Histograms with Frequency Density, Cumulative Frequency Curves & Box Plots
On the GCSE Maths Higher Tier, grouped continuous data with unequal class intervals is displayed using histograms, where the vertical y-axis is Frequency Density ($ ext{FD} = ext{Frequency} div ext{Class Width}$) and the area of each bar (not its height) represents the frequency. To estimate medians, quartiles, and the interquartile range (IQR), grouped frequencies are added as a running total and plotted at the upper endpoint of each class interval to form a cumulative frequency curve. Examiners frequently ask students to construct a box plot directly from a cumulative frequency curve and compare two distributions in context using both a measure of average (the median) and a measure of spread (the IQR).
- Histograms & Frequency Density Formula (Higher Tier)
- Used for continuous grouped data with unequal class widths. Formula: $\text{Frequency Density (FD)} = \frac{\text{Frequency}}{\text{Class Width}}$, and rearranging gives $\text{Frequency} = \text{Class Width} \times \text{Frequency Density}$ (the area of the bar). If the y-axis has no numbers (an unlabelled scale), use one bar whose frequency IS given in the table to work out the Frequency Density of that bar and label the y-axis scale.
- Estimating Proportions Inside a Histogram Bar
- To estimate how many items have a value greater than $x$ when $x$ falls inside a bar running from $a$ to $b$: assume the data is uniformly distributed across the class $[a, b]$ and calculate the area of the sub-rectangle from $x$ to $b$: $\text{Frequency} = (b - x) \times \text{FD}$.
- Cumulative Frequency Tables & Curves (Higher Tier)
- 1. Calculate the cumulative frequency column by keeping a running total (adding each class frequency to the sum of all previous frequencies; the final value must equal the total sample size $N$). 2. Plot each cumulative frequency against the upper endpoint of its class interval (never the midpoint). 3. Join the points with a smooth S-shaped curve (or straight line segments if instructed).
Quadratic Parabola Plotter, Roots, Turning Point & Quadratic Formula
Adjust coefficients a, b, and c in y = ax² + bx + c to see how the parabola, roots (x-intercepts), and turning point move on the graph.
y = x² - 2x - 3
(-2)² − 4(1)(-3) = 16 → 2 real roots (crosses x-axis twice)x = 1.00b in brackets when squaring: (-2)² = 4 (always positive!). Typing -2² into a calculator without brackets gives -4 and loses the accuracy mark.Exam notes: Frequency Density Equation, Unequal Class Widths, Cumulative Frequency & Box Plots
- In a GCSE Higher Tier Histogram with unequal class widths, the y-axis is never Frequency - it is always frequency density, calculated using the equation: Frequency Density = Frequency ÷ Class Width.
- Rearranging the Frequency Density equation gives: Frequency = Frequency Density × Class Width. Geometrically, this means the area OF each bar (width × height) is equal to the frequency (not the height of the bar).
- For a Cumulative Frequency Curve, always plot points at the upper bound of each class interval (never the midpoint) and join with a smooth S-shaped curve. Read the Median at 50% of N, Lower Quartile (LQ) at 25% of N, and Upper Quartile (UQ) at 75% of N; then Interquartile Range (IQR) = UQ − LQ.
- When comparing two Box Plots in a 2-mark exam question, you must make two distinct comparisons in context: (1) compare a measure of average (the Medians, stating which group is higher on average) and (2) compare a measure of spread (the IQRs or Ranges, stating which group is more consistent/less varied).
| Diagram Type | How to Plot (Axes & Coordinates) | What Examiners Ask You to Calculate |
|---|---|---|
| Histogram (Unequal Class Widths) | x-axis = Continuous Variable; y-axis = Frequency Density (FD = Freq ÷ Width) | Find missing frequencies from bar areas (Freq = Width × FD) or estimate totals across partial bars |
| Cumulative Frequency Diagram | x-axis = Upper Class Boundary; y-axis = Running Total (Cumulative Frequency) | Estimate Median (N/2), Lower Quartile (N/4), Upper Quartile (3N/4), IQR, and count above a threshold |
| Box-and-Whisker Plot | 5-number summary: Minimum, Lower Quartile (Q₁), Median (Q₂), Upper Quartile (Q₃), Maximum | Compare two distributions in context: 1 mark for Median (average) + 1 mark for IQR (consistency/spread) |
Key Concepts & Exam Definitions (6)
- Histograms & Frequency Density Formula (Higher Tier)
- Used for continuous grouped data with unequal class widths. Formula: $\text{Frequency Density (FD)} = \frac{\text{Frequency}}{\text{Class Width}}$, and rearranging gives $\text{Frequency} = \text{Class Width} \times \text{Frequency Density}$ (the area of the bar). If the y-axis has no numbers (an unlabelled scale), use one bar whose frequency IS given in the table to work out the Frequency Density of that bar and label the y-axis scale.
