A cumulative frequency graph, also called an ogive, shows how many data values are less than or equal to each value on the horizontal axis. It is useful because it turns a frequency table into a visual summary of the whole distribution. From one curve, you can estimate the median, quartiles, percentiles, and spread.
This makes it easier to compare groups, such as test scores from two classes or heights from two samples.
To draw an ogive, plot cumulative frequency against the upper class boundary or data value, then join the points with a smooth increasing curve or straight line segments. To read a percentile, move horizontally from the chosen cumulative frequency to the curve, then move down to the value axis. The median is found at 50% of the total frequency, while the lower and upper quartiles are found at 25% and 75%.
Comparing two ogives on the same axes shows which distribution tends to have larger values and which has more variability.
Key Facts
- Cumulative frequency is the running total of frequencies up to and including a value or class.
- For total frequency n, median position = n/2 on the cumulative frequency axis.
- Lower quartile position = n/4 and upper quartile position = 3n/4.
- Percentile position = (p/100)n, where p is the percentile number.
- Interquartile range = Q3 - Q1, measuring the spread of the middle 50% of the data.
- An ogive should never decrease because cumulative frequency can only stay the same or increase.
Vocabulary
- Cumulative frequency
- The total number of data values up to and including a particular value or class.
- Ogive
- A graph of cumulative frequency plotted against data values or class boundaries.
- Median
- The middle value of a data set, found at 50% of the total cumulative frequency.
- Quartile
- One of the values that divides an ordered data set into four equal parts.
- Percentile
- A value below which a given percentage of the data lies.
Common Mistakes to Avoid
- Plotting ordinary frequency instead of cumulative frequency is wrong because an ogive uses running totals, not the height of each class alone.
- Using class midpoints when the table gives grouped intervals can be wrong because ogives are usually plotted against upper class boundaries.
- Reading the median directly from the horizontal axis first is wrong because you must start at n/2 on the cumulative frequency axis, go across to the curve, then down.
- Assuming the curve gives exact raw data values is wrong because an ogive from grouped data gives estimates based on interpolation between plotted points.
Practice Questions
- 1 A data set has total frequency 80. On a cumulative frequency graph, what cumulative frequency levels should you use to estimate Q1, the median, and Q3?
- 2 An ogive for 120 students shows that the 30th percentile corresponds to a score of 46. How many students scored about 46 or below?
- 3 Two cumulative frequency graphs are drawn on the same axes. Graph A is mostly to the left of Graph B, and Graph B has a wider distance between Q1 and Q3. Explain what this suggests about the typical values and spread of the two distributions.