Probability and statistics · GCSE Maths
Cumulative frequency and box plots
GCSE Maths cumulative frequency and box plots: plot running totals at upper class boundaries, read the median and quartiles, then compare two distributions with IQR as well as average.
Cumulative frequency is a running total plotted at the upper end of each class. Median is at n/2, Q1 at n/4, Q3 at 3n/4. IQR = Q3 − Q1. Compare average and spread.
The important bits
What you need to know
- 1
Add a cumulative-frequency column: running total of frequencies. The last value equals n, the number of data values.
- 2
Plot cumulative frequency against the upper class boundary, not the midpoint. Join with a smooth curve or polygon as the paper expects.
- 3
Median: go to n/2 on the vertical axis, read across to the curve, down to the data axis. Q1 uses n/4; Q3 uses 3n/4. Some papers use (n+1)/4; follow the question’s convention if stated.
- 4
Interquartile range IQR = Q3 − Q1. It measures the spread of the middle half and is more resistant to outliers than the range.
- 5
A box plot shows minimum, Q1, median, Q3, maximum (or whiskers to a rule the paper gives). The box is Q1 to Q3; a line in the box is the median.
- 6
To compare two distributions, write one sentence about average (medians) and one about spread (IQRs or ranges), using the actual figures. “More consistent” means a smaller IQR.
- 7
The 10th percentile is the value at 0.10n on the cumulative-frequency axis. Reading “how many scored below 45” is a horizontal then vertical read the other way.
- 8
Grouped data give estimated median and quartiles from the graph. They are not exact because you no longer have the raw list. Say “estimate” if the data are grouped.
Quotations worth analysing
Short evidence. Real method.
“Plot at the upper class boundary”
The running total of 0–10, 10–20 is the number who scored up to 20, so the point sits at 20, not at 15. Plotting at midpoints shifts every quartile.
“IQR = Q3 − Q1”
The range uses extremes and can be dominated by one outlier. IQR is the length of the box and is the spread mark schemes want in “compare” questions.
“Class A has a higher median but a larger IQR”
Two facts, both with figures: average and spread. “Class A did better” without numbers scores poorly. Higher median is “typically larger”; larger IQR is “less consistent”.
Go deeper
Running total, then n/2 across and down
Frequencies 4, 10, 16, 8, 2. Cumulative: 4, 14, 30, 38, 40. n = 40. If the classes are 0–10, 10–20, 20–30, 30–40, 40–50, plot (10, 4), (20, 14), (30, 30), (40, 38), (50, 40), and usually (0, 0) if the data start at 0. Median at 20 on the vertical axis (n/2 = 20). Read across to the curve, down to the horizontal: that x-value is the estimated median. Q1 at 10, Q3 at 30. If those reads are 18, 26, 33, then IQR = 15. The graph is an estimate because everyone in 20–30 was treated as spread across that class by the slope of the curve. A table-only linear interpolation in the median class is the same idea without a graph: median class is the first cumulative at least 20, here 20–30, then interpolate.
Go deeper
Box plots are five-number summaries in a picture
Once you have min, Q1, median, Q3, max, draw a number line, a box from Q1 to Q3, a median line, and whiskers to min and max unless the paper uses an outlier rule. Two box plots on the same scale are how you compare. Look at the median lines: the class with the median further right has the higher typical value. Look at the box lengths: the longer box is more spread in the middle 50%. A long whisker on one side shows skew. Comment with numbers: “Team A median 12 s, Team B median 15 s, so Team A is typically faster; Team A IQR 4 s, Team B IQR 9 s, so Team A is more consistent.” Both sentences are required. Comparing only the maximum is not a comparison of the distributions.
Go deeper
Reading the graph backwards and comparing
“How many people scored less than 45?” Go to 45 on the data axis, up to the curve, across to cumulative frequency, say 28. That 28 is an estimate. “What mark did the top 10% beat?” Top 10% starts at 0.90n. For n = 40, 36 on the vertical axis, across and down to a mark. Skew: if the median is closer to Q1 than to Q3, the upper tail is longer (positive skew). Use that language only if you can point to the box plot. When two cumulative-frequency curves are drawn on one grid, the curve that reaches n/2 at a smaller x has the smaller median. The curve that is steeper through the middle has more of the data packed in a smaller interval, hence a smaller IQR. Steepness is spread; left-right position is average.
See the idea in action
n = 80 test scores. Cumulative frequencies plotted at upper boundaries give an estimated Q1 = 42, median = 51, Q3 = 63. The lowest score is 20 and the highest is 78. Compare with a second class whose box plot has Q1 = 40, median = 47, Q3 = 55, min 18, max 90. Step 1: Class 1 IQR = 63 − 42 = 21. Class 2 IQR = 55 − 40 = 15. Step 2: Average: Class 1 median 51 > Class 2 median 47, so Class 1 typically scored higher. Step 3: Spread: Class 2 has the smaller IQR (15 vs 21), so Class 2 is more consistent in the middle 50%. Step 4: Range: Class 2 range 72 vs Class 1 range 58, so Class 2 has a wider overall spread, pulled by a high of 90. A complete comment uses median and IQR with the figures, not “Class 1 did better” alone.
Exam technique
Turn knowledge into marks
Plot cumulative frequency at the upper class boundary. For compare questions, write one sentence about medians and one about IQRs, each with numbers from the graph or box plot.
Common mistakes
Do not give these marks away
- 01
Plotting cumulative frequency at midpoints or at lower boundaries, which shifts the median and quartiles.
- 02
Comparing two classes with only the mean or only the highest score, with no measure of spread.
- 03
Reading Q1 at n/2 instead of n/4, or subtracting Q1 from the median and calling it the IQR.
For 80 values, the median on a cumulative-frequency graph is read at a cumulative frequency of
A40
B80
C20
D60
Show the answer
40. n/2 = 40. 80 is the top of the graph, the total. 20 is n/4 (Q1). 60 is 3n/4 (Q3).
Quick questions
If this is the bit you searched
Where do you plot cumulative frequency?
At the upper class boundary, against the running total. The last point is (largest upper bound, n). Often include (lowest lower bound, 0).
How do you find the median from a cumulative-frequency graph?
Go to n/2 on the cumulative-frequency axis, read across to the curve and down to the data axis. That value is the estimated median.
How do you compare two box plots?
Compare medians for typical value and IQRs for consistency, quoting the figures. You may also mention range or skew if the whiskers show it.
What does a smaller IQR mean?
The middle half of the data is packed into a shorter interval, so that group is more consistent. It is not automatically “better” unless the context says smaller spread is desirable.