**What is Z-Score?**

The Z-Score, often termed as the “Standard Score,” represents** how many standard deviations a data point is from the mean**.

In simpler terms, it’s a way to understand where a particular value stands in relation to the average of a group of values.

**Example**

Imagine you took a math test, and your score was 85. If the class average was 80 with a standard deviation of 5, your Z-Score would be 1.

This means your score was one standard deviation above the average.

**Formula**

- = Z-Score
- = Individual data point
- = Mean of the data set
- = Standard deviation of the data set

**How to Calculate Z-Score: Step by Step**

- Find the mean of the dataset.
- Calculate the standard deviation .
- For each data point, subtract the mean and divide by the standard deviation:

**Calculating the Z-Score: Example**

Let’s consider a simple dataset as an example:**[ 15, 14, 10, 16, 13, 17, 19, 10, 16 ]**

**1. Find the Mean µ**

To determine the mean, sum all the numbers and then divide by the number of data points.

(rounded to two decimal places)

The mean provides a central value for our dataset, which will be used to determine how each data point deviates from this average.

See Also: Mean in Statistics: Difference from Median

**2. Calculate the Standard Deviation σ**

Using our previous method of calculating standard deviation, let’s determine the standard deviation for our dataset.

For brevity, we’ll skip the detailed calculation here and provide the result:

(rounded to two decimal places)

The standard deviation **σ** measures **the average distance between each data point and the mean**. It gives us an idea of the spread or dispersion of our data.

See Also: Standard Deviation in Statistics: Beginner’s Guide

**3. Calculate the Z-Score for Each Data Point**

#### Using the formula:

#### For the first data point** (15)**:

Remember the σ is 2.87 and the µ is 14.44 ( We got it in the second step).

(rounded to two decimal places)

For the second data point (14):

(rounded to two decimal places)

Here we find the Z-Score for the first datapoint “15” is **“-0.15”**

**… and so on for the rest of the data points.**

The Z-Score tells us how many standard deviations away a data point is from the mean.

A Z-Score of 0 means the data point is right at the mean, **a positive Z-Score indicates the data point is above the mean**, and **a negative Z-Score indicates it’s below the mean**.

It’s a way to standardize our data and understand its relative position within the dataset.

**Z-Score vs Percentile**

While both Z-Score and **percentiles** help understand the relative position of a data point, they convey this information differently.

**Z-Score**: Tells how many standard deviations away a data point is from the mean. It can be negative (below the mean) or positive (above the mean).**Percentile**: Indicates the percentage of data points that lie below a particular value. For instance, if you’re in the 90th percentile for height, you’re taller than 90% of people in the group.

**Usages of Z-Score in Business**

Z-Scores are invaluable in various business sectors:

**Finance**: To assess stock volatility. A high Z-Score might indicate that a stock’s price is above its historical average, suggesting it’s overpriced.**Marketing**: To identify outliers in sales data, helping businesses spot unusual sales activities.**Quality Control**: In manufacturing, Z-Scores can help determine if a process is producing items within quality specifications.

**2 Types of Z-Scores**

While the basic concept of a Z-Score remains consistent, there are variations based on the type of data and purpose:

**Population Z-Score**

Derived using the entire population’s mean and standard deviation. It provides a definitive measure for the entire dataset.

**Sample Z-Score**

Derived using a sample’s mean and standard deviation. It’s an estimate, used when the entire population data isn’t available.

In both cases, the formula remains the same, but the values for mean and standard deviation will differ based on whether you’re considering the entire population or just a sample.

**Conclusion**

Remember, the Z-Score is a handy tool in statistics, helping to standardize data and make comparisons more meaningful. Whether you’re comparing test scores, heights, or any other data, the Z-Score can provide valuable insights!