6.2 Statistical measures

Learning objectives

6.2A The concept of averageUnderstand averages for data presented in lists or tables.6.2B The mean, median, mode and range for a discrete datacalculate the mean, median, mode and range for a discrete data set Includes simple problems using these measures 6.2C Estimating the mean from grouped dataCalculate an estimate of the mean from grouped data.6.2D The modal class for grouped dataidentify the modal class for grouped data

An average is a single value used to describe a typical or central feature of a data set. Different averages answer different questions, so the context and data shape determine which is useful.

Measure	What it represents	Sensitive to extreme values?
mean	equal-share value using every observation	yes
median	middle of ordered data	much less
mode	most frequent value or category	no

For a frequency table, frequencies tell how many times each value occurs; expand conceptually or use cumulative counts to locate the middle.

There is no universally 'best' average. A data set may have no mode or several modes, and a mean can be unrepresentative when a few extreme values pull it away from most observations.

For discrete data, calculate each measure from the actual values and their frequencies, keeping centre and spread distinct.

Measure	Rule
mean	$\dfrac{\sum x}{n}$ , or $\dfrac{\sum fx}{\sum f}$ for a frequency table
median	middle ordered value; average the two middle values when $n$ is even
mode	value with greatest frequency
range	maximum $-$ minimum

In reverse-mean problems, first convert a mean to a total: $\text{total}=\text{mean}\times\text{number}$ . Combine or subtract totals before dividing by the new number of values.

Order the data before finding the median. Do not divide by the number of table rows: for a frequency table the divisor is $\sum f$ .

Grouped data do not reveal the exact observations, so represent every value in a class by its class midpoint and calculate an estimated mean.

Step	Calculation
midpoint	$(\text{lower boundary}+\text{upper boundary})/2$
estimated class total	midpoint $\times$ frequency
estimated mean	$\dfrac{\sum(\text{midpoint}\times f)}{\sum f}$

For $100<h\le110$ with frequency 12, use midpoint 105, contributing $105\times12=1260$ to the estimated total.

The answer is an estimate because actual values need not equal the midpoint. Never multiply frequency by class width when estimating the mean.

The modal class is the class interval containing the greatest frequency. State the complete interval, including its boundary convention.

Class	Frequency
$0<d\le4$	9
$4<d\le8$	15
$8<d\le12$	7

The modal class in the example is $4<d\le8$ because 15 is the largest frequency.

The modal class is an interval, not its midpoint. If information is shown only by a histogram with unequal widths, compare frequency density or bar area according to the task rather than assuming the tallest-looking width gives the class frequency.

For total frequency $N$ , the median lies at cumulative frequency $N/2$ . Read horizontally from $N/2$ to the cumulative-frequency curve, then vertically to the data axis.

Step	Action
1	read the final cumulative total $N$
2	calculate $N/2$
3	move from $N/2$ on the vertical axis to the curve
4	move down to estimate the median value

To compare typical values for two groups, compare their medians in context: the group with the larger median tends to have the larger observed value.

Use the cumulative-frequency axis position, not half the horizontal-axis range. The result is an estimate and should not claim more precision than the graph supports.

A measure of spread describes how variable or dispersed the data are. Larger spread means values are less tightly clustered; smaller spread means greater consistency.

Measure	Uses	Sensitivity
range $=\max-\min$	full data width	strongly affected by extremes
interquartile range $=Q_3-Q_1$	middle 50% width	resistant to extremes

Compare groups with both centre and spread: median describes a typical value, while IQR describes consistency of the middle half. Write comparisons in the units and context of the data.

A higher median does not mean greater spread, and a smaller IQR does not mean a smaller typical value; centre and variation answer different questions.

Order the data, locate the lower quartile $Q_1$ and upper quartile $Q_3$ using the course's median-of-halves convention, then calculate $IQR=Q_3-Q_1$ .

Ordered data count	Split for quartiles
odd	exclude the overall median, then find medians of lower and upper halves
even	split into equal lower and upper halves, then find each half's median

For 11 ordered values, the 6th is the overall median; $Q_1$ is the median of values 1–5 and $Q_3$ is the median of values 7–11.

Quartiles come from ordered positions, not one quarter and three quarters of the numerical range. State and use one consistent convention.

For total frequency $N$ , read the lower quartile at cumulative frequency $N/4$ and the upper quartile at $3N/4$ , then subtract.

Quantity	Cumulative frequency position
$Q_1$	$N/4$
median	$N/2$
$Q_3$	$3N/4$
IQR	$Q_3-Q_1$

A larger estimated IQR means the middle half is more spread out; a smaller estimated IQR means it is more consistent. Support the statement with both estimated IQRs when comparing groups.

Do not subtract cumulative frequencies $3N/4-N/4$ ; those positions locate quartile values on the horizontal axis, and the horizontal readings must be subtracted.

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—6.3 ProbabilityRead

6.2 Statistical measures

Learning objectives

Understand what an average represents

Calculate mean, median, mode and range

Estimate the mean from grouped data

Identify the modal class

Estimate the median from cumulative frequency

Interpret measures of spread

Find the interquartile range of discrete data

Estimate interquartile range from cumulative frequency

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