How to Calculate a Z Score: 15 Steps (with Pictures)

2024 Author: Samantha Chapman | [email protected]. Last modified: 2023-12-17 09:13

A Z score allows you to take a sample of data within a larger set and to determine how many standard deviations it is above or below the mean. To find the Z score, you first need to calculate the mean, variance, and standard deviation. Next, you will need to find the difference between the sample data and the mean and divide the result by the standard deviation. Although, from start to finish, there are many steps to follow to find the value of the Z score with this method, still know that it is a simple calculation.

Steps

Part 1 of 4: Calculate the mean

Step 1. Look at your dataset

You will need some key information to find the arithmetic mean of the sample.

• Find how much data makes up the sample. Consider a group consisting of 5 palm trees.

  

• Now give the numbers meaning. In our example, each value corresponds to the height of a palm tree.

  

• Take note of how much the numbers vary. Does the data fall within a small or large range?

  

Step 2. Write down all values

You need all the numbers that make up the data sample to start the calculations.

• The arithmetic mean tells you around which mean value the data that make up the sample are distributed.
• To calculate it, add all the values of the set together and divide them by the number of data that make up the set.
• In mathematical notation, the letter “n” represents the sample size. In the example of the heights of the palms, n = 5, since we have 5 trees.

Step 3. Add all the values together

This is the first part of the calculation to find the arithmetic mean.

• Consider the sample of palm trees whose heights are 7, 8, 8, 7, 5 and 9 meters.
• 7 + 8 + 8 + 7, 5 + 9 = 39, 5. This is the sum of all the data in the sample.
• Check the result to make sure you haven't made a mistake.

Step 4. Divide the sum by the sample size "n"

This last step will give you the average of the values.

• In the example of the palms, you know that the heights are: 7, 8, 8, 7, 5 and 9. There are 5 numbers in the sample, so n = 5.
• The sum of the heights of the palms is 39.5. You have to divide this value by 5 to find the average.
• 39, 5/5 = 7, 9.
• The average height of the palm trees is 7.9 m. The mean is often represented with the symbol μ, so μ = 7, 9.

Part 2 of 4: Finding the Variance

Step 1. Calculate the variance

This value shows how much the sample is distributed around the mean value.

• The variance gives you an idea of how much the values that make up a sample differ from the arithmetic mean.
• Samples with low variance are composed of data that tend to distribute very close to the mean.
• Samples with a high variance are composed of data that tend to be distributed very far from the average value.
• Variance is often used to compare the distribution of two samples or data sets.

Step 2. Subtract the average value from each number that makes up the set

This gives you an idea of how much each value differs from the average.

• Considering the example of palm trees (7, 8, 8, 7, 5 and 9 meters), the average was 7, 9.
• 7 - 7.9 = -0.9; 8 - 7.9 = 0.1; 8 - 7.9 = 0.1; 7, 5 - 7, 9 = -0, 4 and 9 - 7, 9 = 1, 1.
• Redo the calculations to make sure they are correct. It is extremely important that you have not made any mistakes in this step.

Step 3. Square any differences you found

You must raise all values to the power of 2 to calculate the variance.

• Remember that, considering the example of palm trees, we subtracted the average value 7, 9 from each value that makes up the whole (7, 8, 8, 7, 5 and 9) and we obtained: -0, 9; 0, 1; 0, 1; -0, 4; 1, 1.
• Square: (-0, 9)² = 0, 81; (0, 1)² = 0, 01; (0, 1)² = 0, 01; (-0, 4)² = 0, 16 and (1, 1)² = 1, 21.
• The squares obtained from these calculations are: 0, 81; 0.01; 0.01; 0, 16; 1, 21.
• Check that they are correct before proceeding to the next step.

Step 4. Add the squares together

• The squares of our example are: 0, 81; 0.01; 0.01; 0, 16; 1, 21.
• 0, 81 + 0, 01 + 0, 01 + 0, 16 + 1, 21 = 2, 2.
• As for the sample of five palm heights, the sum of the squares is 2, 2.
• Check the amount to be sure it is correct before continuing.

Step 5. Divide the sum of the squares by (n-1)

Remember that n is the number of data that makes up the set. This last calculation gives you the variance value.

• The sum of the squares of the example of the heights of the palms (0, 81; 0, 01; 0, 01; 0, 16; 1, 21) is 2, 2.
• In this sample there are 5 values, so n = 5.
• n-1 = 4.
• Remember that the sum of the squares is 2, 2. To find the variance, divide 2, 2/4.
• 2, 2/4=0, 55.
• The variance of the sample of palm heights is 0.55.

Part 3 of 4: Calculating the Standard Deviation

Step 1. Find the variance

You will need it to calculate the standard deviation.

• The variance shows how far the data in a set is distributed around the mean value.
• The standard deviation represents how these values are distributed.
• In the previous example, the variance is 0.55.

Step 2. Extract the square root of the variance

This way you find the standard deviation.

• In the example of palm trees, the variance is 0.55.
• √0, 55 = 0, 741619848709566. Often you will find values with a long series of decimals when making this calculation. You can safely round the number to the second or third decimal place to determine the standard deviation. In this case, stop at 0.74.
• Using a rounded value, the sample standard deviation of tree heights is 0.74.

Step 3. Check the calculations again for the mean, variance, and standard deviation

By doing so, you are certain that you have not made any mistakes.

• Write down all the steps you followed in performing the calculations.
• Such forethought helps you find any mistakes.
• If during the verification process you find different mean, variance or standard deviation values, then repeat the calculations again with great care.

Part 4 of 4: Calculating the Z Score

Step 1. Use this formula to find the Z score:

z = X - μ / σ. This allows you to find the Z score for each sample data.

• Remember that the Z score measures how many standard deviations each sample value deviates from the mean.
• In the formula, X represents the value you want to examine. For example, if you want to know by how many standard deviations the height 7, 5 differs from the average value, replace X with 7, 5 within the equation.
• The term μ represents the mean. The mean value of the sample in our example was 7.9.
• The term σ is the standard deviation. In the palm sample, the standard deviation was 0.74.

Step 2. Begin the calculations by subtracting the average value from the data you want to examine

In this way proceed with the calculation of the Z score.

• Consider, for example, the Z score of the value 7, 5 of the sample of tree heights. We want to know how many standard deviations it deviates from the mean 7, 9.
• Do subtraction 7, 5-7, 9.
• 7, 5 - 7, 9 = -0, 4.
• Always check your calculations to make sure you haven't made a mistake before continuing.

Step 3. Divide the difference you just found by the standard deviation value

At this point you get the Z score.

• As mentioned above, we want to find the Z score of the data 7, 5.
• We have already subtracted from the mean value and found -0, 4.
• Remember that the standard deviation of our sample was 0.74.
• -0, 4 / 0, 74 = -0, 54.
• In this case the Z score is -0.54.
• This Z score means that the data 7.5 is at -0.54 standard deviations from the mean value of the sample.
• Z scores can be both positive and negative values.
• A negative Z score indicates that the data is lower than the average; on the contrary, a positive Z score indicates that the data taken into consideration is greater than the arithmetic mean.