, cumulative frequency, tally marks, and histogram charts from raw data. Our free frequency table generator handles ungrouped and grouped data instantly.
| Value | Tally | Frequency (f) | Relative Freq (f/n) | Relative Freq % | Cumulative Freq (cf) | Cumulative % |
|---|
Enter class intervals with frequencies. Midpoints, mean, median, standard deviation, and histogram are calculated automatically with full working shown.
| Class Interval | Midpoint (x) | Tally | Frequency (f) | x × f | Relative Freq % | Cumulative f | Cumulative % |
|---|
Two-way frequency tables show the relationship between two categorical variables simultaneously. Enter row and column labels, fill in the cell frequencies, and all joint, marginal, and conditional frequencies are computed automatically.
Paste any list of numbers — from homework, a spreadsheet, or a dataset. Instantly see how many times each value appears, ranked by frequency. Works with decimals, negatives, and any delimiter. Also detects text/word frequency automatically.
What is a Frequency Distribution?
A frequency distribution organises raw data to show how often each value or range of values occurs in a dataset. It turns a messy list of numbers into a structured, readable summary table — one of the most fundamental tools in descriptive statistics. Our frequency distribution calculator and frequency table generator above handle everything automatically, from counting tallies to computing cumulative percentages and drawing the histogram.
Frequency (f) is the raw count of occurrences for each value. Relative frequency expresses each count as a proportion of the total (f ÷ n). Cumulative frequency is a running total — each row adds the current frequency to the sum of all previous frequencies. Cumulative relative frequency expresses that running total as a percentage, always ending at exactly 100%.
Example Frequency Distribution Table — Data: 2, 3, 3, 4, 4, 4, 5, 5, 6 (n = 9)
| Value | Freq (f) | Relative Freq | Relative Freq % | Cumulative f | Cumulative % |
|---|---|---|---|---|---|
| 2 | 1 | 0.111 | 11.1% | 1 | 11.1% |
| 3 | 2 | 0.222 | 22.2% | 3 | 33.3% |
| 4 | 3 | 0.333 | 33.3% | 6 | 66.7% |
| 5 | 2 | 0.222 | 22.2% | 8 | 88.9% |
| 6 | 1 | 0.111 | 11.1% | 9 | 100.0% |
How to Make a Frequency Table — Step by Step
Building a frequency table by hand requires seven systematic steps. Understanding each step helps you verify the calculator's output and answer exam questions correctly. The frequency table generator above completes all seven steps automatically from pasted raw data.
- Collect your raw data values. Example dataset: 5, 3, 7, 5, 2, 3, 5, 7, 3, 5 — total n = 10 observations.
- Identify all unique values and list them in ascending order: 2, 3, 5, 7.
- Tally each occurrence using tally marks grouped in fives for easy counting: 2?I, 3?III, 5?IIII, 7?II.
- Record the frequency f for each value: value 2 ? 1, value 3 ? 3, value 5 ? 4, value 7 ? 2. Check: 1+3+4+2 = 10 = n ?
- Calculate relative frequency = f ÷ n: 1/10 = 0.100, 3/10 = 0.300, 4/10 = 0.400, 2/10 = 0.200. Check: sum = 1.000 ?
- Calculate cumulative frequency by adding each frequency to the running total: 1, 4 (1+3), 8 (4+4), 10 (8+2). Final value must equal n ?
- Calculate cumulative relative frequency %: 10%, 40%, 80%, 100%. Final row must be exactly 100% ?
Shortcut: Paste your raw data into the Frequency Table tool above — all seven steps complete instantly with tally marks, relative frequencies, cumulative columns, step-by-step working, and an automatic histogram.
Relative Frequency Calculator — Formula and Examples
Relative frequency expresses each value's count as a proportion of the total observations. It is always between 0 and 1 (or 0% to 100%), and all relative frequencies in a table must sum to exactly 1.000 (100%). Relative frequency allows fair comparison between datasets of different sizes — a count of 10 means very different things in a dataset of 20 versus 2,000. Use the relative frequency calculator tab above to compute these automatically.
