A** z-score** tells us how many standard deviations away a given value is from the mean. The z-score of a given value is calculated as:

**z-score** = (x – μ) / σ

where:

**x:**individual value**μ:**population mean**σ:**population standard deviation

This tutorial explains how to calculate z-scores on a TI-84 calculator.

**How to Calculate the Z-Score of a Single Value**

Suppose a distribution is normally distributed with a mean of 12 and a standard deviation of 1.4 and we wish to calculate the z-score of an individual value x = 14. To calculate the z-score in a TI-84 calculator, we would simply type in the following formula:

This tells us that an individual value of 14 has a z-score of **1.4286**. In other words, the value 14 lies 1.4286 standard deviations above the mean.

**How to Calculate the Z-Score of Several Values**

Suppose instead that we have a list of data values and that we would like to calculate the z-score for every value in the list. In this case, we can perform the following steps:

**Step 1: Input the data.**

First, we will input the data values. Press Stat and then press EDIT. Enter the following values in column L1:

**Step 2: Find the mean and standard deviation of the data values.**

Next, we will find the mean and the standard deviation of the dataset. Press Stat and then scroll over to **CALC**. Highlight **1-Var Stats** and press Enter.

For **List**, make sure L1 is chosen since this is the column we entered our data in. Leave **FreqList **blank. Highlight **Calculate **and press Enter.

The following output will appear:

We can see that the mean of the dataset is x = **10** and the standard deviation is s_{x} = **5.558**. We will use these two values in the next step to calculate z-scores.

**Step 3: Use a formula to calculate every z-score.**

Next, we will calculate the z-score for every individual value in the dataset. Press Stat and then press EDIT. Highlight L2 and type in the formula (**L1-10) / 5.558** and then press Enter. The z-score of every individual value will automatically appear in column L2:

**Note: **To enter “L1” in the formula, press 2nd and then press 1.

**How to Interpret Z-Scores**

Recall that a z-score simply tells us how many standard deviations away a value is from the mean. A z-score can be positive, negative, or equal to zero:

- A
**positive z-score**indicates that a particular value is greater than the mean. - A
**negative z-score**indicates that a particular value is less than the mean. - A
**z-score of zero**indicates that a particular value is equal to the mean.

In our example, we found that the mean was **10 **and the standard deviation was **5.558**.

So, the first value in our dataset was 3, which had a z-score of (3-10) / 5.558 = **-1.259**. This means that the value “3” is 1.259 standard deviations *below *the mean.

The next value in our dataset, 4, had a z-score of (4-10) / 5.558 = **-1.08**. This means that the value “4” is 1.08 standard deviations *below *the mean.

The further away a value is from the mean, the higher the absolute value of the z-score will be for that value.

For example, the value 3 is further away from the mean compared to 4, which explains why 3 had a z-score with a larger absolute value.