It looks like you're using Internet Explorer 11 or older. This website works best with modern browsers such as the latest versions of Chrome, Firefox, Safari, and Edge. If you continue with this browser, you may see unexpected results.

- StatisticsToggle Dropdown
- Why Study Statistics?
- Descriptive & Inferential Statistics
- Fundamental Elements of Statistics
- Quantitative and Qualitative Data
- Measurement Data Levels
- Collecting Data
- Ethics in Statistics
- Describing Qualitative Data
- Describing Quantitative Data
- Histograms
- Stem-and-Leaf Plots
- Measures of Central Tendency
- Measures of Variability
- Describing Data using the Mean and Standard Deviation
- Measures of Position
- Z-Scores

- ProbabilityToggle Dropdown
- Inferential Statistics

A common random continuous distribution you will encounter in statistics is the **normal distribution**. It is **bell-shaped** and it is sometimes called the **bell-curve**. It is a continuous probability distribution relating to the mean and standard deviation.

The normal distribution plays an important role in inferential statistics.

Probability density function: \[f(x)=\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left[\frac{x-\mu}{\sigma} \right]^2}\] where \(\mu =\) Population mean of the normal random variable \(x\) \(\sigma =\) Population standard deviation \(P(x<a)\) is obtained from a table of normal probabilities. |

The standard normal distribution is a normal distribution with \(\mu =0\) and \(\sigma =1\). A random variable with a standard normal distribution, denoted by the symbol \(z\), is called a standard normal variable. |

Instead of plugging into the function above to find each probability, we can use a **table of z variables** to find the probability.

Example: Find the probability that a standard normal random variable lies

- to the left of 1.23.
- to the right of 1.23.
- in between -0.26 and 1.23.

See the video below for the solution

The z-table refers to the **standard normal distribution** with mean \(\mu=0\) and standard deviation \(\sigma=1\). To apply normal distributions with different means or standard deviations we have to convert the value of \(x\) to a z-score before looking up the table.

If \(x\) is a normal random variable with mean \(\mu\) and standard deviation \(\sigma\), then the random variable \(z\) defined by the formula \[z=\frac{x-\mu}{\sigma}\] has a standard normal distribution. The value \(z\) describes the number of standard deviations between \(x\) and \(\mu\). |

Example 1: Suppose \(x\) is a normally distributed random variable with \(\mu=11\) and \(\sigma=2\). Find the probability that \(x\) is between 7.8 and 12.6.

**Solution**

First, we have to convert the \(x\) values 7.8 and 12.6 into z-scores.

\[z_1=\frac{7.8-11}{2} =-1.6\]

\[z_2=\frac{12.6-11}{2} =0.8\]

So, \(P(7.8<x<12.6)=P(-1.6<z<0.8)\)

Next, we look up the probabilities to the left of \(z_1=-1.6\) and \(z_2=0.8\) and find the difference.

From z-table, \(P(z<-1.6)= 0.0548\) and \(P(z<0.8)=0.7881\).

\begin{align} P(7.8<x<12.6)&=P(-1.6<z<0.8) \\ &= 0.7881 - 0.0548\\ &=0.7333\end{align}

Example 2: Suppose \(x\) is a normally distributed random variable with \(mu=30\) and \(\sigma=8\). Find a value \(x_0\) of the random variable \(x\) such that

- \(P(x<x_0)=0.8\)
- 25% of the values are greater than \(x_0\)

**Solution**

1. To find 0.8 or 80% of the population, we have to look inside the z-table.

We can say 80% or 0.8000 is close to the value of 0.7995. 0.7995 represents the z-score 0.84.

\[P(z<0.84)=0.7995 \approx 0.8\]

We now need to find the \(x_0\) that corresponds to \(z=0.84\) using the formula,

\begin{align} z&=\frac{x-\mu}{\sigma}\\ x&=\mu+z\sigma\\ x&=30+0.84(8)=36.72\end{align}

Therefore, approximately 80% of the data is less than \(x=36.72\).

2. To find \(x_0\) value representing the largest 25% of the data, we once again use the z-table. But since the z-table represents values less than, we are looking up the the value that represents 75% or 0.7500 of the data.

Since 75% or 0.7500 is almost exactly between the z-scores 0.67 and 0.68. We find the midpoint and say it is approximately equal to 0.675.

\[P(z<0.675) \approx 0.75\]

Since we are looking for the largest 25%, we change the relation around to \(P(z>0.675) \) and find the \(x\) value.

\begin{align} x&=\mu+z\sigma\\ x&=30+0.675(8)=35.4\end{align}

Therefore, approximately 25% of the data is greater than \(x=35.4\).

Statistics by Matthew Cheung. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

- Last Updated: Mar 24, 2022 8:28 AM
- URL: https://libraryguides.centennialcollege.ca/c.php?g=717168
- Print Page

chat loading...