Why the Middle Value Can Tell a Truer Story Than the Average

The mean uses every value, so one extreme value can drag it far from typical. Two other measures resist that pull:

- Median — the middle value when data is arranged in order
- Mode — the value that occurs most often

Quartiles go further, splitting ordered data into four equal parts to show how spread out the middle half is. This part covers median, mode, quartiles and problems with a missing value.

**Arrange the data in order; for $n$ odd the median is the $\left(\frac{n+1}{2}\right)$th value, for $n$ even it is the average of the $\frac{n}{2}$th and $\left(\frac{n}{2}+1\right)$th values; for grouped data, the median class contains the $\frac{n}{2}$th observation.

Worked example 1 — odd $n$.** $12,5,9,15,7,10,3$ arranged is $3,5,7,9,10,12,15$. With $n=7$, the $4$th value gives **median $=9$.

Worked example 2 — even $n$.** Adding $20$ gives $3,5,7,9,10,12,15,20$ with $n=8$:

$$\text{median}=\frac{9+10}{2}=9.5$$

Worked example 3 — median class. Frequencies $5,8,12,10,5$ for $0$–$10$, $10$–$20$, $20$–$30$, $30$–$40$, $40$–$50$ have cumulative frequencies $5,13,25,35,40$. With $\frac{n}{2}=20$, the $20$th observation lies where the total first reaches $20$, so **the median class is $20$–$30$.

An everyday example. News reports on house prices often quote the median, since a few luxury flats would inflate the mean.

The substance. Always arrange the data first**; the middle of an unordered list means nothing.

The mode of raw data is the value that occurs most often, and the modal class of grouped data is the class with the highest frequency.

Worked example 1. $3,5,5,7,8,8,8,9$: the value $8$ occurs three times, so **mode $=8$.

Worked example 2.** $2,4,4,6,7,7,9$: both $4$ and $7$ occur twice, so the data is bimodal, with modes $4$ and $7$.

Worked example 3. In the grouped table above, the highest frequency is $12$, so **the modal class is $20$–$30$.

Worked example 4.** $1,3,6,8,10$: every value occurs once, so there is no mode.

An everyday example. A shoe shop stocking a new design orders the most of the modal size, because that is the size most customers ask for.

The substance. A distribution can have one mode, several modes or none, while it always has exactly one mean and one median.

**Arrange the data in order; the lower quartile $Q_1$ marks the first quarter of the data, the upper quartile $Q_3$ marks three-quarters, and the inter-quartile range is $Q_3-Q_1$.

Worked example — $n$ odd.** The data $3,5,7,8,10,12,13,15,18,20,22$ has $n=11$.

$$Q_1=\left(\frac{n+1}{4}\right)\text{th}=3\text{rd value}=7$$
$$\text{median}=\left(\frac{n+1}{2}\right)\text{th}=6\text{th value}=12$$
$$Q_3=\left(\frac{3(n+1)}{4}\right)\text{th}=9\text{th value}=18$$
$$\text{inter-quartile range}=18-7=11$$

From an ogive. For $n$ observations, read across from $\frac{n}{4}$ and $\frac{3n}{4}$ on the cumulative frequency axis to the curve and down to the value axis.

An everyday example. A coaching institute comparing two batches looks at the inter-quartile range of test scores to see which batch has more consistent middle performers.

The substance. The inter-quartile range ignores the lowest and highest quarters, so a few extreme scores do not distort it.

Write the given mean or median as an equation containing the unknown, using the formula for that measure, and solve.

Worked example 1 — missing observation from a mean. The mean of $6,8,x,10,12$ is $9$.

$$\frac{36+x}{5}=936+x=45x=9$$

Worked example 2 — missing frequency. For $x=5,10,15,20,25$ with $f=3,7,p,5,2$, the mean is $14$.

$$\sum f=17+p\sum fx=235+15p$$
$$235+15p=14(17+p)235+15p=238+14pp=3$$

Check: $\sum f=20$, $\sum fx=280$, mean $=14$.

Worked example 3 — missing value from a median. The ordered data $10,12,14,x+2,x+4,20,22,25$ has median $17$.

$$\frac{(x+2)+(x+4)}{2}=172x+6=34x=14$$

The middle values are $16$ and $18$, which keeps the data in order.

An everyday example. A teacher who has lost one student's mark can recover it from the class total or average.

The substance. Always check that the recovered value fits — here it must keep the data in increasing order.