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
- 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 sx = 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.