Statistics: Data Representation and Analysis

At GCSE you extend data representation to cumulative frequency graphs, box plots, histograms with frequency density, and more advanced comparison of distributions. You also study sampling methods and make inferences about populations.

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Averages and Spread (Review)

Mean, median, mode and range from raw data and frequency tables — see KS3 lesson for details.

Mean from a grouped frequency table: Estimated mean = sum(f × midpoint) / sum(f)

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Cumulative Frequency

Cumulative frequency is the running total of frequencies.

Plotting: plot (upper class boundary, cumulative frequency), then join with a smooth S-shaped curve.

Height h (cm)	Frequency	Cumulative Frequency
140 ≤ h < 150	4	4
150 ≤ h < 160	12	16
160 ≤ h < 170	20	36
170 ≤ h < 180	14	50

Reading off the graph:

• Median → read at n/2 = 25th value
• Lower quartile (Q1) → read at n/4 = 12.5th value
• Upper quartile (Q3) → read at 3n/4 = 37.5th value
• Interquartile range (IQR) = Q3 − Q1

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Box Plots (Box-and-Whisker Diagrams)

A box plot displays: Minimum, Q1, Median, Q3, Maximum.

• The box spans from Q1 to Q3 (the middle 50% of data).
• The whiskers extend to the minimum and maximum (or to outlier boundaries).
• IQR = Q3 − Q1 measures the spread of the middle half.

Comparing two distributions using box plots:

• Compare medians (typical value).
• Compare IQRs (consistency / spread).
• Comment on any overlap or skew.

Example comparison: "Group A has a higher median (72 vs 65), so Group A performed better on average. Group A also has a smaller IQR (12 vs 20), suggesting Group A's results are more consistent."

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Histograms [H]

Used for continuous grouped data where class widths may be unequal.

Frequency density = frequency ÷ class width

The area of each bar = frequency (not the height).

Reading a histogram: Frequency = frequency density × class width.

Example:

Time t (mins)	Class width	Freq density	Frequency
0 ≤ t < 10	10	3	30
10 ≤ t < 15	5	8	40
15 ≤ t < 25	10	5	50
25 ≤ t < 40	15	2	30

Total frequency = 150 Estimated mean = use midpoints 5, 12.5, 20, 32.5 with respective frequencies.

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Frequency Polygons

Plot the frequency against the midpoint of each class interval. Join the points with straight lines. Useful for comparing two distributions on the same graph.

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Sampling Methods