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# 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:

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

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.

![](/files/-LM2_-iWfSW3WXpUbbu7)

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

![](/files/-LM2isb9wnYuEdJAlafB)

### Standard normal distribution:&#x20;

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

### 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.

![](/files/-LM2mNfEEnA6QPR-snOO)

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:&#x20;

   $$z\_1 = \frac{19.63 - 21.25}{2} = \frac{-1.62}{2} = -0.81$$

   $$z\_2 = \frac{20.53 - 21.25}{2} = \frac{-0.72}{2} = -0.36$$
2. We want to find the proportion of temperatures between these two z-scores:

   ![](https://cdn.kastatic.org/ka-perseus-graphie/15590e61d44ea2eddd5f740df3b15ebc90f64407.svg)$$z\_1z\_2$$
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:

   ![](https://cdn.kastatic.org/ka-perseus-graphie/e60def34b3e0faadfda0eaf530d40d2015c2044e.svg)$$z\_1$$
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:

   ![](https://cdn.kastatic.org/ka-perseus-graphie/c06e2cbc867d90958150f433a3b07a455d6dc4f7.svg)$$z\_2$$
5. To find the area between $$z\_1$$ and $$z\_2$$we can subtract the area below $$z\_1$$ from the area below $$z\_2$$

   $$0.3594 - 0.2090 = \greenD{0.1504}$$

   ![](https://cdn.kastatic.org/ka-perseus-graphie/15590e61d44ea2eddd5f740df3b15ebc90f64407.svg)$$z\_1​z\_2$$
6. The answer: $$\greenD{0.1504}$$