Edexcel A-level Mathematics (9MA0) · Data presentation and interpretation
Mini-Lesson
Data presentation and interpretation
This mini-lesson covers Topic 2 of Edexcel A-level Mathematics (9MA0): histograms and frequency density, cumulative frequency and linear interpolation, box plots, measures of location and spread (mean, median, quartiles, variance, standard deviation), outliers (the 1.5 × IQR rule and the mean ± 2 standard deviations rule), skewness, and bivariate data: scatter diagrams, correlation, regression lines, interpolation and the dangers of extrapolation.
Frequency density = frequency ÷ class width. With a histogram it is the area, not the height, that is proportional to frequency.
Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Throughout, take g = 9.8 m s−2 unless a question says otherwise. Press Start when you are ready.
Presentation · histograms
Histograms & frequency density
A histogram displays grouped continuous data. Because the classes may have different widths, the vertical axis is frequency density, and it is the area of each bar that is proportional to frequency.
frequency density = frequency ÷ class widthso: frequency = frequency density × class width
Worked example
The class 10 ≤ x < 25 has frequency 45. Class width = 25 − 10 = 15.
Frequency density = 45 ÷ 15 = 3.
Reversing it: a bar over 20 ≤ x < 30 with frequency density 2.4 represents 2.4 × 10 = 24 items.
Careful: "the tallest bar contains the most data" is false in a histogram with unequal widths. A short, wide bar can hold far more.
Calculate
Frequency density
1In a histogram, the class 10 ≤ x < 25 has a frequency of 45. Calculate the frequency density for this class.
Hint: frequency density = frequency ÷ class width = 45 ÷ (25 − 10).
Calculate
Back to a frequency
2On the same histogram, the bar for 20 ≤ x < 30 has a frequency density of 2.4. How many items are in this class?
items
Hint: frequency = frequency density × class width = 2.4 × (30 − 20).
Quick check
Height or area?
?In a histogram with unequal class widths, what does the area of each bar represent?
Presentation · location, spread & box plots
Cumulative frequency, quartiles & box plots
A cumulative frequency diagram plots the running total against the upper class boundary. Read the median at n ÷ 2, Q₁ at n ÷ 4 and Q₃ at 3n ÷ 4, then use linear interpolation to get percentiles from grouped data.
Measures of location: mean x̄ = Σx ÷ n (or Σfx ÷ Σf), median, mode. Measures of spread: range, interquartile range IQR = Q₃ − Q₁, interpercentile ranges, variance and standard deviation.
σ² = Σx²/n − x̄²standard deviation σ = √(Σx²/n − x̄²) · for frequencies use Σfx²/Σf − x̄²
A box plot shows the five-figure summary and any outliers, plotted as crosses beyond the whiskers.
Skew: if Q₂ − Q₁ < Q₃ − Q₂ the data has positive skew (a long tail to the right). Positive skew is also suggested by mean > median.
Calculate
Standard deviation
3Ten values have Σx = 100 and Σx² = 1094. Calculate the standard deviation, to 3 significant figures.
Cleaning data: an outlier may be a genuine extreme value or a recording error. Only remove it if there is a good reason (e.g. an impossible value, or a known mistake) — and say so. Missing entries (like "n/a" in the large data set) must be excluded, not read as zero.
Calculate
The 1.5 × IQR rule
4A data set has Q₁ = 18 and Q₃ = 30. Using the rule that an outlier is more than 1.5 × IQR beyond a quartile, find the upper limit above which a value is an outlier.
?The times have mean 50 and standard deviation 4. An outlier is defined as a value more than 2 standard deviations from the mean. Which of these values is an outlier?
Sort it
Which diagram is it?
Tap a statement, then tap the diagram it describes.
📊 Histogram
🗃️ Box plot
✳️ Scatter diagram
Interpretation · bivariate data
Scatter diagrams, correlation & regression
Bivariate data has two variables per item. Plot it on a scatter diagram. The explanatory (independent) variable goes on the x-axis; the response (dependent) variable on the y-axis.
Correlation describes how close the points lie to a straight line. The product moment correlation coefficient r satisfies −1 ≤ r ≤ 1. r = ±1 means the points lie exactly on a straight line; r near 0 means little or no linear correlation.
The regression line of y on x, y = a + bx, is the least-squares line. b is the change in y for a unit increase in x; a is the value of y when x = 0.
Interpolation — estimating inside the range of the data. This is reasonably reliable.
Extrapolation — estimating outside the range. This is unreliable: there is no evidence the linear relationship continues.
Correlation is not causation. Ice-cream sales and drownings both rise in summer; neither causes the other. Always look for a hidden third variable.
Calculate
Using a regression line
5For data collected for x between 5 and 20, the regression line of y on x is y = 4.2 + 1.6x. Use it to estimate y when x = 12.
Hint: y = 4.2 + 1.6 × 12 = 4.2 + 19.2. (x = 12 is inside the data range, so this is interpolation — a sound estimate.)
Quick check
Interpolation vs extrapolation
?The regression line above was found from data with x between 5 and 20. A student uses it to predict y when x = 60. Why is this unwise?
Quick check
Reading r
?A scatter diagram of two variables gives a product moment correlation coefficient of r = −0.92. How should this be interpreted?
Match it
Formula or phrase → its meaning
Tap an item on the left, then its partner on the right.
Item
Meaning
Presentation · linear interpolation
Linear interpolation for grouped data
With grouped data you do not know the individual values, so you assume the data is spread evenly within each class and interpolate.
estimate = L + ((position − cf below) ÷ f) × class widthL = lower boundary of the class containing the value you want
Worked example — the median
Classes and frequencies: 0–10: 5 · 10–20: 13 · 20–30: 17 · 30–40: 5. Total n = 40.
Cumulative frequencies: 5, 18, 35, 40. The median is the n ÷ 2 = 20th value, which lies in the class 20–30 (the cf jumps from 18 to 35 there).
Percentiles work identically — for the 30th percentile use position 0.3n instead of n ÷ 2, and for Q₃ use 0.75n.
Calculate
Median by interpolation
6Grouped data: 0–10 has frequency 5, 10–20 has 13, 20–30 has 17, 30–40 has 5 (n = 40). Use linear interpolation to estimate the median, to 3 significant figures.
Hint: The median is the 20th value. Cumulative frequency reaches 18 by the end of 10–20, so the median is in 20–30: median = 20 + ((20 − 18) ÷ 17) × 10.
Recap
The big ideas to know
Histogram: frequency density = frequency ÷ class width; area = frequency
Cumulative frequency: plot at upper class boundaries; median at n/2, use linear interpolation
Spread: IQR = Q₃ − Q₁ · σ = √(Σx²/n − x̄²)
Outliers: beyond Q₁ − 1.5 × IQR / Q₃ + 1.5 × IQR, or outside x̄ ± 2σ (the rule is given in the question)
Skew: Q₂ − Q₁ < Q₃ − Q₂ suggests positive skew
Bivariate: −1 ≤ r ≤ 1 · regression y = a + bx · interpolation is safe, extrapolation is not
Correlation ≠ causation, and data must be cleaned before it is analysed
That is the whole of 9MA0 Topic 2 — Data presentation and interpretation. Press Finish to see your score.
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