Z-Score Quiz BA Random Sample Of Pet Owners Found That They Spent The Following Amounts On Their Pets Per Week (in Dollars): 30, 31, 33, 37, 39, 40.Level 1: Find The Mean Of The Sample Set. $\bar{x}$ Or $\mu$ = ______Level 2: Find
Introduction
In statistics, a z-score is a measure of how many standard deviations an element is from the mean. It is a crucial concept in understanding the distribution of data and is widely used in various fields, including mathematics, science, and finance. In this article, we will delve into the concept of z-scores and provide a step-by-step guide on how to calculate them.
What is a Z-Score?
A z-score is a numerical value that represents how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the element and then dividing the result by the standard deviation. The z-score is a dimensionless quantity that can be used to compare the relative position of an element within a dataset.
Calculating the Mean
To calculate the z-score, we first need to find the mean of the sample set. The mean is calculated by summing up all the elements in the dataset and then dividing the result by the number of elements.
Level 1: Find the Mean of the Sample Set
A random sample of pet owners found that they spent the following amounts on their pets per week (in dollars): 30, 31, 33, 37, 39, 40.
To find the mean, we need to sum up all the elements in the dataset and then divide the result by the number of elements.
Step 1: Sum up the elements
30 + 31 + 33 + 37 + 39 + 40 = 210
Step 2: Divide the result by the number of elements
There are 6 elements in the dataset. To find the mean, we divide the sum of the elements (210) by the number of elements (6).
or = 210 / 6 or = 35
Therefore, the mean of the sample set is 35.
Level 2: Find the Standard Deviation
To calculate the z-score, we need to find the standard deviation of the sample set. The standard deviation is a measure of the amount of variation or dispersion of a set of values.
Step 1: Calculate the deviations from the mean
To calculate the deviations from the mean, we subtract the mean (35) from each element in the dataset.
| Element | Deviation |
|---|---|
| 30 | -5 |
| 31 | -4 |
| 33 | -2 |
| 37 | 2 |
| 39 | 4 |
| 40 | 5 |
Step 2: Calculate the squared deviations
To calculate the squared deviations, we square each deviation.
| Element | Deviation | Squared Deviation |
|---|---|---|
| 30 | -5 | 25 |
| 31 | -4 | 16 |
| 33 | -2 | 4 |
| 37 | 2 | 4 |
| 39 | 4 | 16 |
| 40 | 5 | 25 |
Step 3: Calculate the sum of the squared deviations
To calculate the sum of the squared deviations, we add up all the squared deviations.
25 + 16 + 4 + 4 + 16 + 25 = 90
Step 4: Calculate the variance
To calculate the variance, we divide the sum of the squared deviations (90) by the number of elements (6) minus 1.
Variance = 90 / (6 - 1) Variance = 90 / 5 Variance = 18
Step 5: Calculate the standard deviation
To calculate the standard deviation, we take the square root of the variance.
Standard Deviation = √18 Standard Deviation = 4.24
Therefore, the standard deviation of the sample set is 4.24.
Calculating the Z-Score
Now that we have the mean and standard deviation, we can calculate the z-score.
Step 1: Subtract the mean from the element
To calculate the z-score, we subtract the mean (35) from each element in the dataset.
| Element | Z-Score |
|---|---|
| 30 | -5 |
| 31 | -4 |
| 33 | -2 |
| 37 | 2 |
| 39 | 4 |
| 40 | 5 |
Step 2: Divide the result by the standard deviation
To calculate the z-score, we divide the result by the standard deviation (4.24).
| Element | Z-Score |
|---|---|
| 30 | -5 / 4.24 = -1.18 |
| 31 | -4 / 4.24 = -0.94 |
| 33 | -2 / 4.24 = -0.47 |
| 37 | 2 / 4.24 = 0.47 |
| 39 | 4 / 4.24 = 0.94 |
| 40 | 5 / 4.24 = 1.18 |
Therefore, the z-scores for the sample set are -1.18, -0.94, -0.47, 0.47, 0.94, and 1.18.
Conclusion
Introduction
In our previous article, we discussed the concept of z-scores and provided a step-by-step guide on how to calculate them. In this article, we will answer some frequently asked questions about z-scores to help you better understand this important statistical concept.
Q: What is a z-score?
A: A z-score is a measure of how many standard deviations an element is from the mean. It is a dimensionless quantity that can be used to compare the relative position of an element within a dataset.
Q: How do I calculate the z-score?
A: To calculate the z-score, you need to find the mean and standard deviation of the sample set. The z-score is calculated by subtracting the mean from the element and then dividing the result by the standard deviation.
Q: What is the formula for calculating the z-score?
A: The formula for calculating the z-score is:
z = (X - μ) / σ
Where:
- z is the z-score
- X is the element
- μ is the mean
- σ is the standard deviation
Q: What is the difference between a z-score and a standard score?
A: A z-score and a standard score are the same thing. They are both measures of how many standard deviations an element is from the mean.
Q: Can I use the z-score to compare data from different distributions?
A: No, the z-score is only applicable to data that follows a normal distribution. If the data does not follow a normal distribution, you should use a different type of score, such as the t-score.
Q: How do I interpret the z-score?
A: The z-score can be interpreted in the following ways:
- A z-score of 0 means that the element is equal to the mean.
- A positive z-score means that the element is above the mean.
- A negative z-score means that the element is below the mean.
- A z-score of 1 or -1 means that the element is 1 standard deviation away from the mean.
- A z-score of 2 or -2 means that the element is 2 standard deviations away from the mean.
Q: Can I use the z-score to determine the probability of an event?
A: Yes, the z-score can be used to determine the probability of an event. By using a z-table or a calculator, you can find the probability of an event occurring given a certain z-score.
Q: What are some common applications of the z-score?
A: The z-score has many applications in various fields, including:
- Psychology: to determine the probability of a person scoring above or below a certain score on a test.
- Medicine: to determine the probability of a patient responding to a certain treatment.
- Finance: to determine the probability of a stock price increasing or decreasing.
- Engineering: to determine the probability of a machine failing or not failing.
Conclusion
In conclusion, the z-score is a powerful statistical concept that can be used to compare data from different distributions. By understanding how to calculate and interpret the z-score, you can make informed decisions in various fields. We hope that this article has helped you better understand the z-score and its applications.