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Welcome to GCSE Edexcel Maths revision.

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Topic M 42: Histograms, cumulative frequency and box plots.

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This video covers Higher tier.

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A histogram represents continuous grouped data.

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Bars meet at class boundaries; their widths correspond to class widths.

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For unequal classes, area, not height, represents frequency.

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Frequency density equals the fraction with numerator open bracket frequency close bracket and denominator open bracket class width close bracket , end fraction .

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Therefore frequency equals bar width multiplied by frequency density.

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Label the vertical axis Frequency density.

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Unequal-width histogram classes

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Worked example: Class 0 to 10 with frequency 8 has density 0.8.

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Class 10 to 30 with frequency 12 has density 0.6.

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The second bar is shorter but has greater area and more observations.

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Read class boundaries carefully; a class from 10 to 30 has width 20, not 21.

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Use the continuous measurement convention given in the question.

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For a partly filled class, estimating frequency from part of its area assumes values are distributed uniformly within that class.

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State that it is an estimate.

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Cumulative frequency is the running total up to each class boundary.

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Plot cumulative counts against upper class boundaries, usually starting at the lowest boundary with cumulative count 0.

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Reading a median from cumulative frequency

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A cumulative frequency curve never decreases.

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The final cumulative count equals the total sample size; use the class boundaries rather than midpoints on the horizontal axis.

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For a graph with n observations,

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read an estimated median at cumulative frequency the fraction with numerator open bracket n close bracket and denominator open bracket 2 close bracket ,

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end fraction ,

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lower quartile at the fraction with numerator open bracket n close bracket and denominator open bracket 4 close bracket ,

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end fraction and upper quartile at the fraction with numerator open bracket 3 n close bracket and denominator open bracket 4 close bracket ,

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end fraction .

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These graph estimates can differ from exact ordered-data conventions.

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Worked example: With 40 observations, read Q subscript open bracket 1 close bracket at cumulative count 10, median at 20 and Q subscript open bracket 3 close bracket at 30.

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Go horizontally to the curve, then vertically down to the measurement axis.

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To estimate how many values exceed a measurement, subtract the cumulative count at that measurement from the total.

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Keep uncertainty from graph reading and within-class distribution in mind.

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Interquartile range open bracket I Q R close bracket equals Q subscript open bracket 3 close bracket minus Q subscript open bracket 1 close bracket .

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It describes the spread of the middle 50 percent and is less affected by extreme values than the range.

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A standard GCSE box plot marks minimum, lower quartile, median, upper quartile and maximum on a common scale.

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The box extends from Q subscript open bracket 1 close bracket to Q subscript open bracket 3 close bracket , with a median line inside.

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Box plot from a five-number summary

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Worked example: Minimum 4,

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Q subscript open bracket 1 close bracket equals 10,

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median 16,

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Q subscript open bracket 3 close bracket equals 22 and maximum 35 give I Q R equals 12 and range equals 31.

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The box width does not show sample size.

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Compare medians for typical values and IQRs for consistency.

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A group with smaller I Q R has less variation in its middle half, which is not necessarily the same as smaller overall range.

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For exact ungrouped data, order the observations and follow the stated quartile convention; small datasets can use different conventions.

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For grouped continuous data, quartiles read from a curve are estimates.

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If a task supplies an outlier rule, apply it precisely.

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The often-used limits Q subscript open bracket 1 close bracket minus 1.5 I Q R and Q subscript open bracket 3 close bracket plus 1.5 I Q R flag unusual values,

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but do not by themselves prove those values are errors.

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That completes Histograms, cumulative frequency and box plots.

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Revisit the notes and test yourself on the revision website.