6.2 - Mean, median, mode and range

6.2 - Mean, median, mode and range

Use mean, median and mode to describe a typical value, and range to describe spread. Calculate from lists and frequency tables, recover missing information and choose a summary suited to the question.

Mean as an equal share

Can five waits have a mean of 2020 minutes when four of them are close to 1010? We will find out why one average can give a different impression from another.

You already know

Add all observations to find a total. Divide by the number of observations to share that total equally. A frequency tells you how many times a value occurs.

An average is one value used to represent the centre or a typical value of a dataset. Mean, median and mode are different averages. None lists all the observations, so each loses some information.

Mean

The value each observation would have if the total were shared equally among all observations.

The mean uses every value. A repeated value counts each time it appears, and a zero still counts as an observation.

Mean

mean=sum of all valuesnumber of values\text{mean}=\frac{\text{sum of all values}}{\text{number of values}}

Not in the Foundation or Higher formulae sheet: learn it.

Try the equal-share meaning with a small list before using a longer table.

Worked example 1: share the total

Question. Four pupils read 3,5,5,73,5,5,7 books. Calculate the mean.

Plan. Add all four counts and divide by four pupils.

total=3+5+5+7=20include both fivesmean=20÷4=5equal share per pupil\begin{aligned} \text{total}&=3+5+5+7=20 &&\text{include both fives}\\ \text{mean}&=20\div4=5 &&\text{equal share per pupil} \end{aligned}

Why divide by 44, not 33? There are four pupils even though only three different counts occur.

Answer. The mean is 55 books per pupil.

Check. 4×5=204\times5=20, recovering the total.

Change only the last count from 77 to 00. The pupil remains in the group.

Worked example 2: include a zero

Question. Calculate the mean of 3,5,5,03,5,5,0 books.

Plan. Recalculate the total but keep the count of four pupils.

total=3+5+5+0=13zero adds no booksmean=13÷4=3.25zero still counts as a pupil\begin{aligned} \text{total}&=3+5+5+0=13 &&\text{zero adds no books}\\ \text{mean}&=13\div4=3.25 &&\text{zero still counts as a pupil} \end{aligned}

Why can a whole-number dataset have a decimal mean? Equal sharing is a calculation, not a claim that anyone read exactly that many books.

Answer. 3.253.25 books per pupil.

Check. 4×3.25=134\times3.25=13; the result lies between 00 and 55.

A mean has the same unit as the data and lies between the smallest and largest observations. It need not be one of them. Keep exact values until any requested rounding.

Watch out

The line mean=13÷3\text{mean}=13\div3 leaves out the pupil who read zero books. Correct it to 13÷4=3.2513\div4=3.25.

The next list includes repeated observations too. Count them all.

Median, mode and range

Ordering the data lets you locate its middle and see the full distance between its endpoints.

Median, mode and range

The median is the middle value of ordered data; with two middle values, take their mean. The mode is the value occurring most often. The range is the largest value minus the smallest; it measures spread, rather than average.

For an odd count nn, the median is the value in position (n+1)/2(n+1)/2. For an even count, average values in positions n/2n/2 and n/2+1n/2+1. Positions tell you where to look, not what the data value is. These rules and the range relationship are not in either tier's formulae sheet.

If several values tie for highest frequency, report all the modes. If every value occurs just once, conventionally report no mode. Categories such as favourite colour can have a mode but no numerical mean.

Worked example 3: an even-sized list

Question. Find the median, mode and range of 8,2,5,5,11,38,2,5,5,11,3.

Plan. Order all six observations, then use middle positions, repetitions and endpoints.

ordered list:2,3,5,5,8,11smallest to largestmedian=5+52=5third and fourth valuesmode=5occurs twicerange=11−2=9largest minus smallest\begin{aligned} \text{ordered list}&:2,3,5,5,8,11 &&\text{smallest to largest}\\ \text{median}&=\frac{5+5}{2}=5 &&\text{third and fourth values}\\ \text{mode}&=5 &&\text{occurs twice}\\ \text{range}&=11-2=9 &&\text{largest minus smallest} \end{aligned}

Why is the mode not 22? Two is the frequency of the modal value, not the modal value itself.

Answer. Median 55, mode 55, range 99.

Check. Three observations occupy each half of the list; the range is positive.

Change only the number of observations by adding a single value, 2020. The list now has one middle position.

Worked example 4: add one observation

Question. Find the median, mode and range after adding 2020 to the same list.

Plan. Insert 2020 into order and identify the fourth of seven observations.

ordered list:2,3,5,5,8,11,20new maximum at the endmedian position=7+12=4one middle positionmedian=5fourth valuemode=5still occurs most oftenrange=20−2=18new maximum, same minimum\begin{aligned} \text{ordered list}&:2,3,5,5,8,11,20 &&\text{new maximum at the end}\\ \text{median position}&=\frac{7+1}{2}=4 &&\text{one middle position}\\ \text{median}&=5 &&\text{fourth value}\\ \text{mode}&=5 &&\text{still occurs most often}\\ \text{range}&=20-2=18 &&\text{new maximum, same minimum} \end{aligned}

Why did the range change but not the median? The endpoint moved, while the middle value stayed 55.

Answer. Median 55, mode 55, range 1818.

Check. There are three values before and three after the fourth position, including repeats.

The range uses only two endpoints, so one unusual observation can change it greatly. It has the data's original unit and is zero when all values are equal.

Watch out

The line median=3.5\text{median}=3.5 for six observations gives a middle position, not a value. Average the values in positions 33 and 44.

Use the labels median, mode and range in your answer so each result has a clear meaning.

Averages from frequency tables

A frequency table compresses a list by recording how many times each exact value occurs.

For discrete data, such as numbers of books, each listed value is exact. Value xx occurring ff times contributes fxfx to the total. The symbol ∑\sum means “add all the entries”.

Mean from a frequency table

mean=∑fx∑f\text{mean}=\frac{\sum fx}{\sum f}

Here ∑f\sum f counts observations and ∑fx\sum fx totals their values. Not in the Foundation or Higher formulae sheet: learn it.

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