Example 1 — Simple Dataset
A dataset has 20 observations. Value X appears 5 times. Relative frequency = 5 ÷ 20 = 0.250 = 25.0%. This means value X accounts for one quarter of all observations in the dataset. The remaining 15 observations make up the other 75%.
Example 2 — Class Survey
A class of 30 students was asked their favourite subject. 12 chose mathematics. Relative frequency = 12 ÷ 30 = 0.400 = 40.0%. Using relative frequency instead of raw counts allows fair comparison with other classes of different sizes — if another class of 50 had 18 maths-preferrers, their RF = 18/50 = 36%, making the first class more maths-inclined despite the smaller absolute count.
Example 3 — Quality Control
A factory inspected 500 items and found 15 defective. Defect relative frequency = 15 ÷ 500 = 0.030 = 3.0%. This is the relative frequency distribution of defects — expressed as a proportion it can be directly compared to industry benchmarks regardless of batch size. The relative frequency table calculator above computes this for any dataset with a full relative frequency distribution table output.
Cumulative Frequency Calculator — Formula and Examples
Cumulative frequency is a running total of frequencies from the first class through to the current class. Each row's cumulative frequency equals the sum of all individual frequencies up to and including that row. The final row's cumulative frequency always equals n exactly — if it does not, there is an arithmetic error. The cumulative frequency calculator above verifies this automatically.
Cumulative Relative Frequency % = (cf ÷ n) × 100
Cumulative frequency has several important applications: finding the median (the value at the 50th percentile), locating quartiles (Q1 at 25%, Q3 at 75%), identifying percentiles, and building the ogive — the S-shaped graph of cumulative frequency that visually shows the distribution of data. Use the Cumulative Ogive toggle on the histogram above to see this curve.
Worked Example — Test Scores (n = 30)
| Class | Freq (f) | Cumulative Freq (cf) | Cumulative % |
|---|---|---|---|
| 50–59 | 3 | 3 | 10.0% |
| 60–69 | 7 | 10 | 33.3% |
| 70–79 | 12 | 22 | 73.3% |
| 80–89 | 6 | 28 | 93.3% |
| 90–99 | 2 | 30 | 100.0% |
Finding the median: n/2 = 15. The first class where cf = 15 is 70–79 (cf = 22). So the median lies in the 70–79 class. Using the interpolation formula: Median = 70 + ((15 - 10) / 12) × 10 = 70 + 4.17 = 74.2
Grouped Frequency Distribution Calculator
Use a grouped frequency distribution when your dataset is large, contains continuous measurements, or spans a wide numerical range where listing every individual value would be impractical. Raw values are sorted into equal-width class intervals (also called bins or classes). The class midpoint — calculated as (lower bound + upper bound) ÷ 2 — represents all values within that class and is used to estimate the mean, median, and standard deviation.
When choosing class width, aim for 5 to 20 classes. Too few classes lose important detail; too many classes defeat the purpose of grouping. A common guideline is class width ˜ Range ÷ vn. The Grouped Data tab above calculates all statistics automatically with full formula working shown.
Median = L + ((n/2 - cfbefore) / fmedian) × w
s = v[Sf(x - x¯)² / Sf]
Worked Example — Heights of 35 Students (class width = 5 cm)
| Class (cm) | Midpoint (x) | f | x × f | (x - x¯)² | f(x - x¯)² | cf |
|---|---|---|---|---|---|---|
| 150–155 | 152.5 | 4 | 610.0 | 102.01 | 408.04 | 4 |
| 155–160 | 157.5 | 8 | 1260.0 | 27.04 | 216.32 | 12 |
| 160–165 | 162.5 | 13 | 2112.5 | 2.07 | 26.91 | 25 |
| 165–170 | 167.5 | 7 | 1172.5 | 27.10 | 189.70 | 32 |
| 170–175 | 172.5 | 3 | 517.5 | 102.13 | 306.39 | 35 |
S(x·f) = 5672.5 | Sf = 35 | Mean = 5672.5 / 35 = 162.07 cm
n/2 = 17.5 ? median class = 160–165 (cf before = 12, f = 13, w = 5)
Median = 160 + ((17.5 - 12) / 13) × 5 = 160 + 2.12 = 162.12 cm
Sf(x-x¯)² = 1147.36 | Variance = 1147.36/35 = 32.78 | s = v32.78 = 5.73 cm
Two-Way Frequency Table Calculator
A two-way frequency table (also called a contingency table or cross-tabulation) simultaneously cross-tabulates two categorical variables, revealing the relationship between them. Each cell in the body of the table shows the joint frequency — the number of observations that fall into both that row category and that column category. Row totals and column totals show the marginal frequencies for each variable independently.
Three types of relative frequencies are derived from a two-way table: joint relative frequencies (each cell divided by the grand total), marginal relative frequencies (row or column totals divided by grand total), and conditional relative frequencies (cell divided by its row total for row %, or divided by its column total for column %). The Two-Way Table tab above calculates all four tables automatically.
Worked Example — 100 Students: Gender vs Favourite Subject
| Science | Maths | English | Row Total | |
|---|---|---|---|---|
| Male | 18 | 22 | 10 | 50 |
| Female | 12 | 15 | 23 | 50 |
| Col Total | 30 | 37 | 33 | 100 |
Joint RF (Male, Science) = 18/100 = 0.18 = 18%
Marginal RF (Male) = 50/100 = 0.50 = 50%
Conditional RF — Row % (Male who prefer Science) = 18/50 = 0.36 = 36%
Conditional RF — Col % (Science preferrers who are Male) = 18/30 = 0.60 = 60%
Number Frequency Counter
The number frequency counter is the fastest way to count how many times each unique value appears in any list of numbers. It is the highest-volume search term on this page because students constantly need to count repeated values from homework datasets, exam papers, and lab results without tedious manual tallying.
Simply paste your numbers into the Number Counter tab above — the tool handles any mix of commas, spaces, and new lines. It immediately produces a ranked frequency table showing each value, its count, relative frequency percentage, cumulative frequency, and cumulative percentage. A visual horizontal bar chart shows which values appear most frequently at a glance, with the mode (most frequent value) highlighted in gold.
The tool also detects non-numeric text automatically and switches to word frequency counting — paste a paragraph and instantly see which words appear most often, useful for text analysis and linguistics exercises.
Example use case: Paste 50 exam scores ? instantly see that 85 appears 7 times (14%), 78 appears 5 times (10%), mode = 85. Teachers share this tool with students because it eliminates manual counting errors entirely.
Worked Examples
1. How to create a frequency distribution table from raw data
List all unique values in ascending order, count how many times each appears using tally marks, record the frequency f for each value, then compute relative frequency as f/n and add cumulative columns. For data 2, 3, 3, 4, 4, 4 (n=6): unique values are 2, 3, 4 with frequencies 1, 2, 3. Relative frequencies are 1/6=0.167, 2/6=0.333, 3/6=0.500. Cumulative frequencies are 1, 3, 6. Cumulative percentages are 16.7%, 50.0%, 100.0%.
2. How to calculate relative frequency from a frequency table
Relative frequency = f ÷ n, where f is the frequency of that value and n is the total number of observations. The formula is: RF = f/n. If value 7 appears 4 times in a dataset of 25 observations: RF = 4/25 = 0.160 = 16.0%. All relative frequencies in the complete table must sum to exactly 1.000 (100%). Apply a rounding correction to the final row if needed to absorb any decimal remainder.