- Estimating Proportions Inside a Histogram Bar
- To estimate how many items have a value greater than $x$ when $x$ falls inside a bar running from $a$ to $b$: assume the data is uniformly distributed across the class $[a, b]$ and calculate the area of the sub-rectangle from $x$ to $b$: $\text{Frequency} = (b - x) \times \text{FD}$.
- Cumulative Frequency Tables & Curves (Higher Tier)
- 1. Calculate the cumulative frequency column by keeping a running total (adding each class frequency to the sum of all previous frequencies; the final value must equal the total sample size $N$). 2. Plot each cumulative frequency against the upper endpoint of its class interval (never the midpoint). 3. Join the points with a smooth S-shaped curve (or straight line segments if instructed).
- Reading Median, LQ, UQ & Interquartile Range (IQR)
- For total cumulative frequency $N$: 1. Median ($Q_2$): read across from $\frac{1}{2}N$ (50%) on the cumulative frequency y-axis to the curve, and down to the x-axis. 2. Lower Quartile (LQ, $Q_1$): read across from $\frac{1}{4}N$ (25%). 3. Upper Quartile (UQ, $Q_3$): read across from $\frac{3}{4}N$ (75%). 4. Interquartile Range: $\text{IQR} = \text{UQ} - \text{LQ}$ (measures the spread of the middle 50% of the data, unaffected by extreme outliers).
- Box Plots (Box-and-Whisker Diagrams) (Higher Tier)
- Drawn to scale above a number line using five summary statistics: Minimum value (left whisker), Lower Quartile (left edge of box), Median (vertical line inside box), Upper Quartile (right edge of box), and Maximum value (right whisker). Each of the four sections of a box plot represents exactly 25% (one quarter) of the population.
- Comparing two box plots / distributions
- Always write two structured sentences in context: 1. Average comparison: 'On average, Group A had higher [scores/times/weights] than Group B because Group A has a higher median ([value] vs [value]).' 2. Spread/consistency comparison: 'Group B had more consistent (less varied) [scores/times/weights] than Group A because Group B has a smaller interquartile range ([value] vs [value]).'
Quick-Fire Flashcards (6)
Click any card to check the answer, then mark it as Known to track your progress.
State the formula for Frequency Density and explain how to find the frequency from a histogram bar.
In a histogram, the class interval 20 < t ≤ 35 has a frequency of 45. What is the class width and what is the frequency density plotted on the y-axis?
For the class interval 10 < w ≤ 30 with cumulative frequency 42, at what x-coordinate do you plot the point (x, 42) on a cumulative frequency graph?
If a cumulative frequency graph has a total of N = 80 people, at what values on the vertical axis do you read off the Lower Quartile, Median, and Upper Quartile?
State the formula for the Interquartile Range (IQR) and explain why it is often better than the Range.
What two things must you compare (in context) when comparing two box plots for 2 marks?
Exam-Style Questions & Mark Schemes (4)
Write or think through your answer first, then click Check Mark Scheme to see the exact points examiners award marks for.
- Q1
An incomplete histogram and table show the times t (in minutes) taken by commuters to travel to work. The class 0 < t ≤ 10 has frequency 15; 10 < t ≤ 25 has frequency 45; 25 < t ≤ 30 has frequency 30; and 30 < t ≤ 50 has a bar of frequency density 1.8. (a) Calculate the frequency densities for the first three classes. (b) Work out the total number of commuters in the survey, and estimate how many commuters took more than 28 minutes. (Higher Tier) [5 marks]
Click “Check Mark Scheme” to compare your answer, 1. (a) Calculate Frequency Density = Frequency ÷ Class Width for the first three bars: for 0 < t ≤ 10 (width 10), FD = 15 ÷ 10 = 1.5; for 10 < t ≤ 25 (width 15), FD = 45 ÷ 15 = 3.0; for 25 < t ≤ 30 (width 5), FD = 30 ÷ 5 = 6.0 [2 marks]. 2. (b) Find the frequency of the 4th class (30 < t ≤ 50, width 20) using Area = Width x FD = 20 x 1.8 = 36 commuters [1 mark]. 3. Total number of commuters = 15 + 45 + 30 + 36 = 126 commuters [1 mark]. 4. Commuters taking more than 28 minutes = slice of the 25-30 bar from 28 to 30 (width 2, FD 6.0) plus all of the 30-50 bar: (2 x 6.0) + 36 = 12 + 36 = 48 commuters [1 mark].
