We use Z Scores to transform a given standard distribution into something that is easy for us to calculate probabilities on. Why? So we can determine the likelihood of some event happening.
This is a common transformation, so there is a reference chart that allows us to look up values. Those values correlate to the value under the normal distribution curve – in other words, what’s the chance of an event happening. We use the Z table to find the percent chance.
Sometimes, depending on the wording of the problem, we look for different things. I find it’s easiest if we draw the problem and decide what exactly we are looking for.
There are 3 types of Z Score Questions:
- The percent chance of an event happening beyond a certain point.
- This is the number under the curve beyond the z value.
- The percent chance of an event happening under a certain point.
- This is the number under the curve up to the z value.
- The percent chance of an event happening between two points.
- This could be the number under the curve bounded by two points.
- Sometimes one of those points is the mean – or the center of the distribution.
- In this case, Z scores are used to determine how far off a particular point in a distribution is from the mean.
- This could be the number under the curve bounded by two points.
Depending on the information given in the problem, there are also 2 different ways you go about solving the questions asked. One is by using the Z table to find the Z score. The other is by calculating the Z score. These are really the same thing just done in reverse. We will cover both types.
As you might guess, there are a number of ways of asking these kinds question. We will cover the ones I’ve seen most often. If I miss one, just let me know in the comments.
Before we get into how to solve problems, let’s cover the Z table.
The Normalized Z Table
The Z table (a table based on Normal Standard Distribution) is used to solve how far off a point in a distribution is likely to be from the center (mean).
A common statistical way of standardizing data on one scale so a comparison can take place is using a z-score. The z-score is like a common yard stick for all types of data. – from here.
Both z and t distributions are symmetric and bell-shaped, and both have a mean of zero.
By using the Z transformation, we can convert any normal distribution into a normal distribution with a mean of 0 and a standard deviation of 1. Thus, we can use a single normal table to find probabilities. – Pyzdeck
How to Create Your Own Z Table
Don’t have a reference chart handy? Have Excel? Great!
If you know your Z score, you can find the percentage by using the formula: 1-NORMSDIST(Z), where Z is your calculated Z Score.
How to Use a Z Table to find a Z Score
Be careful. Pay attention to what side of the z you should be on.
Step 1: Pick the right Z row by reading down the right column
Step 2: Read across the top to find the decimal space.
Step 3: Find the intersection and multiply by 100.
Ex This means that 6.18% of the normal curve resides to the right of Z. This also means that 93.82% resides to the left of the normal curve.
What if the Z Score is off the Chart?
Don’t panic. Z score tables sometimes only go up to 3. But depending on the spread of your population, z scores could go on for a while. A Z score of 3 refers to 3 standard deviations. That would mean that more than 99% of the population was covered by the z score. There’s not a lot left, but there is some. You can use Excel to find the actual value if your table doesn’t go that high.
How do you Calculate a Z Score?
Calculating a Z Score for a Population
Great example of using a Z score to determine how well you scored on a test (compared to the rest of the field) here.
How many parts in a population will be longer or greater than some number?
Z score examples using standard deviation
Example 1: Longer than
Hospital stays for admitted patients at a certain hospital are measured in hours and were found to be normally distributed with an average of 200 hours and a standard deviation of 75 hours. How many of these stays can be expected to last for longer than 300 hours? <from book>
Example 2: Less than
Hospital stays, for admitted patients at a certain hospital are measured in hours and were found to be normally distributed with an average of 200 hours and a standard deviation of 75 hours. How many of these stays can be expected to last less than 75 hours?
Example 3: Both less than AND greater than. (Percentage outside the range)
The mean inside diameter of a sample of 200 washers produced by a machine is 0.502 inches and the standard deviation is 0.005 inches. The purpose for which these washers are intended allows a maximum tolerance in the diameter of 0.496 to 0.508 inches, otherwise the washers are considered defective. Determine the percentage of defective washers produced by the machine, assuming the diameters are normally distributed.
Example 4: Both upper and lower bound. (Percentage OUTSIDE the range.)
The weights of 500 American men were taken and the sample mean was found to be 194 pounds with a standard deviation of 11.2 pounds. What percentages of men have weights between 175 and 225 pounds.
Example 5: What is the Area Under the Normal Curve?
A common Z value test question is to find the area under a normal curve.
What is the area under the curve between +.7 and +1.3 Standard deviations?
Ans. Show example curve. Look up both on Z table. Subtract the 2. The remainder is the area.
Z Transformation Example with Standard Deviation
A batch of batteries with an average of 60v and a Standard Deviation of 4v. If 9 batteries are selected at random, what is the probability that the total voltage of the batteries is greater than 530?
With an average voltage of 60, you’d expect the total to be 540v. This ends up being a standard deviation of a sample problem.
First, we find the standard deviation of the sample. To do this we use the variance equation:
Thus, Z = 530 – 540 / Sqrt(9*4^2) = -10 / Sqrt(144) = -10/12 = -0.833
Using the Z table, we find the area to the right of the Z is 0.7976. So, there is a 79.76%.
What is the probability that the average voltage is less than 62?
The expected value would be 60.
Z = 62-60 / (4 / Sqrt(9)) = 2 / (4/3) = 3/2 = 1.5.
The area to the left of Z is 1 – Z. Thus 1 – 0.0668 = 0.9332. Or 93.32%.
Z Test for Two Proportion
What do you do when the sample size is less than 20?
Great question! You’d apply student t-scores.
Six Sigma Green Belt Z Score Questions
Question: This formula Z = (X – μ)/σ is used to calculate a Z score that, with the appropriate table, can tell a Belt what ____________________________________.
A) Ratio the area under the curve is to the total population
B) Number of Standard Deviations are between X and μ
C) The Median of the sample population is
D) Proportion of the data is between X and μ
Answer: Proportion of the data is between X and μ. This is a definition question. See Z Scores.
Question: A battery manufacturer was considering changing suppliers for a particular part. The purchasing manager required that the average cost of the part be less than or equal to $32 in order to stay within budget. A sample of the 32 initial deliveries had a Mean of the new product upgrade price of $28 with an estimated Standard Deviation of $3. Based on the data provided, the Z value for the data assuming a Normal Distribution is?
Answer: 1.33 This is an easy algebra question. Z = (X – μ)/σ = 28-32 / 3 = 4/3 = 1.33. See Z Scores.