Why Two Batters With the Same Average Can Be Completely Different

Two batters can have the same average — one scoring steadily, the other swinging between big scores and ducks. The mean hides that difference. Measures of dispersion describe how spread out the values are, which is often what a decision really depends on.

This part covers the range, the mean deviation about the mean, and interpreting dispersion to compare data sets.

The range is the difference between the largest and smallest values — for ungrouped data it uses the actual observations, and for grouped data it is the upper limit of the highest class minus the lower limit of the lowest class.

Formulae:

- Ungrouped: range = largest value $-$ smallest value
- Grouped: range = upper limit of the highest class $-$ lower limit of the lowest class
- Coefficient of range $=\frac{L-S}{L+S}$, for comparing data in different units

Worked example (ungrouped). Daily maximum temperatures over a week: 31, 34, 29, 36, 33, 30 and 35 °C. The range is $36-29=7$ °C, and the coefficient of range is $\frac{7}{65}\approx 0.108$.

Worked example (grouped). Scores grouped as 10–20, 20–30, 30–40 and 40–50 have range $50-10=40$.

An everyday example. Tomato prices at a mandi varying from ₹20 to ₹60 per kg in a week show a range of ₹40, a quick first sense of how much prices swing.

The substance. The range depends only on the two extreme values — one unusual observation can make it misleadingly large.

**The mean deviation about the mean is the average of the absolute deviations of the observations from their mean: $\frac{1}{n}\sum |x_i-\bar{x}|$ for ungrouped data and $\frac{1}{N}\sum f_i|x_i-\bar{x}|$ for grouped data, using class marks for continuous classes.

Worked example (ungrouped).** Find the mean deviation of 6, 7, 10, 12, 13, 4, 8 and 12.

- Mean $\bar{x}=\frac{72}{8}=9$
- Absolute deviations: 3, 2, 1, 3, 4, 5, 1 and 3, with sum 22
- Mean deviation $=\frac{22}{8}=2.75$

Worked example (grouped).

- Classes 0–10, 10–20, 20–30 and 30–40 with frequencies 5, 8, 15 and 2
- Class marks 5, 15, 25 and 35, with $N=30$
- $\sum f_i{x}_i=25+120+375+70=590$, so $\bar{x}=\frac{590}{30}\approx 19.67$
- $\sum f_i|x_i-\bar{x}|=5(14.67)+8(4.67)+15(5.33)+2(15.33)\approx 221.3$
- Mean deviation $\approx \frac{221.3}{30}\approx 7.38$

An everyday example. A school canteen tracking the plates it sells each day uses mean deviation to see how far a typical day strays from the average.

The substance. The absolute values are essential — without them the deviations from the mean always add up to zero.

When two data sets have similar means, the one with the smaller range or mean deviation is less variable and more consistent, and when the means or units differ, a relative measure such as mean deviation divided by the mean is compared instead.

Worked example. Runs scored by two batters in 5 matches:

- Batter A: 30, 35, 40, 45 and 50, with mean 40
- Batter B: 5, 10, 40, 70 and 75, with mean 40

Mean deviations:

- A: $\frac{10+5+0+5+10}{5}=6$
- B: $\frac{35+30+0+30+35}{5}=26$

Both average 40, but A is far more consistent.

Relative comparison. A third batter averaging 60 with mean deviation 9 has ratio $\frac{9}{60}=0.15$, the same as A's ratio $\frac{6}{40}=0.15$, so the two are equally consistent relative to their averages.

An everyday example. Choosing between two bus routes to school with the same average travel time — the route with the smaller mean deviation is the one you can rely on.

The substance. Dispersion compares spread, not level — a steady batter averaging 20 still scores less than an erratic one averaging 40.