S4 - Averages and range

S4 - Averages and range

Describe a data set using its mean, median, mode and range. Distinguish data types, find a missing value from a mean, and explain how an outlier can change the story an average tells.

What kind of data have you collected?

Five people wait 1,2,2,3,421,2,2,3,42 minutes for a bus. Their mean wait is 1010 minutes, yet four people wait less than 1010 minutes. Can an average be correct and still give a misleading impression? Keep that question beside you as we choose what to measure.

You already know

A frequency counts how often a value occurs. Tables and charts organise observations, but the population and the way a sample was selected still matter.

Start by identifying the data. A pupil counting passengers and timing buses collects two kinds of number. The passenger count jumps in whole people. A waiting time can, in principle, lie anywhere between two recorded times, even if a stopwatch rounds it to seconds.

Discrete and continuous data

Discrete data take separate possible values, such as the number of passengers. Continuous data are measurements that can take any value in an interval, such as waiting time, mass or length. Recording a continuous measurement to the nearest whole number does not turn the underlying quantity into a count.

A value can also be a category: a bus route name or a favourite colour. Arithmetic on numerical labels is not automatically meaningful. The mean of shirt numbers does not describe a typical player's ability.

Primary and secondary data

Primary data are collected first-hand for the investigation, such as timing buses yourself. Secondary data were already collected by someone else, such as a published timetable or an existing survey. Secondary data can be useful, but check their date, definitions and whether they answer your question.

Neither label guarantees quality. Your own tiny biased survey can be worse than a carefully collected existing dataset. These definitions are not on the formulae sheet: learn them.

Key idea

Ask two different questions: “Where did these data come from?” and “What sort of values can this variable take?”

Rounding is a reporting choice, so it does not settle the second question.

The mean and missing values

The mean shares the total equally. If four jars contain 4,6,8,104,6,8,10 counters, their 2828 counters could be shared as 77 per jar. The original jars need not contain 77.

Mean

The total of the numerical values divided by the number of values. It uses every value. For example, 22 and 33 have mean 2.52.5, even though a count cannot itself be 2.52.5.

That last example is a small surprise: an average of whole-number counts can be a number nobody could actually report as their own count. The mean describes the group through a fair share, not necessarily one member.

Mean and total

For a non-empty numerical data set,

mean=totalnumber of valuestotal=mean×number of values\begin{aligned} \text{mean}&=\frac{\text{total}}{\text{number of values}}\\ \text{total}&=\text{mean}\times\text{number of values} \end{aligned}

The mean has the same units as the data.

Not on the formulae sheet: learn it.

Work with the total first. It is also the bridge back from a given mean to a missing value.

Worked example 1: an equal share

Question. Work out the mean of 4,6,8,104,6,8,10. Non-calculator.

Plan. Add the four values, then divide by four.

total=4+6+8+10=28include every valuemean=28÷4=7share between four\begin{aligned} \text{total}&=4+6+8+10=28 &&\text{include every value}\\ \text{mean}&=28\div4=7 &&\text{share between four} \end{aligned}

Why this step? Divide by the number of observations, not by their largest value or by the number of different digits.

Answer. The mean is 77.

Check. 44 and 1010 balance around 77, as do 66 and 88.

Change only what is missing: now the mean is known, but one observation is hidden.

Worked example 2: recovering a missing value

Question. Five numbers have mean 88. Four of them are 4,6,9,114,6,9,11. Work out the fifth. Non-calculator.

Plan. Recover the required total and subtract the known total.

required total=5×8=40undo division by fiveknown total=4+6+9+11=30add known observationsmissing number=40−30=10complete the total\begin{aligned} \text{required total}&=5\times8=40 &&\text{undo division by five}\\ \text{known total}&=4+6+9+11=30 &&\text{add known observations}\\ \text{missing number}&=40-30=10 &&\text{complete the total} \end{aligned}

Why this step? The mean of five numbers fixes their total, even though it does not fix each number separately.

Answer. 1010.

Check. (4+6+9+11+10)÷5=40÷5=8(4+6+9+11+10)\div5=40\div5=8.

Zeros count as observations too: 0,0,90,0,9 has mean 33, not 99.

Watch out

Wrong: divide the total by the number of non-zero values. Right: count all observations. Zero adds nothing to the total but still occupies one place in the group.

The next mean fixes a total for six values.

The median and mode

The median looks at position rather than total. Put the observations in order before trying to find its middle.

Median and mode

The median is the middle value of an ordered data set; with an even number of observations, it is the mean of the two middle values. The mode is the most frequent value or category. A set may have two or more modes, or no mode if no value occurs more often than the others.

For 2,2,3,4,5,8,92,2,3,4,5,8,9, the middle observation is the fourth, 44. The mode is 22, because it appears twice. The median need not be the mode. A category can have a mode, but finding an arithmetic mean of colours makes no sense.

Locating the median

For nn ordered observations, an odd nn gives the median in position n+12\frac{n+1}{2}. For an even nn, average the observations in positions n2\frac n2 and n2+1\frac n2+1.

Not on the formulae sheet: learn it.

Keep a repeated observation in the list each time it occurs. Frequency affects position.

Worked example 3: an odd number of observations

Question. Work out the median and mode of 8,2,5,2,9,4,38,2,5,2,9,4,3. Non-calculator.

Plan. Order all seven observations, then find the middle and the most frequent value.

ordered data:2,2,3,4,5,8,9sort every observationmiddle position=(7+1)÷2=4three on each sidemedian=4fourth observationmode=2occurs twice\begin{aligned} \text{ordered data}&:2,2,3,4,5,8,9 &&\text{sort every observation}\\ \text{middle position}&=(7+1)\div2=4 &&\text{three on each side}\\ \text{median}&=4 &&\text{fourth observation}\\ \text{mode}&=2 &&\text{occurs twice} \end{aligned}

Why this step? Sorting changes positions but not the data. The middle of the original unsorted list has no special meaning.

Answer. Median 44, mode 22.

Check. Three observations lie on either side of the fourth entry.

Change only the number of observations: remove the 99, leaving an even-sized set.

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