Notes
Z-scores and percentiles
- $\displaystyle z = \frac{x - \mu}{\sigma}$ counts standard deviations from the mean. To find a value, use $x = \mu + z\sigma$.
- A percentile is the percent of values at or below a given value. In a normal distribution, $z = 0$ is the 50th percentile.
The normal distribution
- Normal distributions are symmetric and bell-shaped, with mean = median.
- 68-95-99.7 rule: about $68\%$, $95\%$ and $99.7\%$ of values lie within 1, 2 and 3 standard deviations of the mean.
Normal calculations
- Standardize, then use a table of areas to the left.
- Right area $= 1 -$ left area. Area between $=$ larger left area $-$ smaller left area.
- Z-scores allow fair comparisons between different distributions.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle z = \frac{58 - 50}{4} = 2$.
720 and 880 are 2 standard deviations below and above the mean.
A percentile is the percent of the data at or below the value.
$1 - 0.8944 = 0.1056$.
$x = 60 + (-1.5)(8) = 48$.
Normal distributions are symmetric. Only the standard normal distribution has mean 0 and standard deviation 1.
Free response
Write your answer first, then open the scoring guide.
Free-response question
The weights of apples from an orchard are approximately normal with mean 150 grams and standard deviation 10 grams.
(a) Find the z-score of a 170 gram apple. About what percent of apples weigh more than 170 grams?
Show the answer and scoring guide
$\displaystyle z = \frac{170 - 150}{10} = 2$. By the 68-95-99.7 rule, about $2.5\%$ are above $\mu + 2\sigma$.
Check off the points you earned:
(b) Use the table values: area left of $z = 1$ is $0.8413$, and area left of $z = -2$ is $0.0228$. What proportion of apples weigh between 130 and 160 grams?
Show the answer and scoring guide
130 g has $z = -2$ and 160 g has $z = 1$. Proportion $= 0.8413 - 0.0228 = 0.8185$.
Check off the points you earned:
(c) Using the 68-95-99.7 rule, the heaviest $16\%$ of apples weigh more than about how many grams?
Show the answer and scoring guide
About $16\%$ of values are above $\mu + \sigma$, so the cutoff is $150 + 10 = 160$ grams.
Check off the points you earned: