Modeling data distribution

z-score - how many $\sigma$away from the mean $\mu$

example value 65

$\frac{value - \mu}{\sigma} \Longrightarrow \frac{65-81}{6.3}=-2.54$

A z-score measures exactly how many standard deviations above or below the mean a data point is. Here's the formula for calculating a z-score:

$z = \frac{data point−mean}{standard deviation} \Longrightarrow z = \frac{x - \mu}{\sigma}$

Here are some important facts about z-scores:

• A positive z-score says the data point is above average.

• A negative z-score says the data point is below average.

• A z-score close to 000 says the data point is close to average.

• A data point can be considered unusual if its z-score is above 333 or below -3−3minus, 3.

Standard Deviation and IQR change only with multiplication and devision, but not with addition and subtraction. The mean and Median do change either way.

Normal distribution: Empirical Rule (68-95-99.7%)

Standard normal distribution:

$\mu = 0 (\text{mean})\newline \sigma = 1 (\text{standard deviation})$

What is a normal distribution?

Early statisticians noticed the same shape coming up over and over again in different distributions—so they named it the normal distribution.

Normal distributions have the following features:

• symmetric bell shape

• mean and median are equal; both located at the center of the distribution

• ≈ 68% of the data falls within 1 standard deviation of the mean

• ≈ 95% of the data falls within 2 standard deviations of the mean

• ≈ 99.7% of the data falls within 3 standard deviations of the mean

Quiz

A set of average city temperatures in August are normally distributed with a mean of $21.25 ^\circ$C and a standard deviation of $2 ^\circ$C.

$\large \frac{value - \mu}{\sigma} = \text{z-score}$

What proportion of temperatures are between $19.63^\circ$ C and $20.53^\circ$ C? You may round your answer to four decimal places.

1. Let's find the z-score for$19.63^\circ$C and $20.53^\circ$C:

   $z_1 = \frac{19.63 - 21.25}{2} = \frac{-1.62}{2} = -0.81$

   $z_2 = \frac{20.53 - 21.25}{2} = \frac{-0.72}{2} = -0.36$

2. We want to find the proportion of temperatures between these two z-scores:

   $z_1z_2$

3. Looking up $z_1 = -0.81$ on the z-table, we see that $0.2090$ of temperatures are below $\blueD{19.63}^\circ$C:

   $z_1$

4. Looking up $z_2 = -0.36$ on the z-table, we see that $0.3594$ of temperatures are below $\goldD{20.53}^\circ$C:

   $z_2$

5. To find the area between $z_1$ and $z_2$we can subtract the area below $z_1$ from the area below $z_2$

   $0.3594 - 0.2090 = \greenD{0.1504}$

   $z_1​z_2$

6. The answer: $\greenD{0.1504}$