3. How to find cumulative frequency step by step
Cumulative frequency is built row by row by adding each frequency to the running total. Formula: cfi = fi + cfi-1. For frequencies 3, 5, 7, 4, 2: cumulative frequencies are 3, 8 (3+5), 15 (8+7), 19 (15+4), 21 (19+2). The final cumulative frequency (21) must equal n (total observations = 21). Cumulative relative frequency % = (cf/n)×100 = 14.3%, 38.1%, 71.4%, 90.5%, 100.0%.
4. How to calculate mean from a frequency distribution table
The mean from a frequency table uses the formula x¯ = S(x × f) / Sf. Multiply each value by its frequency, sum all products to get S(x·f), then divide by the total frequency Sf. Example — values 2, 3, 5 with frequencies 1, 3, 2: S(x·f) = 2(1)+3(3)+5(2) = 2+9+10 = 21; Sf = 1+3+2 = 6; x¯ = 21/6 = 3.500. For grouped data, substitute the class midpoint for x in the same formula.
5. How to find the median from a grouped frequency distribution
First find n/2 (the target cumulative frequency for the median). Locate the median class — the first class where cumulative frequency reaches or exceeds n/2. Then apply the interpolation formula: Median = L + ((n/2 - cfbefore) / fmedian) × w. For n=40, median class 160–165 with L=160, cfbefore=13, fm=14, w=5: Median = 160 + ((20-13)/14)×5 = 160 + 2.50 = 162.50. The Grouped Data tab calculates this with full substituted working shown.
6. How to make a frequency chart (histogram) from data
Draw a horizontal axis labelled with values or class intervals and a vertical axis labelled with frequency. Draw a bar above each value or class, with the bar's height equal to the frequency for that value. For continuous data, bars touch each other with no gaps; for discrete data, small gaps may separate bars. The Y-axis should start at zero and extend to the maximum frequency plus 10% headroom. The Frequency Table tool above generates a professional histogram automatically with gridlines, hover tooltips, and optional overlays for the frequency polygon and cumulative ogive.
7. How to create a two-way frequency table
List one categorical variable as rows and the second as columns. Count how many observations fall into each combination of row and column categories and enter these joint frequencies into the cells. Sum each row for row totals (marginal frequencies), sum each column for column totals, and confirm the sum of all row totals equals the sum of all column totals equals the grand total N. For a survey of 80 students (Male/Female × Maths/Science/English): if Male-Maths=15, Male-Science=12, Male-English=8 ? row total Male = 35.
8. How to calculate cumulative relative frequency
Cumulative relative frequency = (cumulative frequency ÷ n) × 100%. It represents the percentage of observations at or below each value. Formula: CRF% = (cf/n) × 100. For a dataset with n=50 and cumulative frequency 35 at value X: CRF% = (35/50)×100 = 70.0% — meaning 70% of all observations are equal to or less than value X. The final row always gives exactly 100.0%.
9. How to find the mode from a frequency distribution
The mode is the value with the highest frequency in the frequency distribution. In the frequency table, scan the frequency column (f) and identify the largest value. The corresponding data value is the mode. For a frequency table with values 3?f=4, 5?f=7, 6?f=3, 8?f=7: the dataset is bimodal (two modes: 5 and 8, both with f=7). For grouped data, the modal class is the class interval with the highest frequency. The Number Counter tool highlights the mode automatically in gold.
10. How to convert a frequency table to a relative frequency table
Divide each frequency f by the total n to get the relative frequency as a decimal: RF = f/n. Multiply by 100 for percentage form. Apply a rounding correction to the final row: compute the sum of all rounded relative frequencies; if it does not equal exactly 100%, add or subtract the difference from the last row. For example, with n=7 and frequencies 1,2,4: RF = 0.143, 0.286, 0.571; percentages = 14.3%, 28.6%, 57.1% — sum = 100.0% ?. The frequency table generator above applies this correction automatically.