## Practical Applications of the Standard Normal Model

The standard normal distribution could help you figure out which subject you are getting good grades in and which subjects you have to exert more effort into due to low scoring percentages. Once you get a score in one subject that is higher than your score in another subject, you might think that you are better in the subject where you got the higher score. This is not always true.

You can only say that you are better in a particular subject if you get a score with a certain number of standard deviations above the mean. The standard deviation tells you how tightly your data is clustered around the mean; It allows you to compare different distributions that have different types of data — including different means.

For example, if you get a score of 90 in Math and 95 in English, you might think that you are better in English than in Math. However, in Math, your score is 2 standard deviations above the mean. In English, it’s only one standard deviation above the mean. It tells you that in Math, your score is far higher than most of the students (your score falls into the tail).

Based on this data, you actually performed better in Math than in English!

### Probability Questions using the Standard Model

Questions about standard normal distribution probability can *look* alarming but the key to solving them is understanding what the area under a standard normal curve represents. The total area under a standard normal distribution curve is 100% (that’s “1” as a decimal). For example, the left half of the curve is 50%, or .5. So the probability of a random variable appearing in the left half of the curve is .5.

Of course, not all problems are quite *that* simple, which is why there’s a z-table. All a z-table does is measure those probabilities (i.e. 50%) and put them in standard deviations from the mean. The mean is in the center of the standard normal distribution, and a probability of 50% equals zero standard deviations.

### Standard normal distribution: How to Find Probability (Steps)

**Step 1:** Draw a bell curve and shade in the area that is asked for in the question. The example below shows z >-0.8. That means you are looking for the probability that z is greater than -0.8, so you need to draw a vertical line at -0.8 standard deviations from the mean and shade everything that’s greater than that number.

*shaded area is z > -0.8*

**Step 2:** Visit the normal probability area index and find a picture that looks like your graph. Follow the instructions on that page to find the z-value for the graph. The z-value *is *the probability.

**Tip: **Step 1 is technically optional, but it’s *always* a good idea to sketch a graph when you’re trying to answer probability word problems. That’s because most mistakes happen not because you can’t do the math or read a z-table, but because you subtract a z-score instead of adding (i.e. you imagine the probability under the curve in the wrong direction. A sketch helps you cement in your head exactly what you are looking for.

### Normal Distribution Key-Problems

This video shows one example of a normal distribution key-problem. For more examples, read on below:

When you tackle normal distribution in a statistics class, you’re trying to find the area under the curve. The total area is 100% (as a decimal, that’s 1). **Normal distribution problems** come in **six **basic types. How do you know that a key-problem involves normal distribution? Look for the key phrase “*assume the **variable** is normally distributed*” or “*assume the variable is approximately normal*.” To solve a key-problem you need to figure out which type you have.

- “Between”: Contain the phrase “between” and includes an upper and lower limit (i.e. “find the number of houses priced between $50K and 200K”).
- “More Than” or “Above”: contain the phrase “more than” or “above”.
- “Less Than”.
- Lower Cut Off Example, Upper Cut Off Example, and Middle Percent Example

**1. “Between”**

This how-to covers solving problems that contain the phrase “between” and includes an upper and lower limit (i.e. “find the number of houses priced between $50K and 200K”. Note that this is different from finding the “middle percentage” of something.

### Key-problems with normal distribution: “Between”: Steps

**Step 1:** *Identify the parts of the key-problem*. The key-problem will identify:

- The mean (average or μ).
- Standard deviation(σ).
- Number selected (i.e. “choose one at random” or “select ten at random”).
- X: the numbers associated with “between” (i.e. “between $5,000 and $10,000” would have X as 5,000 and as $10,000).

In addition, you will be given EITHER:

- Sample size(i.e. 400 houses, 33 people, 99factories, 378 plumbers etc.). OR
- You might be asked for a probability (in which case your sample size will most likely be everyone, i.e. “Journeyman plumbers” or “First year pilots.”

**Step 2:** *Draw a graph*. Put the mean you identified in Step 1 in the center. Put the number associated with “between” on the graph (take a guess at where the numbers would fall–it doesn’t have to be exact). For example, if your mean was $100, and you were asked for “hourly wages between $75 and $125”) your graph will look something like this:

**Step 3: ***Figure out the **z-scores*. Plug the first X value (in my graph above, it’s 75) into the z value formula and solve. The μ (the mean), is 100 from the sample graph. You can get these figures (including σ, the standard deviation) from your answers in step 1 :

*Note: if the formula confuses you, all this formula is asking you to do is:

- subtract the mean from X
- divide by the standard deviation.

