Two Students With the Same Average Can Be Very Different

Riya scored $60,62,64,66,68$ in five tests; Kabir scored $40,50,64,78,88$. **Both have a mean of $64$, yet Riya is steady and Kabir swings wildly.

A measure of central tendency tells you where the data sits; a measure of dispersion tells you how spread out** it is.

This part covers the range, mean deviation for raw data, mean deviation for frequency distributions, and the limitations of mean deviation.

The range is the largest observation minus the smallest; it gives a quick first idea of spread but depends only on the two extreme values.

Worked example.

$$\text{Riya: }68-60=8\text{Kabir: }88-40=48$$

The larger range shows Kabir's marks are far more spread out.

Why it is not enough. Take $10,50,50,50,90$ and $10,20,50,80,90$. Both have range $80$, but the first set is tightly packed around $50$ except for two values, while the second is spread evenly. The range ignores everything between the extremes.

An everyday example. A weather report giving the highest and lowest temperature of a day reports its range — a hill station with warm afternoons and cold nights has a wider range than a coastal city.

The substance. A single unusual value can change the range completely, so a better measure uses every observation.

**Mean deviation about a central value $a$ is the average of the absolute distances of the observations from $a$: $\text{M.D.}(a)=\frac{1}{n}\sum |x_i-a|$, with $a$ taken as the mean $\bar{x}$ or the median $M$.

Absolute values are needed because plain deviations from the mean always add to zero.

Worked example.** Data: $2,4,5,6,8,11,13$, so $n=7$.

About the mean. $\bar{x}=\frac{49}{7}=7$.

$$|x_i-7|:5,3,2,1,1,4,6\sum =22\text{M.D.}(\bar{x})=\frac{22}{7}\approx 3.14$$

About the median. The data is already in order, so $M=6$.

$$|x_i-6|:4,2,1,0,2,5,7\sum =21\text{M.D.}(M)=\frac{21}{7}=3$$

An everyday example. The mean deviation of the heights of students in a class line is the typical distance of a student's height from the class average.

The substance. For raw data, the sum of absolute deviations is smallest about the median, so M.D. about the median never exceeds M.D. about the mean.

**For a frequency distribution, $\text{M.D.}(a)=\frac{1}{N}\sum f_i|x_i-a|$ with $N=\sum f_i$; for continuous data, $x_i$ is the class mid-point, and the mean is found quickly by step-deviation: $\bar{x}=A+\frac{\sum f_i{d}_i}{N}\times h$ with $d_i=\frac{x_i-A}{h}$.

Discrete example.** $x$: $10,20,30,40,50$ with $f$: $3,5,8,3,1$, so $N=20$.

$$\sum f_i{x}_i=30+100+240+120+50=540\bar{x}=27$$

$$\sum f_i|x_i-27|=51+35+24+39+23=172\text{M.D.}(\bar{x})=\frac{172}{20}=8.6$$

Cumulative frequencies are $3,8,16,19,20$; the 10th and 11th values are $30$, so $M=30$.

$$\sum f_i|x_i-30|=60+50+0+30+20=160\text{M.D.}(M)=8$$

Continuous example. Classes $0$-$10$ to $40$-$50$ with $f$: $5,8,15,16,6$, so $N=50$. Mid-points $5,15,25,35,45$; take $A=25$, $h=10$.

$$d_i:-2,-1,0,1,2\sum f_i{d}_i=-10-8+0+16+12=10\bar{x}=25+\frac{10}{50}\times 10=27$$

$$\sum f_i|x_i-27|=110+96+30+128+108=472\text{M.D.}(\bar{x})=\frac{472}{50}=9.44$$

An everyday example. Grouping students' daily travel times to school in 10-minute classes gives exactly this kind of continuous table.

The substance. Step-deviation changes only the arithmetic, not the answer — the mean is the same as $\frac{\sum f_i{x}_i}{N}$.

Mean deviation uses every observation, but it drops signs by taking absolute values, which makes further algebra awkward, and its value depends on whether the mean or median is used.

- Absolute values cannot be expanded or combined easily, so combining the M.D. of two groups is not straightforward
- About the mean, the sum of absolute deviations is generally larger than about the median, so M.D. about the mean is not the least possible
- For highly variable data, the median may not represent the centre well, so M.D. about the median can mislead

A quick illustration. In the raw data above, M.D. about the mean was $3.14$ and about the median $3$ — two different answers to "how spread out is it?"

An everyday example. Two batters with the same batting average may differ in consistency; M.D. shows this, but squaring the deviations handles it more cleanly.

The substance. These drawbacks are why variance and standard deviation, which square deviations instead, are the preferred measures.