Standard Normal Distribution
Even if two variables can both be described by a normal distribution, it can also be difficult to ______these variables if their mean and or standard deviations are ______, for example heights in centimeters and weights in kilograms. As the empirical rule suggests, all normal distributions share many properties. In fact, all normal distributions are ______if we measure in units of size ___ from the mean μ as the center. To solve this conflict, we can create a new variable, ___, which defines a ______value that can apply to ______normal distributions.
The standard normal distribution is a normal distribution with a mean of ____ and a standard deviation of ___. This can be expressed as Z. Our new variable, z, gives a measure of ______the variable is from the mean (x−μ)then "______" it by dividing by the standard deviation (σ). This new variable gives us a way of comparing different variables. The ______tells us ______
______or "how many sigmas" the variable is from its respective ______. In essence, we are just "______" the original distribution to fit the properties of a ______distribution.
Example 1. The percentages scored in an exam are normally distributed with a mean of 70% and a standard deviation of 10%.
- Victoria scored 90% on the exam. Calculate her z-score and explain what it means.
- Ethan scored 55% on the exam. Calculate his z-score and explain what it means.
Example 2.The table shows Emma’s midyear exam results. The exam results for each subject are normally distributed with the mean and standard deviation shown in the table.
Subject / Emma’s grade / /English / 48 / 40 / 4.4
Mandarin / 81 / 60 / 9
Geography / 84 / 55 / 18
Biology / 68 / 50 / 20
Algebra / 84 / 50 / 15
- Find the z-score for each of Emma’s subjects.
Subject / z-score
English
Mandarin
Geography
Biology
Algebra
- Relatively speaking, in what subject did Emma get the “best” grade? The “worst”?
Using the Normal Distribution
The normal distribution is useful when finding an ______mean or standard deviation for a normal distribution. You may be given ______and be asked to find the mean (if is known) or the standard deviation (if is known).
Example 3. Suppose X is normally distributed with a mean of 40, and P(X 45) = 0.9. Find the standard deviation.
We will need to convert our data to a ______curve to figure out the standard deviation.
Original distribution/ Standard distribution
Since z = ______, we know that 45 is ______standard deviations away from the mean of 40. To find the standard deviation, we will use the formula .
Example 4. Sacks of potatoes with a mean weight of 5 kg are packed by an automatic loader. In a test, it was found that 15% of the bags were over 5.2 kg. Use this information to find the standard deviation of the process.
Original distribution/ Standard distribution
Since z = ______, we know that 5.2 is ______standard deviations away from the mean of 5. To find the standard deviation, we will use the formula .