Z Score Formula: Definition, Calculation & Interpretation

Z Score is the standard deviation of a raw score from its mean. Z score is also referred to as the standard score. It is used to represent the number of standard deviations by which a raw score is above or below the mean. Z score is one of the most important numerical measurements used in Statistics. Z score is used as part of a z test to derive interpretations about population data. It eventually helps to compare data from different normal distributions. 

Z Score Formula is given as Z Score = (x − x̅ )/σ. Here, x denotes the standardized random variable, x̅ refers to the mean, and σ refers to the standard deviation. Z scores have a distribution with a mean of 0 and a standard deviation of 1. Depending upon the position of the raw score with respect to the mean, a z score can be positive, negative, or zero.

Key Terms: Z Score, Z Score Formula, Mean, Standard Deviation, Statistics, Hypothesis, Random variable, Normal Distribution

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What is Z Score?

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Z Score is a statistical measurement that depicts how far a raw score is from the mean of a distribution. Z score is used in a z test for hypothesis testing. Z score is used to predict the intervals to determine the probability of a random variable falling between a range of values. Z scores can either be positive or negative or null (zero). 

• The positive value of the Z score indicates that the Z score is above the mean.
• The negative value of the Z score indicates that the Z score is below the mean. 
• The zero or null value of Z score indicates that the Z score is the same as the mean.

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Z Score Formula

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One must have knowledge about the concepts of mean and standard deviation to calculate the z score.

When the population mean, and population standard deviation are given, then the z score formula is as follows: 

Where

• x: Standardized Random Variable
• x̅: Mean
• Σ: Standard Deviation

Z score can also be estimated using the sample mean and standard deviation when the parameters of the population are unknown. The modified z score formula will be as follows: 

• x̅: Sample Mean
• S: Sample Standard Deviation
• x: Raw Score

Z Score Formula

Read More: T Distribution

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How to Calculate Z Score?

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Z score gives an idea of how far a raw score is from the mean of a distribution. Consider an example that a student scores 1100 marks in a test. The mean score of the test is 1026 and the population standard deviation is given as 209. Thus, to find out how well the student scored with respect to the score of the average test taker, we need to determine the z score. 

Here are the steps to calculate the z score: 

• Step 1: Put the value of the raw score, the mean, and the standard deviation in the z score formula as follows: z = (1100-1026)/209
• Step 2: Compute the values to calculate the z score. z = (1100-1026)/209 = 0.345
• Step 3: We will use the z score table to find the percentage of the test takers that are below the score of the given student. Using the first two digits of the z score, find out the row containing these digits in the z table. 
• Step 4:  Using the 2nd digit after the decimal, determine the corresponding column. The value would be the intersection of the obtained row and the column. The value will be 0.6368 for the given example.
• Step 5: Now, multiply the value obtained by 100 to get the required percentage. 0.6368 * 100 = 63.68%. 

Thus, it shows that 63.68% of scores are lesser than the given raw score of the student.

Z Score Table

Z Score Interpretation

The interpretations for the z score are as follows: 

• If a z score is 3, it means that the raw score is 3 standard deviations above the mean.
• If a z score is -3, it means that the raw score is 3 deviations below the mean.
• The z score helps to determine where the raw score will be on a normal distribution curve.

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Z Score Confidence Intervals

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Confidence Interval is also a statistical measurement used to depict the probability that a certain parameter will fall between a range of values. In the case of normally distributed data, around 68% of the data would lie between a standard deviation of 1 and -1. Around 25% of the data lies between 2 and -2 standard deviations from the mean and 99% lies between 3 and -3. 

To find the z score using confidence intervals, the following steps need to be followed:

• First, convert the confidence interval into decimals.
• Using the confidence interval, find the alpha level as α = 1 - confidence interval.
• Divide the value by 2 to get the actual alpha level.
• Now, subtract the alpha level from 1 to get the necessary area.
• Lastly, find the corresponding z value from the z score table using the area obtained.

Read More: Confidence Interval Formula

Z Score for 99 Confidence Interval

Z score for the 99% confidence interval indicates that 99% of the observations lie between the standard deviations of 3 and -3. It is given as

• The 99% confidence level converted in decimals is 0.99.
• Alpha Level: α = (1 - 0.99) / 2 = 0.005
• Area: 1 - 0.005 = 0.995
• The z score for a 99% confidence interval is 2.57.

Z Score for 95 Confidence Interval

Z score for 95% confidence interval will also be calculated using the same steps. It indicates that 95% of the observations will lie between 2 and -2 on the normal distribution curve.

• The 95% confidence level in decimals is 0.95.
• Alpha Level: α = (1 - 0.95) / 2 = 0.025
• Area: 1 - 0.025 = 0.975
• The z score for a 95% confidence interval is 1.96.

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Z Score Formula Solved Examples

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Here are a few examples of the z score formula to understand the concept better: 

Example 1: Anita scored 70 marks on a maths test. The mean score of her class was 60 with a standard deviation of 15. What will be the z score for the marks secured by Anita?

Solution: It is given that, 

• Marks Scored by Anita x = 70
• Standard Deviation σ = 15
• Mean Marks μ = 60

Using the Z Score Formula,

Z Score =(x − x̅ )/σ

Z Score for secured marks = z = (70-60)/15

= 10/15

= 0.6667

Thus, the z score for Anita’s marks is 0.6667.

Example 2: The mean temperature of 60 railway stations was recorded to be 65 degrees with a standard deviation of 5 degrees. If a railway station records a temperature of 68 degrees what percentage of temperatures lie below this value?

Solution: Given that,

• x = 68
• μ = 65
• σ = 5

Using the Z Score formula, 

z score = (68−65)/5

 = 0.6

Now, using the z table, the corresponding value is found as 0.72575.

Converting the value into percentages, we get 72.575%.

Thus, around 72.6% of temperatures lie below 68 degrees.

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Things to Remember 

• Z score is a statistical measurement tool used to determine the distance of a raw score from the mean through standard deviation.
• Z score can be positive, negative, or zero depending upon the position of the raw score with respect to the mean. 
• The z score formula is Z Score = (x − x̅ )/σ, where x is the standardized random variable, x̅ is the mean σ is the standard deviation.
• Z scores have a distribution with a mean of 0 and a standard deviation of 1. 
• The z score table is used to find the percentile of a z score.
• Confidence interval shows the probability that a certain parameter will fall between a range of values.