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# Z-Score-Definition, Calculation, Interpretation, and Examples

December 8, 2023
written by Azhar Ejaz

A Z-score is also known as a standard score. It is a statistical measurement that describes a value’s relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean.

A Z-score of 0 indicates that the data point’s score is identical to the mean score, while a Z-score of +1.0 indicates a score that is one standard deviation from the mean, and a Z-score of -1.0 indicates a score that is one standard deviation from the mean in the opposite direction.

## Z-Score Formula

Z-Score Formula

The Z-score formula is given by:

$Z = \frac{x – \mu}{\sigma}$

Where:

• ( Z ) is the Z-score (standard score),
• ( x ) is the observed value,
• ( \mu ) is the mean of the sample, and
• ( \sigma ) is the standard deviation of the sample.

## How to Calculate Z-Score?

Here are the steps to calculate z score:

### Steps

1. Identify the data point (x): This is the individual value you want to find the z-score for.
2. Identify the mean (Î¼): This is the average value of the entire data set.
3. Identify the standard deviation (Ïƒ): This is a measure of how spread out the data is around the mean.
4. Plug the values into the formula: z = (x – Î¼) / Ïƒ
5. Calculate the z-score: Simplify the expression to get the final z-score.

To calculate a z-score, use formula:

For instance, suppose you scored 75 on a test, the class average is 70, and the standard deviation is 5. The calculation would be:

So, your z-score is 1. This means your score is one standard deviation above the average. If the z-score were -1, it would indicate one standard deviation below the average. The z-score helps you understand where your score stands relative to the average and how spread out the scores are.

## Z-Score Interpretation

Here is the detailed description to interpret Z-score:

### Interpreting value of a z-score:

• 0: A z-score of 0 indicates that the data point is exactly at the mean of the distribution.
• Positive: A positive z-score signifies that the data point is above the mean.
• The higher the z-score, the further away from the mean and the more unusual the data point is.
• For example, a z-score of +1 signifies one standard deviation above the mean, while a z-score of +2 signifies two standard deviations above the mean.
• Negative: A negative z-score indicates that the data point is below the mean.
• The lower the z-score, the further away from the mean and the more unusual the data point is.
• For example, a z-score of -1 signifies one standard deviation below the mean, while a z-score of -2 signifies two standard deviations below the mean.

Negative and Positive Z-Score Table are given at the end.

### Interpreting z-scores using standard normal distribution

• Area under the curve: The area under the curve of the standard normal distribution represents the probability of a data point falling within a certain range of z-scores.
• Height of the curve: The taller the curve at a particular z-score, the higher the probability of a data point falling within that range.
• Tails of the curve: The tails of the curve represent the less likely z-scores, indicating more unusual data points.

### Interpreting z-scores using Standard deviations

• A z-score equal to 1 represents an element, which is 1 standard deviation greater than the mean.
• A z-score equal to 2 signifies 2 standard deviations greater than the mean, etc.
• A z-score equal to -1 represents an element, which is 1 standard deviation less than the mean.
• A z-score equal to -2 signifies 2 standard deviations less than the mean, etc.

### Other concepts to Interpret Z-scores

1. Less than 0: A z-score of less than 0 represents an element less than the mean.
2. Greater than 0: A z-score of greater than 0 represents an element greater than the mean.
3. Equal to 0: A z-score of 0 represents an element equal to the mean.
4. Large sets: If the number of elements in the set is large, approximately:
• 68% of the elements have a z-score between -1 and 1.
• 95% have a z-score between -2 and 2.
• 99% have a z-score between -3 and 3.

## Understand Z-scores with Real-world Examples

Z-scores help us understand how individual data points relate to the overall average and spread of a dataset. They are especially useful when comparing data sets with different units or scales. Here are some real-world examples:

### Example 1: Exam Scores

Imagine a student scored 75 points on a math exam with an average score of 70 and a standard deviation of 5. To compare their score with the class performance, we calculate the z-score:

A z-score of 1 indicates the student scored 1 standard deviation above the average.

### Example 2: Plant Heights

Suppose you measure the height of sunflower plants in your garden. You find their average height is 20 inches with a standard deviation of 2 inches. One sunflower stands out at 24 inches tall. We can calculate its z-score:

A z-score of 2 signifies this sunflower is 2 standard deviations taller than the average plant, indicating a potentially significant difference.

### Example 3: Stock Prices

Assume you’re analyzing stock market data. The average stock price for a certain company is $100 with a standard deviation of$10. You observe a price of \$115 for a specific stock. Calculating the z-score gives:

A z-score of 1.5 suggests the stock price is 1.5 standard deviations higher than the average, potentially indicating a period of growth or increased interest.

## Z-Scores and Their Corresponding Probabilities/Percentiles:

1. Z-score 0.05: Approximately 1.645 standard deviations below the mean (84.134% probability)
2. Z-score 0.025: Approximately 1.960 standard deviations below the mean (97.5% probability)
3. Z-score 0.5: Mean (50% probability)
4. Z-score 0: Mean (50% probability)
5. Z-score 0.01: Approximately 2.326 standard deviations above the mean (98.764% probability)
6. Z-score 0.005: Approximately 2.576 standard deviations above the mean (99.379% probability)
7. Z-score 0.95: Approximately 1.645 standard deviations above the mean (84.134% probability)
8. Z-score 0.1: Approximately 1.282 standard deviations below the mean (89.442% probability)
9. Z-score 0.25: Approximately 0.675 standard deviations below the mean (77.337% probability)

### Z-Scores and Percentiles

1. Z-score 1: Approximately 84th percentile
2. Z-score 1.5: Approximately 93rd percentile
3. Z-score 1: 84th percentile
4. Z-score 1.28: Approximately 89th percentile
5. Z-score 1.96: Approximately 97.5th percentile
6. Z-score 1: Approximately 84th percentile
7. Z-score 1.25: Approximately 89th percentile
8. Z-score -1.28: 10th percentile
9. Z-score 1.2: Approximately 88th percentile
10. Z-score -1.28: 10th percentile