10F

The empirical rule

The 68–95–99.7 empirical rule for normal distributions, finding percentages and predicting numbers of data values, identifying unusual values, and checking whether data is roughly normal.

The 68–95–99.7 rule

The empirical rule

The shaded areas are about 68%, 95% and 99.7% for k = 1, 2 and 3.

Try this

What percentage is left outside for $k = 2$, and how much of it is in each tail?

By symmetry, each half is split equally. For example, $34\%$ lie between $\mu$ and $\mu + \sigma$, and $2.5\%$ lie above $\mu + 2\sigma$.

Region	$\mu$ to $\mu + \sigma$	$\mu + \sigma$ to $\mu + 2\sigma$	$\mu + 2\sigma$ to $\mu + 3\sigma$	above $\mu + 3\sigma$
Percentage	$34\%$	$13.5\%$	$2.35\%$	$0.15\%$

Using the rule

Example 1●●

Test scores are normal with mean $50$ and standard deviation $5$. Find the percentage of scores between $45$ and $60$.

$45$ is $z = -1$ and $60$ is $z = 2$.

$34\% + 47.5\% = 81.5\%$.

Example 2: predicting data●●

$400$ students sat that test. About how many scored above $60$?

Above $z = 2$ is $2.5\%$.

$0.025 \times 400 = 10$ students.

Flashcards

Within 1, 2, 3 standard deviations

$68\%$, $95\%$, $99.7\%$

Between $mu$ and $mu + sigma$

$34\%$

Above $mu + 2sigma$

$2.5\%$

Practice

Scores have mean $50$ and sd $5$. What percentage lie between $45$ and $60$?
$400$ students sat that test. About how many scored above $60$?
What percentage of scores are above $\mu + \sigma$?
What percentage are between $\mu$ and $\mu + 2\sigma$?
What percentage are below $\mu - 3\sigma$?
Heights have mean $170$ and sd $8$. What percentage lie between $154$ and $186$?
For the same heights, what percentage are below $154$?
Scores have mean $50$ and sd $5$. Which score is most unusual?
About what percentage of values lie within one standard deviation of the mean?