Frequency Polygon Calculator
Enter your class midpoints and their frequencies below. The calculator plots the frequency polygon, computes the mean, median class, mode class, standard deviation and variance, and builds a full cumulative frequency table so you can see the ogive at a glance. Use the class-count selector to add or remove rows.
What is a frequency polygon?
A frequency polygon is a line graph that shows how data are distributed across class intervals. To draw one, you plot the midpoint of each class on the horizontal axis against the class frequency on the vertical axis, then connect adjacent points with straight lines. The polygon is typically anchored to the horizontal axis at both ends by adding a zero-frequency class before the first interval and after the last. Because a polygon uses lines rather than bars, it makes it easy to overlay several distributions on the same axes for direct comparison, which is harder to read with a histogram.
Key statistics computed from grouped data
When data are grouped into class intervals, exact values are no longer known, so calculations use each class midpoint as a representative value. The grouped mean is the sum of (midpoint times frequency) divided by the total frequency. The population variance averages the squared deviations of each midpoint from the mean, weighted by frequency, and the standard deviation is the square root of that. The median class is found by accumulating frequencies until the running total reaches half the total, and the modal class is simply the class with the highest frequency. These approximations are exact when all observations within a class are equal to the midpoint, and they are good estimates when class widths are narrow relative to the spread of the data.
The ogive: cumulative frequency polygon
The ogive (pronounced oh-jive) is the cumulative version of the frequency polygon. Instead of plotting class frequency on the vertical axis, it plots the running total of frequencies up to and including each class. The resulting S-shaped curve (when the underlying distribution is approximately normal) rises steeply through the middle classes and flattens near the extremes. The ogive is useful for reading off percentiles: find the frequency that corresponds to, say, 75% of the total on the vertical axis, then read across to the horizontal axis to find the approximate 75th-percentile value.
How to use this calculator
Choose how many class intervals your data has from the selector (3 to 8 classes), then enter each class midpoint and its frequency. The midpoint is the average of the lower and upper class boundaries: for the interval 20-30 the midpoint is 25. Enter the frequencies as whole numbers; the calculator does not require them to be pre-sorted, but sorting midpoints in ascending order makes the polygon chart easier to read. Results update instantly as you type. The live chart shows both the frequency polygon and the ogive, and the table below gives the full cumulative frequency breakdown.
Distribution shape guide
| Shape | Relationship | Polygon appearance |
|---|---|---|
| Symmetric (normal) | Mean = Median = Mode | Bell-shaped, balanced peak |
| Right-skewed (positive) | Mode < Median < Mean | Long right tail, peak shifted left |
| Left-skewed (negative) | Mean < Median < Mode | Long left tail, peak shifted right |
| Uniform | All classes have equal frequency | Flat horizontal line |
| Bimodal | Two modes | Two distinct peaks |
The relative positions of mean, median and mode give a quick visual diagnosis of distribution shape.
Frequently asked questions
What is the difference between a frequency polygon and a histogram?
A histogram uses bars; a frequency polygon uses connected line segments. Both show the same frequency distribution, but the polygon makes it easier to compare two or more datasets on the same graph because overlapping lines are much easier to read than overlapping bars. Histograms give a more immediate sense of the area each class represents, while polygons emphasise the shape and trend of the distribution.
How do I find the midpoint of a class interval?
Add the lower and upper boundaries of the class and divide by 2. For the interval 20-30, the midpoint is (20 + 30) / 2 = 25. For the interval 15-25, the midpoint is (15 + 25) / 2 = 20. If your classes have equal width w, midpoints will be evenly spaced w apart, which makes the polygon easier to interpret.
Why does the frequency polygon start and end at zero?
The convention is to anchor the polygon to the horizontal axis at both ends. This is done by adding a zero-frequency class just before the first real class and just after the last real class. Without this, the polygon would appear to "float" and would not enclose a clear area. The area under the polygon is proportional to the total frequency, so anchoring it at zero ensures that relationship is visually clear.
What is an ogive and how do I use it to find percentiles?
An ogive is a cumulative frequency polygon. It plots the total frequency accumulated up to the end of each class against the class midpoint. To find the 50th percentile (the median), multiply the total frequency by 0.50, find that value on the vertical axis, and read across to the horizontal axis. The same approach works for any percentile: multiply the total by the decimal percentile (0.25 for the 25th, 0.75 for the 75th), and read the corresponding horizontal value.
Are the statistics here exact or estimates?
They are estimates. When data are grouped, only the class midpoints are available, not the individual values. Using the midpoint as the representative value for all observations in that class introduces rounding error. The estimates improve as class width decreases relative to the range of the data. For precise statistics you need the raw, ungrouped data. These grouped calculations are widely used in education and in situations where only summary tables are published.
How do I tell if my distribution is skewed from the frequency polygon?
Look at the shape of the peak. A symmetric polygon has a single peak near the centre with roughly equal tails on both sides. A right-skewed polygon has a peak shifted toward the left with a long tail stretching to the right, and the mean will be larger than the median and the mode. A left-skewed polygon has the peak shifted right with a tail to the left, and the mean is smaller than the median and mode. This calculator reports the mean, median class and modal class so you can check these relationships directly.