Frequency Distribution Calculator

A frequency distribution lists every value, class or category of a data set together with how often it occurs. Counting the observations in each row turns a long list into a table that can be read at a glance, and it is the first step toward a histogram, a bar chart or a pie chart.

The frequency f of a row is the number of observations in it, and the frequencies add up to n, the total number of observations. The other columns put the counts in proportion: the relative frequency f / n, the percent, and the cumulative columns, which add up the rows from the top.

The relative frequencies of all rows add up to 1 and the percentages to 100, and the last cumulative frequency equals n. The table shows rounded numbers, so a rounded column can add up to slightly more or less; the calculator says so whenever it happens.

Twenty students were asked how many siblings they have: 2, 1, 3, 0, 2, 1, 2, 4, 1, 2, 3, 2, 0, 1, 2, 3, 1, 2, 1, 0. Choose Numbers: one row for each value and Load example to fill in these answers.

The most frequent answer is 2 siblings: 7 of the 20 students, a relative frequency of 7 / 20 = 0.35. The cumulative frequency of the value 1 is 9, so 9 students (45%) have at most one sibling. The frequencies add up to 20, the relative frequencies to 1 and the last cumulative frequency is 20.

Twenty exam scores range from 58 to 100: 72, 85, 91, 68, 77, 95, 88, 64, 70, 100, 81, 79, 66, 74, 83, 90, 58, 62, 87, 76. Choose Numbers: grouped into classes, then Load example, which sets the first class to start at 50 and the class width to 10.

The scores 70, 90 and 100 sit on class boundaries. Each class includes its lower boundary, so they are counted in 70 to 80, 90 to 100 and 100 to 110. The modal class is 70 to 80 with 6 scores (30%). The cumulative frequency of that class is 11, so 11 of the 20 scores (55%) are below 80. To estimate the mean, median and standard deviation from a table like this one, without the raw scores, use the grouped data calculator.

• Equal widths, no gaps, no overlaps. Every value must belong to exactly one class, and the bars of a histogram are only comparable when the classes are equally wide.
• Start at or below the smallest value. The first class must reach the smallest value. A round number such as 50 or 0 makes the boundaries easy to read.
• Pick a number of classes, then a width. Textbooks usually suggest between 5 and 20 classes. Sturges' rule gives k = ⌈log₂ n⌉ + 1 classes (the default of R's hist function) and the square-root choice gives ⌈√n⌉. For 20 values that is 6 and 5 classes. Divide the range by k and round up to a convenient width.
• Look at the shape. Too few classes hide it, too many make it ragged. Try another width when the picture changes a lot.

The class width calculator turns a range and a number of classes into a width and lists the boundaries, and the range calculator finds the range of your data.

Every class here includes its lower boundary and excludes its upper one, written [lower, upper). A value that equals a boundary goes to the class that starts there, so 70 with classes of width 10 is counted in 70 to 80. The last class is never closed: if the largest value sits on a boundary, one more class is added for it. Other tools follow other rules, and the counts differ when values sit on the boundaries:

Whichever rule you use, state it with the table, and give the class boundaries rather than only the words "60-70" when the data are continuous.

Relative frequency lets you compare data sets of different sizes: 7 of 20 students and 35 of 100 students are the same share, 0.35. Cumulative frequency answers questions of the form "how many values are below this point?". For a class table the cumulative frequency of a class counts the values below its upper boundary, and the cumulative relative frequency is the share of values below that boundary.

Plotting the cumulative relative frequency against the upper class boundaries gives an ogive, from which you can read approximate percentiles. For exact percentiles of the raw values use the percentile calculator, and for the quartiles and the extremes the five number summary.

For words or labels, enter one item per observation. Each different spelling is one category, so "Yes" and "yes" are two categories: use one spelling throughout. The rows can be listed in the order the categories first appear or from the highest frequency down, which puts the most common category first and makes the cumulative column show how much the top categories cover.

To test whether the observed counts fit an expected pattern, take the frequencies to the chi-square goodness of fit calculator.

Most decimals have no exact binary form, so a spreadsheet or program that computes the class of a value as FLOOR((x − start) / width) can move a value on a boundary into the wrong class. In double precision 0.3 / 0.1 is 2.9999999999999996, which would put 0.3 in the class before the one that starts at 0.3. This calculator reads every number as an exact decimal and compares and divides them exactly, so 0.3 is counted in [0.3, 0.4).

Values written differently are the same value: 1, 1.0 and 1e0 form one row, and numbers with up to 60 digits are accepted. A number that is not a number, a value below the first class and a table that would need too many classes are reported with a message instead of being guessed at.

• Enter every observation. The calculator counts for you and does not read a column of counts: to enter a value that occurs 5 times, write it 5 times.
• Grouping hides detail. A table of classes no longer shows the individual values, so a mean or median calculated from it is an approximation.
• Rounded columns. Rounded percentages can add up to 99.99 or 100.01. The Total row shows the exact totals.
• Compare with relative frequency. Two groups of different sizes cannot be compared by their counts.
• Limits. A table is built from at most 10,000 observations and shows up to 500 different values or categories, or up to 200 classes. The chart is drawn for up to 60 rows.