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E9.3 Averages and measures of spread

Syllabus
0580–2028–2029
Topic
E9.3
Level
Extended

Learning objectives

E9.3.1Calculate the mean, median, mode, quartiles, range and interquartile• Calculate the mean, median, mode, quartiles, range and interquartile range for individual data and distinguish between the purposes for which these are used.E9.3.2Calculate an estimate of the mean for grouped discrete or grouped• Calculate an estimate of the mean for grouped discrete or grouped continuous data.E9.3.3Identify the modal class from a grouped frequency distribution.• Identify the modal class from a grouped frequency distribution.

Calculate and choose averages and measures of spread

Mean, median and mode describe centre; quartiles divide ordered data into quarters; range and interquartile range describe spread.

Measure Calculation Main use
Mean sum of values ÷\div number of values uses every numerical value; affected by extremes
Median middle of ordered data resistant to extremes
Mode most frequent value identifies the most common value or category
Quartiles Q1Q_1 and Q3Q_3 mark the lower and upper quarter positions locate the middle half of ordered data
Range maximum −- minimum full spread; sensitive to extremes
IQR Q3−Q1Q_3-Q_1 spread of the middle half; resistant to extremes

For ordered data 2,4,5,6,7,8,9,112,4,5,6,7,8,9,11, the median is (6+7)÷2=6.5(6+7)\div2=6.5. The lower half is 2,4,5,62,4,5,6, so Q1=(4+5)÷2=4.5Q_1=(4+5)\div2=4.5; the upper half gives Q3=(8+9)÷2=8.5Q_3=(8+9)\div2=8.5. Thus IQR =8.5−4.5=4=8.5-4.5=4, while range =11−2=9=11-2=9.

Use centre and spread together: median with IQR is often more representative when extremes are present; mean with range may expose the influence of the full data. State the measure used rather than saying only 'average' or 'variation'.

Order the data before finding median or quartiles. Apply one consistent quartile convention, do not confuse a quartile value with a quarter of the numerical range, and remember that IQR ignores the outer quarters.

Estimate the mean of grouped data

For grouped data, estimate the mean by representing every value in a class by that class midpoint.

estimated mean=∑fm∑f,m=lower boundary+upper boundary2\text{estimated mean}=\frac{\sum fm}{\sum f},\qquad m=\frac{\text{lower boundary}+\text{upper boundary}}{2}

Class Frequency ff Midpoint mm fmfm
0<x≤200<x\leq20 3 10 30
20<x≤4020<x\leq40 5 30 150
40<x≤6040<x\leq60 2 50 100
Total 10 280

The estimate is 280÷10=28280\div10=28. Use the frequency, not the class width, as the multiplier. The same midpoint method works for grouped discrete or grouped continuous intervals when their frequencies are known.

The result is an estimate because the exact values inside each class are unknown and need not all equal the midpoint. Preserve units and avoid rounding midpoints or intermediate totals unnecessarily.

Identify the modal class

The modal class is the class interval with the greatest frequency in a grouped frequency distribution.

Height hh (m) Frequency
1.3<h≤1.41.3<h\leq1.4 6
1.4<h≤1.51.4<h\leq1.5 11
1.5<h≤1.61.5<h\leq1.6 18
1.6<h≤1.71.6<h\leq1.7 9

The greatest frequency is 1818, so the modal class is 1.5<h≤1.61.5<h\leq1.6 m. Give the complete interval with its inequality signs and unit, not the frequency 1818.

Grouped data do not reveal the exact most common value inside the interval, so report a modal class rather than inventing a mode. Do not choose the widest class or the interval with the largest boundary.