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

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**

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**

**Identify the data point (x):**This is the individual value you want to find the z-score for.**Identify the mean (μ):**This is the average value of the entire data set.**Identify the standard deviation (σ):**This is a measure of how spread out the data is around the mean.**Plug the values into the formula:**z = (x – μ) / σ**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**

**Z-score 0 = mean. Positive z-score = above mean, each unit increase +1 standard deviation. Negative z-score = below mean, each unit decrease -1 standard deviation.**

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.

- The higher the z-score, the further away from the mean and the
**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.

- The lower the z-score, the further away from the mean and the

**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**

**Less than 0**: A z-score of less than 0 represents an element**less than the mean**.**Greater than 0**: A z-score of greater than 0 represents an element**greater than the mean**.**Equal to 0**: A z-score of 0 represents an element**equal to the mean**.**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:

```
```**z = (75 - 70) / 5 = 1
**

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:

```
```**z = (24 - 20) / 2 = 2
**

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:

```
```**z = (115 - 100) / 10 = 1.5
**

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:**

- Z-score 0.05: Approximately 1.645 standard deviations below the mean (84.134% probability)
- Z-score 0.025: Approximately 1.960 standard deviations below the mean (97.5% probability)
- Z-score 0.5: Mean (50% probability)
- Z-score 0: Mean (50% probability)
- Z-score 0.01: Approximately 2.326 standard deviations above the mean (98.764% probability)
- Z-score 0.005: Approximately 2.576 standard deviations above the mean (99.379% probability)
- Z-score 0.95: Approximately 1.645 standard deviations above the mean (84.134% probability)
- Z-score 0.1: Approximately 1.282 standard deviations below the mean (89.442% probability)
- Z-score 0.25: Approximately 0.675 standard deviations below the mean (77.337% probability)

**Z-Scores and Percentiles**

- Z-score 1: Approximately 84th percentile
- Z-score 1.5: Approximately 93rd percentile
- Z-score 1: 84th percentile
- Z-score 1.28: Approximately 89th percentile
- Z-score 1.96: Approximately 97.5th percentile
- Z-score 1: Approximately 84th percentile
- Z-score 1.25: Approximately 89th percentile
- Z-score -1.28: 10th percentile
- Z-score 1.2: Approximately 88th percentile
- Z-score -1.28: 10th percentile

**Negative Z-Score Table**

Z-Score | Probability |
---|---|

-3.0 | 0.0013 |

-2.9 | 0.0019 |

-2.8 | 0.0026 |

-2.7 | 0.0035 |

-2.6 | 0.0047 |

-2.5 | 0.0062 |

-2.4 | 0.0082 |

-2.3 | 0.0107 |

-2.2 | 0.0139 |

-2.1 | 0.0179 |

-2.0 | 0.0228 |

-1.9 | 0.0287 |

-1.8 | 0.0359 |

-1.7 | 0.0446 |

-1.6 | 0.0548 |

-1.5 | 0.0668 |

-1.4 | 0.0808 |

-1.3 | 0.0968 |

-1.2 | 0.1151 |

-1.1 | 0.1357 |

-1.0 | 0.1587 |

-0.9 | 0.1841 |

-0.8 | 0.2119 |

-0.7 | 0.2420 |

-0.6 | 0.2743 |

-0.5 | 0.3085 |

-0.4 | 0.3446 |

-0.3 | 0.3821 |

-0.2 | 0.4207 |

-0.1 | 0.4602 |

-0.09 | 0.4681 |

-0.08 | 0.4761 |

-0.07 | 0.4840 |

-0.06 | 0.4920 |

-0.05 | 0.5000 |

-0.04 | 0.5080 |

-0.03 | 0.5160 |

-0.02 | 0.5240 |

-0.01 | 0.5320 |

-0.009 | 0.5331 |

-0.008 | 0.5341 |

-0.007 | 0.5351 |

-0.006 | 0.5361 |

-0.005 | 0.5371 |

-0.004 | 0.5381 |

-0.003 | 0.5391 |

-0.002 | 0.5401 |

-0.001 | 0.5411 |

-0.000 | 0.5413 |

**Positive Z-Score Table**

Z-Score | Probability |
---|---|

0.000 | 0.5413 |

0.001 | 0.5423 |

0.002 | 0.5433 |

0.003 | 0.5443 |

0.004 | 0.5453 |

0.005 | 0.5463 |

0.006 | 0.5473 |

0.007 | 0.5483 |

0.008 | 0.5493 |

0.009 | 0.5503 |

0.01 | 0.5513 |

0.02 | 0.5591 |

0.03 | 0.5669 |

0.04 | 0.5747 |

0.05 | 0.5826 |

0.06 | 0.5904 |

0.07 | 0.5982 |

0.08 | 0.6061 |

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