Question

How is Z value calculated?

Answer 1

The "Z value" or "Z-score" is a statistical measurement that describes a value's relationship to the mean of a group of values. It tells you how many standard deviations a data point is from the mean. Here's how you calculate it:

### Formula:
\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- **Z** = Z-score
- **X** = the value you're measuring
- **μ (mu)** = the mean (average) of the data set
- **σ (sigma)** = the standard deviation of the data set

### Steps:
1. **Find the mean (μ)** of the data set: Add up all the data points and divide by the number of data points.
   \[
   \mu = \frac{ \sum X_i }{N}
   \]
   where \(X_i\) are the individual data points, and \(N\) is the number of data points.

2. **Find the standard deviation (σ)**: This measures how spread out the values are from the mean. It’s calculated using the formula:
   \[
   \sigma = \sqrt{\frac{ \sum (X_i - \mu)^2 }{N}}
   \]
   where \(X_i\) is each data point, and \(\mu\) is the mean.

3. **Subtract the mean (μ) from your value (X)** to find the difference from the mean.
4. **Divide that difference by the standard deviation (σ)** to find the Z-score.

### Example:

Let’s say you have the following data set:  
\[ 5, 7, 8, 9, 10 \]

1. **Find the mean (μ)**:
   \[
   \mu = \frac{(5 + 7 + 8 + 9 + 10)}{5} = \frac{39}{5} = 7.8
   \]

2. **Find the standard deviation (σ)**:
   First, calculate the squared differences from the mean:
   \[
   (5 - 7.8)^2 = 7.84, \quad (7 - 7.8)^2 = 0.64, \quad (8 - 7.8)^2 = 0.04, \quad (9 - 7.8)^2 = 1.44, \quad (10 - 7.8)^2 = 4.84
   \]
   Now, calculate the average of these squared differences:
   \[
   \frac{7.84 + 0.64 + 0.04 + 1.44 + 4.84}{5} = 2.96
   \]
   Finally, take the square root of that result:
   \[
   \sigma = \sqrt{2.96} \approx 1.72
   \]

3. **Calculate the Z-score for a value (X = 10)**:
   \[
   Z = \frac{10 - 7.8}{1.72} \approx 1.29
   \]
   So, the Z-score for the value 10 is approximately 1.29, which means it's 1.29 standard deviations above the mean.

### Key Points:
- A Z-score of **0** means the value is exactly at the mean.
- A positive Z-score means the value is above the mean.
- A negative Z-score means the value is below the mean.

Let me know if you want further clarification!