MYP Extended Mathematics 8.3 Interpreting Standard Deviation Notes

Interpreting "Within 1 Standard Deviation"

1. If a dataset has mean $\mu$ and standard deviation $\sigma$, then:
   1. Values within 1 standard deviation are in the interval $[\mu - \sigma, \mu + \sigma]$.
   2. Values more than 1 standard deviation away are outside that interval.
2. This is a practical interpretation tool:
   1. $\mu \pm \sigma$ gives a range of values you can call "fairly typical" in many contexts.
   2. Values well beyond $\mu \pm 2\sigma$ are often "unusual", especially if the distribution is approximately normal.

Standard Deviation and the Normal Distribution

1. Many natural measurements (such as height, mass, and age) are often approximately normally distributed: symmetric, bell-shaped, with most values near the mean.
2. When data follow a normal distribution, you can infer:
   1. About 68% of values lie within 1 standard deviation of the mean $$\mu-\sigma \leq X \leq \mu+\sigma$$
   2. About 95% of values lie within 2 standard deviations of the mean $$\mu-2\sigma \leq X \leq \mu+2\sigma$$
   3. About 99.7% of values lie within 3 standard deviations of the mean $$\mu-3\sigma \leq X \leq \mu+3\sigma$$

Comparing Groups with Standard Deviation

1. Standard deviation is especially meaningful when comparing two groups.
2. Suppose two companies have starting salaries:
   1. Company A: mean €45000, standard deviation €5000
   2. Company B: mean €41000, standard deviation €8000
3. Interpretation:
   1. Company A pays more on average (higher mean).
   2. Company B's salaries vary more (higher standard deviation), meaning there is more uncertainty. Some employees may earn much more than the mean, but many may also earn much less.
4. So "which is better" depends on your preference:
   1. If you value predictability/consistency, you prefer the smaller standard deviation.
   2. If you accept risk for a chance at a higher-than-average salary, you might accept the larger standard deviation.

Common Interpretation Statements

Given $\mu$ and $\sigma$, you should be able to write clear conclusions such as:

1. "The data are tightly clustered around the mean, so results are consistent." (small $\sigma$)
2. "There is a wide spread in the data, indicating high variability." (large $\sigma$)
3. "If the distribution is normal, about 68% of values lie between $\mu-\sigma$ and $\mu+\sigma$."
4. "A value of $x$ is about 2 standard deviations above the mean, so it is relatively unusual."