- Q2
The heights of 60 plants were recorded in a grouped table: 0 < h ≤ 10 (freq 4), 10 < h ≤ 20 (freq 11), 20 < h ≤ 30 (freq 23), 30 < h ≤ 40 (freq 15), and 40 < h ≤ 50 (freq 7).Write down the coordinates of all five points that should be plotted to draw a cumulative frequency graph, and explain how to use the graph to estimate the number of plants taller than 34 cm. (Higher Tier) [4 marks]
Click “Check Mark Scheme” to compare your answer, 1. Calculate the running total cumulative frequencies: 4, 4 + 11 = 15, 15 + 23 = 38, 38 + 15 = 53, and 53 + 7 = 60 [1 mark]. 2. Plot each cumulative frequency at the upper endpoint of its class interval: (10, 4), (20, 15), (30, 38), (40, 53), and (50, 60) [1 mark]. 3. To estimate the number of plants taller than 34 cm, find 34 cm on the horizontal height axis, go vertically up to the cumulative frequency curve, and read across to the vertical cumulative frequency axis (say C, which represents plants ≤ 34 cm) [1 mark]. 4. Subtract that reading C from the total number of plants (60 - C) because 'taller than 34 cm' is the upper tail above 34 cm [1 mark].
- Q3
For a set of 120 test scores, the lowest score is 18, the lower quartile is 42, the median is 58, the interquartile range is 25, and the range is 74. (a) Work out the upper quartile and the highest score needed to draw a box plot. (b) In a second class of 120 students, the median score is 63 and the interquartile range is 19. Compare the distributions of the test scores in the two classes. (Higher Tier) [4 marks]
Click “Check Mark Scheme” to compare your answer, 1. (a) Since Interquartile Range = UQ - LQ, Upper Quartile (UQ) = LQ + IQR = 42 + 25 = 67 [1 mark]. 2. Since Range = Highest: Lowest, Highest score = Lowest + Range = 18 + 74 = 92 [1 mark]. 3. (b) Average comparison in context: On average, students in the second class achieved higher test scores than the first class because their median score was higher (63 compared to 58) [1 mark]. 4. Spread comparison in context: The test scores in the second class were more consistent (less varied/spread out) than in the first class because their interquartile range was smaller (19 compared to 25) [1 mark].
- Q4
A box plot shows the weights of 240 parcels at a sorting office, where Minimum = 0.5 kg, LQ = 1.8 kg, Median = 3.2 kg, UQ = 5.0 kg, and Maximum = 8.5 kg. Work out the number of parcels that weigh more than 1.8 kg, and explain why you cannot tell from the box plot how many parcels weigh exactly 4.0 kg. (Higher Tier) [3 marks]
Click “Check Mark Scheme” to compare your answer, 1. 1.8 kg is the Lower Quartile (LQ), which marks the 25% point of the distribution [1 mark]. 2. Therefore, 75% (three-quarters) of the parcels weigh more than 1.8 kg: 0.75 x 240 = 180 parcels [1 mark]. 3. A box plot only displays the five summary quartile values (showing that 25% of parcels lie between 3.2 kg and 5.0 kg), and does not show how the individual parcel weights are distributed within that interval [1 mark].
Common Exam Questions (4)
What is the golden rule of Histograms on the Higher Tier?
In a histogram: 1. The vertical y-axis is always Frequency Density ($ ext{FD} = ext{Frequency} div ext{Class Width}$). 2. The frequency is equal to the area of the bar ($ ext{Frequency} = ext{Class Width} imes ext{Frequency Density}$), never the height of the bar. If a question asks for the number of people between two values that cut part-way through a bar, simply calculate the area of that slice of the bar.
Where do you plot the points on a Cumulative Frequency graph vs a Frequency Polygon?
Do not mix these two up. 1. On a cumulative frequency graph, you always plot the running total cumulative frequency at the upper bound (end-point) of each class interval (because by the end of the interval, all those items have accumulated). 2. On a frequency polygon, you plot the individual class frequency at the midpoint of each class interval and join the points with straight ruler lines.
What are the 5 key values plotted on a Box Plot (Box-and-Whisker diagram)?
From left to right, the 5 vertical lines on a box plot are: 1. Lowest value (Minimum), 2. Lower Quartile (LQ, at 25% of total $N$), 3. Median ($Q_2$, at 50% of total $N$), 4. Upper Quartile (UQ, at 75% of total $N$), and 5. Highest value (Maximum). The width of the central box is the Interquartile Range ($ ext{IQR} = ext{UQ} - ext{LQ}$), which contains the middle 50% of the data.
How do you score full marks on a 2-mark 'Compare the two distributions' question?
You must make two distinct comparisons, and both must be in the context of the question (e.g., 'weights of apples' or 'test scores', not just 'Group A is higher'): 1. Compare the medians (state which group has a higher median on average, quoting the values). 2. Compare the interquartile ranges (or ranges) (state which group has a smaller IQR and explain that a smaller IQR means the data is more consistent / less varied).
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