Z Score

Z score is also known as a standard score and is used to represent the number of standard deviations by which a raw score is above or below the mean. A z score is usually used as part of a z test to draw interpretations about population data. This score helps to compare data from different normal distributions.

A z score can be positive, negative, or zero depending upon the position of the raw score with respect to the mean. To determine a z score the knowledge of the population mean and the standard deviation is required. In this article, we will learn more about a z score, its formula, and how to calculate it.

A z score is a type of statistical measurement that gives an idea of how far a raw score is from the mean of a distribution. A z score is used in a z test for hypothesis testing. It is also used in prediction intervals to determine the probability of a random variable falling between a range of values.

Z Score Definition

A z score can be defined as a measure of the number of standard deviations by which a score is below or above the mean of a distribution. In other words, it is used to determine the distance of a score from the mean. If the z score is positive it indicates that the score is above the mean. If it is negative then the score will be below the mean. However, if the z score is 0 it denotes that the data point is the same as the mean.

To calculate a z score, knowledge of the mean and standard deviation is required.

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

z = \(\frac{x-\mu}{\sigma}\)

\(\mu\) = population mean

\(\sigma\) = population standard deviation

x = raw score

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

z = \(\frac{x-\overline{x}}{S}\)

\(\overline{x}\) = sample mean

S = sample standard deviation

x = raw score

A z score will help in understanding where a particular observation will lie in a distribution. Suppose on a GRE test a score of 1100 is obtained. The mean score for the GRE test is 1026 and the population standard deviation is 209. In order to find how well a person scored with respect to the score of an average test taker, the z score will have to be determined. The steps to calculate the z score are as follows:

• Step 1: Write the value of the raw score in the z score equation. z = \(\frac{1100-\mu}{\sigma}\)
• Step 2: Write the mean and standard deviation of the population in the z score formula. z = \(\frac{1100-1026}{209}\)
• Step 3: Perform the calculations to get the required z score. z = \(\frac{1100-1026}{209}\) = 0.345
• Step 4: A z score table can be used to find the percentage of test-takers that are below the score of the person. Using the first two digits of the z score, determine the row containing these digits of the z table. Now using the 2^nd digit after the decimal, find the corresponding column. The intersection of this row and column will give a value. As shown below, this value will be 0.6368 for the given example.
• Step 5: Use the value from step 5 and multiply it by 100 to get the required percentage. 0.6368 * 100 = 63.68%. This shows that 63.68% of test-takers scores are lesser than the given raw score.

Z Score Interpretation

• If a z score is 3 it implies that the raw score is 3 standard deviations above the mean.
• A z score of -3 indicates that the raw score is 3 deviations below the mean.
• The z score also shows where the raw score will be on a normal distribution curve.

A confidence interval is a statistical measurement that is used to show the probability that a certain parameter will fall between a range of values. For normally distributed data around 68% of the data will lie between a standard deviation of 1 and -1. 25% lies between 2 and -2 standard deviations from the mean and 99% lies between 3 and -3. The steps to find the z score using confidence intervals are as follows:

• Convert the confidence interval into decimals.
• Find the alpha level using the confidence interval as \(\alpha\) = 1 - confidence interval.
• Now divide this value by 2 to get the actual alpha level.
• Subtract the alpha level from 1 to get the required area.
• Find the corresponding z value from the z score table using the area obtained in the previous step.

Z Score for 99 Confidence Interval

The z score for 99% confidence interval implies that 99% of the observations lie between the standard deviations of 3 and -3. This is given as follows:

• 99% confidence level in decimals is 0.99.
• Alpha level: \(\alpha\) = (1 - 0.99) / 2 = 0.005
• Area: 1 - 0.005 = 0.995
• z score for 99% confidence interval = 2.57

Z Score for 95 Confidence Interval

The z score for 95% confidence interval can be calculated using the same steps. It will lie between 2 and -2 on the normal distribution curve.

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

Related Articles:

• Probability and Statistics
• Data Handling
• Data
• Z Score Calculator

Important Notes on Z Score

• Z score is a statistical measure that is used to determine the distance of a raw score from the mean by measuring the standard deviations.
• A z score can be positive, negative, or zero.
• The z score formula is given as \(\frac{x-\mu}{\sigma}\).
• To find the percentile of a z score, a z table is used.

What is a Z Score in Statistics?

In statistics, a z score can be defined as a measurement that is used to denote the number of standard deviations by which a particular raw score will be above or below the mean of that distribution.

What is the Z Score Formula?

The z score formula is z = \(\frac{x-\mu}{\sigma}\) when the population mean and standard deviation are known. If the sample mean and standard deviation are known then the z score is \(\frac{x-\overline{x}}{S}\).

Can the Z Score be Negative?

Yes, the z score can be negative. This implies that the raw score lies below the mean. To find the corresponding percentile, the negative z table must be used.

What Does a Z score of 2.2 Mean?

A z score of 2.2 implies that the raw score is 2.2 standard deviations further from the mean. As the score is positive it denotes that is raw score is above the mean.

How is Z Score Calculated?

To calculate the z score the steps are as follows:

• Subtract the mean from the raw score.
• Divide this value by the standard deviation to get the z score.

Why is Z Score Used?

A z score helps to find the probability of occurrence of a raw score in the given normal distribution. It is also very useful in comparing scores from different normal distributions.

What are the Applications of Z Score?

A z score is used in a z test and can be used to conduct hypothesis testing to check whether the null hypothesis should be rejected or not. It also helps in determining the probability of a random variable falling in between an interval.