What is the formula to solve for z-score?
2 Answers
The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean. The formula to calculate the z-score is:
\[
z = \frac{(X - \mu)}{\sigma}
\]
Where:
- \( z \) = z-score
- \( X \) = the value you want to standardize
- \( \mu \) = the mean (average) of the dataset
- \( \sigma \) = the standard deviation of the dataset
### Breaking Down the Formula
1. **Mean (\( \mu \))**: This is the average of all the values in your dataset. You calculate it by summing all the values and dividing by the number of values.
\[
\mu = \frac{\sum X}{N}
\]
Where \( N \) is the number of values in the dataset.
2. **Standard Deviation (\( \sigma \))**: This measures the amount of variation or dispersion in a set of values. A low standard deviation means the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
The formula for standard deviation is:
\[
\sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}}
\]
3. **Calculating the z-score**: Once you have the mean and standard deviation, you can substitute those values into the z-score formula.
### Example
Let's say you have a dataset of test scores: 70, 75, 80, 85, and 90.
1. **Calculate the mean (\( \mu \))**:
\[
\mu = \frac{70 + 75 + 80 + 85 + 90}{5} = \frac{400}{5} = 80
\]
2. **Calculate the standard deviation (\( \sigma \))**:
\[
\sigma = \sqrt{\frac{(70-80)^2 + (75-80)^2 + (80-80)^2 + (85-80)^2 + (90-80)^2}{5}} = \sqrt{\frac{100 + 25 + 0 + 25 + 100}{5}} = \sqrt{\frac{250}{5}} = \sqrt{50} \approx 7.07
\]
3. **Calculate the z-score for a specific score, say \( X = 85 \)**:
\[
z = \frac{(85 - 80)}{7.07} \approx \frac{5}{7.07} \approx 0.71
\]
### Interpretation
A z-score of 0.71 means that the score of 85 is 0.71 standard deviations above the mean score of 80.
### Summary
The z-score helps you understand how far a particular value lies from the mean in terms of standard deviations, allowing for comparison across different datasets or distributions.
\[
z = \frac{(X - \mu)}{\sigma}
\]
Where:
- \( z \) = z-score
- \( X \) = the value you want to standardize
- \( \mu \) = the mean (average) of the dataset
- \( \sigma \) = the standard deviation of the dataset
### Breaking Down the Formula
1. **Mean (\( \mu \))**: This is the average of all the values in your dataset. You calculate it by summing all the values and dividing by the number of values.
\[
\mu = \frac{\sum X}{N}
\]
Where \( N \) is the number of values in the dataset.
2. **Standard Deviation (\( \sigma \))**: This measures the amount of variation or dispersion in a set of values. A low standard deviation means the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
The formula for standard deviation is:
\[
\sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}}
\]
3. **Calculating the z-score**: Once you have the mean and standard deviation, you can substitute those values into the z-score formula.
### Example
Let's say you have a dataset of test scores: 70, 75, 80, 85, and 90.
1. **Calculate the mean (\( \mu \))**:
\[
\mu = \frac{70 + 75 + 80 + 85 + 90}{5} = \frac{400}{5} = 80
\]
2. **Calculate the standard deviation (\( \sigma \))**:
\[
\sigma = \sqrt{\frac{(70-80)^2 + (75-80)^2 + (80-80)^2 + (85-80)^2 + (90-80)^2}{5}} = \sqrt{\frac{100 + 25 + 0 + 25 + 100}{5}} = \sqrt{\frac{250}{5}} = \sqrt{50} \approx 7.07
\]
3. **Calculate the z-score for a specific score, say \( X = 85 \)**:
\[
z = \frac{(85 - 80)}{7.07} \approx \frac{5}{7.07} \approx 0.71
\]
### Interpretation
A z-score of 0.71 means that the score of 85 is 0.71 standard deviations above the mean score of 80.
### Summary
The z-score helps you understand how far a particular value lies from the mean in terms of standard deviations, allowing for comparison across different datasets or distributions.
The Z-score is a measure that describes how far a data point is from the mean of a data set, measured in standard deviations. The formula to calculate the Z-score for a given value \(X\) is:
\[ Z = \frac{X - \mu}{\sigma} \]
Where:
- \( Z \) is the Z-score.
- \( X \) is the value of the data point.
- \( \mu \) (mu) is the mean of the data set.
- \( \sigma \) (sigma) is the standard deviation of the data set.
### Steps to Calculate a Z-Score:
1. **Find the Mean (\(\mu\))**: Add up all the values in the data set and divide by the number of values.
2. **Calculate the Standard Deviation (\(\sigma\))**:
- **Find the Variance**: Subtract the mean from each data point, square the result, and then find the average of these squared differences.
- **Take the Square Root** of the variance to get the standard deviation.
3. **Apply the Z-Score Formula**: Subtract the mean from the data point and then divide by the standard deviation.
### Example:
Suppose you have a data point \(X = 75\), a mean \(\mu = 70\), and a standard deviation \(\sigma = 10\).
1. Subtract the mean from the data point:
\[ 75 - 70 = 5 \]
2. Divide the result by the standard deviation:
\[ \frac{5}{10} = 0.5 \]
So, the Z-score for the data point 75 is \(0.5\). This means the value is 0.5 standard deviations above the mean.
\[ Z = \frac{X - \mu}{\sigma} \]
Where:
- \( Z \) is the Z-score.
- \( X \) is the value of the data point.
- \( \mu \) (mu) is the mean of the data set.
- \( \sigma \) (sigma) is the standard deviation of the data set.
### Steps to Calculate a Z-Score:
1. **Find the Mean (\(\mu\))**: Add up all the values in the data set and divide by the number of values.
2. **Calculate the Standard Deviation (\(\sigma\))**:
- **Find the Variance**: Subtract the mean from each data point, square the result, and then find the average of these squared differences.
- **Take the Square Root** of the variance to get the standard deviation.
3. **Apply the Z-Score Formula**: Subtract the mean from the data point and then divide by the standard deviation.
### Example:
Suppose you have a data point \(X = 75\), a mean \(\mu = 70\), and a standard deviation \(\sigma = 10\).
1. Subtract the mean from the data point:
\[ 75 - 70 = 5 \]
2. Divide the result by the standard deviation:
\[ \frac{5}{10} = 0.5 \]
So, the Z-score for the data point 75 is \(0.5\). This means the value is 0.5 standard deviations above the mean.