**Step 4:** *Repeat step 3 for the second X*.

**Step 5:** *Take the numbers from step 3 and 4 and use them to find the area in the **z-table**.*

If you were asked to find a probability in your question, go to step 6a. If you were asked to find a number from a specific given sample size, go to step 6b.

**Step 6a: ***Convert the answer from step 5 into a percentage.*

For example, 0.1293 is 12.93%.

That’s it–skip step 6b!

**Step 6b: ***Multiply the sample size (found in step 1) by the z-value you found in step 4*. For example, 0.300 * 100 = 30.

That’s it!

**2. “More Than” or “Above”**

This how-to covers solving normal distribution problems that contain the phrase “**more than**” (or a phrase like “above”).

**Step 1:** *Break up the key-problem into parts. Find:*

- The mean(average or μ)
- Standard deviation(σ)
- A number (for example, “choose fifty at random” or “select 90 at random”)
- X: the number associated with the “less than” statement. For example, if you were asked to find “under $9,999” then X is 9,999.

**Step 2:** Find the sample from the problem. You’ll have either a specific size (like “1000 televisions”) or a general sample (“Every television”).

*Draw a picture if the problem with the mean and the area you are looking for. *For example, if the mean is $15, and you were asked to find what dinners cost more than $10, your graph might look like this:

**Step 3:** *Calculate the **z-score* (plug your values into the z value formula and solve). Use your answers from step 1 :

Basically, all you are doing with the formula is subtracting the mean from X and then dividing that answer by the standard deviation.

**Step 4:** *Find the area using the z-score from step 3. Use the **z-table**.* Not sure how to read a z-table? See Step 1 of this post for an example: Area under a curve.).

**Step 5:** *Go to Step 6a to find a probability OR go to step 6b to calculate a certain number or amount.*

**Step 6a: ** *Turn step 5’s answer into a percentage.*

For example, 0.1293 is 12.93%.

Skip step 6b: you’re done!

**Step 6b: ***Multiply the sample size from Step 1 by the z-score from step 4*. For example, 0.500 * 100 = 50.

You’re done!

**3. Less Than**

This how-to covers solving **normal distribution key-problems** that have the phrase “**less than**” (or a similar phrase such as “fewer than”).

Normal distribution key-problems less than: Steps

**Step 1:** *Break up the key-problem into parts*:

- The mean (average or μ)
- Standard deviation(σ)
- Number selected (i.e. “choose one at random” or “select ten at random”)
- X: the number that goes with “less than” (i.e. “under $99,000” would list X as 99,000)

Plus, you will have EITHER:

- A specific sample size. For example, 500 boats, 250 sandwiches, 100 televisions etc.
- Everyone in the sample (you’ll be asked to find a probability). For example “first year medical students,” “Cancer patients” or “Airline pilots.”

**Step 2:** *Draw a picture* to help you visualize the problem. The following graph shows a mean of 15, and an area “under 4”):

**Step 3:** *Find the z value* by plugging the given values into the formula. The “X” in our sample graph is 4, and the μ (or mean) is 15. You can get these figures (including σ, the standard deviation) from your answers in step 1, where you identified the parts of the problem:

All you have to do to solve the formula is:

- Subtract the mean from X.
- Divide by the standard deviation.

**Step 4:** *Take the number from step 3, then use the **z-table* to find the area.

**Step 5: ***To **find a probability**, go to step 6a. To find a number from a specific given sample size, go to step 6b.*

**Step 6a: ***Change the number from step 5 into percentage.*

For example, 0.1293 is 12.93%.

That’s it!

**Step 6b: ***Multiply the sample size (found in step 1) by the z-value you found in step 4*. For example, 0.300 * 100 = 30.

That’s it!

**4. Lower Cut Off, Upper Cut Off, and Meddle Percent**

Sometimes on **a normal distribution key-problem** you’ll be asked to find a “lower limit of an upper* percentage*” of something (i.e. “find the cut-off point to pass a certain exam where the upper 40% of test takers pass”). A lower cut off point is the point where scores will fall below that point. For example, you might want to find where the cut off point is for the bottom 10% of test takers.