How Standard Deviation Reveals Which Data Set Is Truly Consistent

Mean deviation uses absolute values, which are awkward in algebra. Squaring the deviations instead gives the variance and its square root, the standard deviation — a measure that combines neatly across data sets and underlies most of advanced statistics.

This part covers variance and standard deviation by the direct method, the short-cut and step-deviation methods, combined mean and standard deviation, and the coefficient of variation.

**Variance is the mean of the squared deviations from the mean, ${\sigma }^2=\frac{1}{n}\sum (x_i-\bar{x}{)}^2$ for ungrouped data or $\frac{1}{N}\sum f_i(x_i-\bar{x}{)}^2$ for grouped data, and the standard deviation $\sigma$ is its positive square root.

Useful equivalent form.** ${\sigma }^2=\frac{\sum f_i{x}_i^2}{N}-{\bar{x}}^2$.

Worked example (ungrouped). For 6, 8, 10, 12 and 14:

- $\bar{x}=10$
- Squared deviations: 16, 4, 0, 4 and 16, with sum 40
- ${\sigma }^2=\frac{40}{5}=8$ and $\sigma =\sqrt{8}\approx 2.83$

Worked example (grouped). Values 2, 4, 6 and 8 with frequencies 1, 3, 4 and 2, so $N=10$:

- $\sum f_i{x}_i=2+12+24+16=54$, so $\bar{x}=5.4$
- $\sum f_i{x}_i^2=4+48+144+128=324$
- ${\sigma }^2=32.4-5.4^2=32.4-29.16=3.24$ and $\sigma =1.8$

An everyday example. A dairy cooperative checking the fat content of milk from many farmers uses standard deviation to see how uniform the supply is.

The substance. Standard deviation has the same units as the data, while variance has squared units — which is why $\sigma$, not ${\sigma }^2$, is quoted in reports.

**The short-cut method takes deviations $d_i=x_i-A$ from a convenient value A, giving $\sigma =\sqrt{\frac{\sum f_i{d}_i^2}{N}-{\left(\frac{\sum f_i{d}_i}{N}\right)}^2}$, and the step-deviation method also divides these by the class width h and multiplies the result back by h.

Step-deviation formula.** With $u_i=\frac{x_i-A}{h}$,

$$\sigma =h\sqrt{\frac{\sum f_i{u}_i^2}{N}-{\left(\frac{\sum f_i{u}_i}{N}\right)}^2}$$

Worked example. Classes 0–10, 10–20, 20–30, 30–40 and 40–50 have frequencies 5, 8, 15, 16 and 6, so $N=50$. Take $A=25$ and $h=10$:

- Class marks 5, 15, 25, 35 and 45 give $u_i=-2,-1,0,1,2$
- $\sum f_i{u}_i=-10-8+0+16+12=10$
- $\sum f_i{u}_i^2=20+8+0+16+24=68$

$$\sigma =10\sqrt{\frac{68}{50}-{\left(\frac{10}{50}\right)}^2}=10\sqrt{1.36-0.04}=10\sqrt{1.32}\approx 11.49$$

The mean is $25+10\times \frac{10}{50}=27$.

An everyday example. A coaching centre summarising the test scores of 50 students in class intervals saves a great deal of arithmetic with step deviations.

The substance. **Changing the origin does not change $\sigma$, but changing the scale does** — which is why the result is multiplied back by h but not adjusted for A.

**For groups of sizes $n_1$ and $n_2$ with means ${\bar{x}}_1$, ${\bar{x}}_2$ and standard deviations ${\sigma }_1$, ${\sigma }_2$, the combined mean is $\bar{x}=\frac{n_1{\bar{x}}_1+n_2{\bar{x}}_2}{n_1+n_2}$ and the combined variance is ${\sigma }^2=\frac{n_1({\sigma }_1^2+d_1^2)+n_2({\sigma }_2^2+d_2^2)}{n_1+n_2}$, where $d_1={\bar{x}}_1-\bar{x}$ and $d_2={\bar{x}}_2-\bar{x}$.

Why the $d^2$ terms appear. Each group spreads around its own mean, and the group means themselves sit away from the overall mean; both kinds of spread add to the combined variance.

Worked example.** Section A has 40 students with mean 50 and standard deviation 6; Section B has 60 students with mean 60 and standard deviation 8.

- Combined mean: $\frac{40\times 50+60\times 60}{100}=56$
- $d_1=-6$ and $d_2=4$
- Combined variance: $\frac{40(36+36)+60(64+16)}{100}=\frac{2880+4800}{100}=76.8$
- Combined standard deviation: $\sqrt{76.8}\approx 8.76$

An everyday example. A school combining the results of two sections can merge their means and standard deviations this way without re-entering every score.

The substance. The combined standard deviation can exceed both group values — here 8.76 is larger than 6 and 8, because the two means differ.

**The coefficient of variation is $\text{C.V.}=\frac{\sigma }{\bar{x}}\times 100$, the standard deviation as a percentage of the mean, and the data set with the smaller C.V. is the more consistent, even when the means or units differ.

Worked example.** Two batters over a season:

- Batter P: mean 50 runs and $\sigma =15$, so $\text{C.V.}=\frac{15}{50}\times 100=30$
- Batter Q: mean 30 runs and $\sigma =6$, so $\text{C.V.}=\frac{6}{30}\times 100=20$

P scores more on average, but Q is more consistent, since its C.V. is smaller.

Worked example 2. A group's heights have mean 150 cm and $\sigma =6$ cm, and their weights have mean 45 kg and $\sigma =5.4$ kg. The C.V. values are 4 and 12, so heights vary less than weights relative to their averages, even though the units differ.

An everyday example. Comparing the price stability of onions and rice at a local market needs the C.V., because their average prices are very different.

The substance. The C.V. becomes meaningless when the mean is close to zero — dividing by a tiny mean makes it blow up.