Statistics (Grade 11 NSC Matric Mathematics): Revision Notes

Variance and Standard Deviation

Introduction to dispersion measures

When we analyse data, measures of central tendency (mean, median, mode) tell us the typical value in a dataset.

However, they do not tell us how spread out the data is.

This is why we use measures of dispersion — variance and standard deviation — which describe how much data values vary from the mean.

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Understanding variance

Definition of variance

Variance is the average squared distance between each data value and the mean.

For a population of $n$ elements
$\{x_1, x_2, \ldots, x_n\}$ with mean $\bar{x}$:

$$\sigma^{2} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^{2}}{n}$$

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Properties of variance

• Always non-negative
  Squaring deviations ensures the result cannot be negative.

• Measured in squared units
  If data is in $\text{cm}$, variance is in $\text{cm}^{2}$.

• Shows spread

  • Large variance ($\sigma^{2}$ large) → data widely spread
  • Small variance ($\sigma^{2}$ small) → data clustered close to the mean

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Worked example: Calculating variance

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Understanding standard deviation

Definition of standard deviation

Standard deviation is the square root of the variance, so it has the same units as the original data:

$$\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^{2}}{n}}$$

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Properties of standard deviation

• Same units as data
• Always positive (or $0$ if all values are identical)
• Measures typical distance from the mean
• Easy to interpret (small $\sigma$ = low spread, large $\sigma$ = high spread)

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Using a table

A table helps avoid arithmetic errors:

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Worked example: Variance & SD of a die roll

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Interpretation and applications

Comparing datasets

Three datasets may share the same mean but differ in spread.

• Small SD $(\sigma = 8.97)$ → tightly clustered
• Medium SD $(\sigma = 17.75)$ → moderately spread
• Large SD $(\sigma = 21.23)$ → widely scattered

(Visuals shown above.)

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Real-world applications

• Quality control: consistency of manufactured products
• Scientific measurement: low SD → high precision
• Finance: SD measures volatility (risk)
• Education: SD shows consistency of scores

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