Cumulative Frequency

1Summary 重點摘要

Learn how to draw cumulative frequency polygon, find median and quartiles.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Computing cumulative frequencyCumulative frequency is the running total of frequencies starting from the first class: the first cumulative value is the first frequency, and each later value is the sum of all frequencies up to that class. For example, with class frequencies 5, 8, 12, 10, the cumulative frequencies are 5, 13, 25, 35, and the last value 35 equals the total number of data N. Why accumulate? Cumulative frequency answers questions such as how many data lie below a value. A common mistake is skipping one class while adding, or confusing the frequency column with the cumulative column.
2. Step 2: Plotting the pointsTo draw a cumulative frequency polygon, plot each point using the upper class boundary as the x-coordinate and the cumulative frequency as the y-coordinate. For example, the class 0-10 has upper boundary 10 and cumulative frequency 5, giving the point (10, 5); with boundaries 0-10, 10-20, 20-30, 30-40 the points are (10, 5), (20, 13), (30, 25), (40, 35). Why the upper boundary rather than the midpoint? Because by the upper boundary all data of the class have been counted. A common mistake is plotting at midpoints, shifting the whole curve.
3. Step 3: Joining the pointsConnect the points with straight segments in order to form the cumulative frequency polygon; first add the point (0, 0), meaning no data lie below zero. The finished curve rises from the lower left to the upper right, never falling, and ends at the total N. In the example the curve runs from (0, 0) through (10, 5), (20, 13), (30, 25) to the end point (40, 35). Why does it never fall? Accumulation only grows and never shrinks. A common mistake is forgetting the point (0, 0), or drawing a smooth curve that strays from the given boundary points.
4. Step 4: Reading the median from the graphWith N values in total, the median sits at position N2. Read it in three steps: mark N2 on the vertical axis, for example 17.5 when N = 35; draw a horizontal line to meet the cumulative frequency curve; from that point drop a vertical line to the horizontal axis, and the reading is the estimated median. Why is the median at N2? The curve arranges the data by position, and the height at any point counts the data below that value. A common mistake is reversing the direction, treating N2 as a cumulative frequency to read across.
5. Step 5: Reading the quartiles from the graphThe lower quartile Q1 sits at position N4 and the upper quartile Q3 at 3N/4, read exactly as the median. For example, with N = 40, Q1 is at the 10th value and Q3 at the 30th: mark 10 and 30 on the vertical axis, draw horizontal lines to the curve, then drop vertically to read the horizontal axis. Why learn the quartiles? They form the interquartile range, an important measure of spread that resists extreme values. A common mistake is misremembering the position of Q3, or confusing Q1 with the minimum value of the data.
6. Step 6: The interquartile rangeThe interquartile range = upper quartile − lower quartile, that is Q3 − Q1, another measure of spread showing the range of the middle half of the data. For example, if Q1 = 22 and Q3 = 31, the interquartile range = 31 − 22 = 9. Why is it unaffected by extremes? It uses only the values at positions N4 and 3N/4, so even an extreme score of 100 or 0 leaves Q1 and Q3 almost unchanged. A common mistake is adding Q3 and Q1 instead of subtracting, or confusing the interquartile range with the range; only the range is stretched by extremes.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Cumulative frequency = running total from smallest to largest
• Graph uses upper class boundary vs cumulative frequency
• Median at N2 position
• Lower quartile Q1 at N4, upper quartile Q3 at 3N/4
• IQR = Q3 - Q1 (measure of spread, not affected by extremes)