CCEA GCE Mathematics (2210) · Data presentation and interpretation
Mini-Lesson
Data presentation and interpretation
This mini-lesson covers histograms (frequency density), box plots, quartiles and outliers, mean and standard deviation, linear interpolation from grouped data, coding, and scatter diagrams, correlation and regression lines.
Where this sits:Unit AS 2: Applied Mathematics — data presentation and interpretation: measures of location and spread, histograms, box plots, outliers, correlation, regression and interpretation in context (with a large data set).
Work through each screen, answer the questions as you go (some are reasoning, some are calculations) and collect ⭐ stars. Press Start when you are ready.
Histograms
Frequency density
A histogram uses area to represent frequency, so classes of unequal width still compare fairly. The vertical axis is frequency density:
frequency density = frequency ÷ class widthfrequency = frequency density × class width = the AREA of the bar
Worked example
Class 20 ≤ x < 40 (width 20) has frequency 30.
Frequency density = 30 ÷ 20 = 1.5
Histogram ≠ bar chart: a histogram is for continuous data, has no gaps, and its bar heights are densities, not frequencies.
Calculate
Frequency density
1In a histogram, the class 20 ≤ x < 40 contains 30 items. Find the frequency density.
Hint: Frequency density = frequency ÷ class width = 30 ÷ (40 − 20) = 30 ÷ 20.
The other common rule is more than 2 standard deviations from the mean. The question will tell you which to use — use the one given.
Worked example
Q₁ = 12, Q₃ = 20 ⇒ IQR = 8
Upper fence = 20 + 1.5 × 8 = 20 + 12 = 32
Lower fence = 12 − 12 = 0. So any value above 32 or below 0 is an outlier.
An outlier is not automatically an error — it may be genuine. You should say whether it should be removed (a mistake in recording) or kept (a real extreme value).
Calculate
Outlier boundary
2A data set has Q₁ = 12 and Q₃ = 20. Using the 1.5 × IQR rule, find the value above which a data point is an outlier.
mean x̄ = Σx / n · variance = Σx²/n − x̄² · sd = √variancefor grouped data use Σfx/Σf and Σfx²/Σf − x̄²
Worked example
n = 10, Σx = 100, Σx² = 1200.
Mean = 100/10 = 10
Variance = 1200/10 − 10² = 120 − 100 = 20
Standard deviation = √20 = 4.47 (3 s.f.)
Median vs mean: the median resists outliers; the mean does not. For skewed data quote the median and IQR; for roughly symmetric data quote the mean and standard deviation.
Calculate
Standard deviation
3For a data set with n = 10, Σx = 100 and Σx² = 1200, find the standard deviation to 3 significant figures.
4Grouped data: 0–10 has frequency 8, 10–20 has frequency 14, 20–30 has frequency 18 (n = 40). Estimate the median by linear interpolation, to 2 decimal places.
Hint: The median is the (40 ÷ 2) = 20th value. The first class holds 8, so we need 12 more from the 14 in the class 10–20: median = 10 + (12 ÷ 14) × 10 = 10 + 8.571.
Coding
Coding data
Coding rescales messy numbers. If y = (x − a)/b then:
x̄ = a + b ȳ · σₓ = b σᵧthe SUBTRACTION shifts the mean but does NOT affect the spread
Worked example
y = (x − 20)/5, with ȳ = 3.2 and σᵧ = 1.4.
Mean of x = 20 + 5(3.2) = 36
Standard deviation of x = 5 × 1.4 = 7 (the −20 has no effect on spread)
Calculate
Coding and spread
5Data are coded using y = (x − 20)/5. The coded data have standard deviation 1.4. Find the standard deviation of x.
Hint: Adding or subtracting a constant does not change spread; only the divisor matters. σₓ = 5 × σᵧ = 5 × 1.4.
Bivariate data
Correlation and regression
Scatter diagram — plots pairs (x, y) to reveal a relationship.
Correlation — the strength and direction of a linear relationship (positive, negative, or none).
Regression line y = a + bx — the line of best fit for predicting y from x. Here b is the change in y per unit increase in x.
Two warnings that earn marks: (1) correlation does not imply causation — a third variable may drive both; (2) predicting outside the range of the data (extrapolation) is unreliable, while predicting inside it (interpolation) is reasonable.
Sort it
Which diagram shows it?
Tap a feature, then tap the diagram that displays it.
📊 Histogram
📦 Box plot
🔵 Scatter diagram
Quick check
Skew
?A box plot shows Q₁ = 20, median = 25, Q₃ = 40. What does this suggest?
Quick check
Correlation and causation
?Ice-cream sales and drowning incidents are strongly positively correlated. What is the correct conclusion?
Match it
Match the statistic
Tap the term on the left, then its meaning on the right.
Statement
Answer
Quick check
Comparing distributions
?You are asked to compare two data sets. What must a full answer include?
Quick check
Histogram bar area
?In a histogram, a bar has frequency density 2.5 and covers the class 10 ≤ x < 30. How many items are in that class?
Examiner traps
Data pitfalls
Histogram heights are densities, not frequencies. Frequency = density × width = the bar's area.
Use the outlier rule you are given (1.5 × IQR, or 2 standard deviations) — do not switch.
Comparisons need location AND spread, both in context.
Coding: adding/subtracting a constant does not change the standard deviation; only the multiplier does.
Never claim causation from correlation, and never extrapolate beyond the data range.
Recap
The big ideas to know
Histogram: frequency density = frequency ÷ class width; frequency = area of the bar
Outliers: Q₁ − 1.5×IQR and Q₃ + 1.5×IQR (or beyond 2 sd from the mean)
Mean and sd: x̄ = Σx/n · variance = Σx²/n − x̄² · sd = √variance
Interpolation: estimate the median inside a grouped class · Coding: y = (x − a)/b ⇒ σₓ = bσᵧ
Correlation ≠ causation; extrapolation beyond the data is unreliable
Data interpretation marks are won by saying what a number MEANS in context, not just calculating it. Press Finish to see your score.
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