OCR A-level Mathematics A (H240) · 2.02 Data Presentation and Interpretation
Mini-Lesson
Data presentation
OCR section 2.02 asks you to present, summarise and above all interpret data: histograms, box plots, measures of location and spread, outliers, correlation and regression.
Questions here are often set in the context of OCR's pre-release large data set. Work through each screen, answer the questions as you go and collect ⭐ stars. Press Start when you're ready.
Data · histograms
Histograms and frequency density
A histogram is not a bar chart. The bars touch, the widths can differ, and the crucial rule is:
AREA = frequency ⇒ frequency density = frequency ÷ class widththe vertical axis is frequency DENSITY, never frequency
Worked example
The class 20 ≤ x < 30 has frequency 45.
Class width = 30 − 20 = 10.
Frequency density = 45 ÷ 10 = 4.5.
Reading a histogram backwards: to find a frequency, compute area = frequency density × class width. And to estimate the frequency in part of a bar, assume the data is evenly spread across the class and take that fraction of the area.
Calculate
Your turn — frequency density
1A histogram class 20 ≤ x < 30 has a frequency of 45. Find its frequency density.
Hint: frequency density = frequency ÷ class width = 45 ÷ 10.
Quick check
What does the bar mean?
?On a histogram with unequal class widths, what represents the frequency?
Data · location and spread
Measures of location and spread
Location:mean (uses every value; distorted by outliers), median (the middle; resistant to outliers), mode (most common).
Spread:range, interquartile range IQR = Q₃ − Q₁ (resistant to outliers), variance and standard deviation.
variance = (Σx²)/n − (mean)² · standard deviation = √variancethe standard deviation has the same units as the data; the variance does not
Worked example
n = 10, Σx = 250, Σx² = 6500.
mean = 250/10 = 25.
variance = 6500/10 − 25² = 650 − 625 = 25.
standard deviation = √25 = 5.
A useful check: the variance can never be negative. If Σx²/n comes out less than the mean squared, you have made an arithmetic error.
Calculate
Your turn — standard deviation
2For a data set with n = 10, Σx = 250 and Σx² = 6500, find the standard deviation.
Hint: mean = 25. Variance = 6500/10 − 25² = 650 − 625 = 25. Then take the square root.
Data · outliers
Outliers — two rules
An outlier is a value that is far from the rest. OCR uses two definitions, and the question will tell you which:
Rule 1: below Q₁ − 1.5 × IQR, or above Q₃ + 1.5 × IQRRule 2: more than 2 standard deviations from the mean, i.e. outside mean ± 2sd
Worked example — the IQR rule
Q₁ = 14, Q₃ = 26. So IQR = 26 − 14 = 12, and 1.5 × IQR = 18.
Upper fence = Q₃ + 18 = 26 + 18 = 44.
Lower fence = Q₁ − 18 = 14 − 18 = −4.
So any value above 44 or below −4 is an outlier. If the data cannot be negative (a time, a length), the lower fence catches nothing — perfectly normal.
Worked example — the mean plus or minus 2sd rule
mean = 50, sd = 6. Accepted range = 50 ± 2(6) = 50 ± 12, i.e. 38 to 62.
Anything above 62 or below 38 is flagged as an outlier.
Do not just delete outliers. An outlier may be a genuine extreme value or a recording error. OCR wants you to identify it, then comment on whether it should be removed and why.
Calculate
Your turn — upper fence
3A data set has Q₁ = 14 and Q₃ = 26. Using the 1.5 × IQR rule, find the value above which a data point is an outlier.
4Same data set (Q₁ = 14, Q₃ = 26). Using the 1.5 × IQR rule, find the value below which a data point is an outlier.
Hint: lower fence = Q₁ − 1.5 × IQR = 14 − 18. The answer is negative.
Calculate
Your turn — the 2sd rule
5A data set has mean 50 and standard deviation 6. Using the mean ± 2 standard deviations rule, find the upper limit of the accepted (non-outlier) range.
Hint: 50 + 2 × 6 = 50 + 12.
Data · box plots
Box plots and skew
A box plot displays five numbers: minimum, Q₁, median, Q₃, maximum — with outliers marked separately as crosses.
The shape reveals the skew:
Symmetric — the median sits centrally in the box and the whiskers are of similar length.
Positive skew — the median is closer to Q₁, and the upper whisker is longer. The tail stretches to the right. (Typically mean > median.)
Negative skew — the median is closer to Q₃, and the lower whisker is longer.
Comparing two box plots earns marks for one comment on location and one on spread, both in context: “Group A has a higher median (so was typically faster) and a smaller IQR (so was more consistent).”
Quick check
Reading the skew
?On a box plot the median lies much closer to Q₁ than to Q₃, and the upper whisker is long. The distribution is:
Data · correlation
Correlation and regression
A scatter diagram shows two variables together. The product moment correlation coefficient r measures the strength of the linear relationship:
−1 ≤ r ≤ 1r near +1: strong positive linear · r near −1: strong negative linear · r near 0: no LINEAR correlation
The regression line of y on x, y = a + bx, is the best-fit line for predicting y from x. Here x is the explanatory (independent) variable and y the response (dependent) variable.
Worked example — a regression line
Data collected for x between 2 and 12 gives the regression line y = 3.2 + 1.8x.
Predicting at x = 5 (inside the data range): y = 3.2 + 1.8(5) = 12.2. This is interpolation and is reliable.
Predicting at x = 40 (far outside the range): y = 3.2 + 72 = 75.2. This is extrapolation and is unreliable — you have no evidence the linear pattern continues that far.
Interpreting b: for every 1 unit increase in x, y increases by about 1.8 units.
Correlation is not causation. Ice cream sales correlate with drownings; neither causes the other. Both are driven by a third variable — hot weather. OCR loves this point.
Quick check
Interpolation or extrapolation?
?Data was collected for x between 2 and 12, giving the regression line y = 3.2 + 1.8x. A student uses it to predict y when x = 40. What is the problem?
Quick check
Interpreting r
?A data set has a product moment correlation coefficient of r = −0.92. This indicates:
Sort it
Location, spread or neither?
Tap a statistic, then tap what it measures.
📍 Location
↔️ Spread
❌ Neither
Match it
Definitions
Tap a term on the left, then its definition or formula.
Term
Definition
Recap
The big ideas to know
Histograms: area = frequency; height = frequency density = frequency ÷ class width
Location: mean (uses all data, distorted by outliers), median (resistant), mode