Normal Distribution & Empirical Rule

Section 1 of 2

Shape & the Normal Distribution

Three shapes

A normal distribution is a symmetric bell — mean, median and mode all in the middle. Skewed left (tail to the left, e.g. age for reading glasses) or skewed right (tail to the right, e.g. family size).

You try

In a normal distribution where do mean, median and mode lie? Which way is a distribution with a long tail to the right skewed?

Symmetric bell → all together; tail to the right = positive skew.

All in the middle

Skewed right (positive)

middle; skewed right

Section 2 of 2

The Empirical Rule

68 – 95 – 99.7

For a normal distribution: $68\%$ within $1$ standard deviation of the mean, $95\%$ within $2$, $99.7\%$ within $3$.

Worked example — weights

Weights: mean $53$ kg, $\sigma = 2$ kg. Find the $68\%$, $95\%$ and $99.7\%$ ranges.

$68\%$: $51$ to $55$

$95\%$: $49$ to $57$

$99.7\%$: $47$ to $59$

$51\text{-}55$; $49\text{-}57$; $47\text{-}59$

Worked example — IQ scores

IQ scores: mean $100$, $\sigma = 10$. Find the $68\%$, $95\%$ and $99.7\%$ ranges.

$68\%$: $90$ to $110$

$95\%$: $80$ to $120$

$99.7\%$: $70$ to $130$

$90\text{-}110$; $80\text{-}120$; $70\text{-}130$

Worked example — test scores

Mean score $56$, $\sigma = 4$. Find the $68\%$, $95\%$ and $99.7\%$ ranges.

$68\%$: $52$ to $60$

$95\%$: $48$ to $64$

$99.7\%$: $44$ to $68$

$52\text{-}60$; $48\text{-}64$; $44\text{-}68$

You try

$68\%$ of a population have a score between $60$ and $70$. Find the mean and standard deviation.

$68\%$ is $\bar{x} \pm \sigma$, so the mean is the middle of $60$ and $70$, and $\sigma$ is the half-width.

$\bar{x} = \tfrac{60+70}{2} = 65$

$\sigma = 70 - 65 = 5$

$\bar{x} = 65,\ \sigma = 5$

That’s Class 9.

The normal distribution, skew, and the $68\text{-}95\text{-}99.7$ rule (both ways). Class 10: margin of error, confidence intervals and hypothesis testing.