How to Use Z Table

Z-score is a very useful tool in statistics and is calculated from the horizontal scale of a standard normal distribution. If you have a difficulty understanding z-table and Z-score, you are at the right place!

• What is a z-score?
• Why do we need a z-score?
• What is z- tables?
• How a value ‘X’ from a normal distribution is converted into z-score?
• How to use z-tables to calculate z-score?

In the end, some practices questions are given along with the solution.

What is a z-score?

A z-score is the ‘standard score’. It indicates the number of standard deviations a random value ‘X’ lies from the population mean, μ. A positive z-score lies above the mean, whereas a negative z-score lies below the mean.

z-score allows you to calculate the probability of occurrence of a certain event. Moreover, it enables you to compare two scores that are from different normal distributions.

Why do we need a z-score?

Data sets can be sampled on different normal distribution curves. The analysis of data is dependent on the population mean, μ, and the population standard deviation, σ. For ease of calculations, the values are ‘Standardized’ from normal distribution to standard normal distribution.

The converted and standardized value is called ‘z-score’. For comparison of two scores, the scores in a normal distribution are converted into z-scores.

The Empirical Rule of 68%-95% and 99.7% subdivides the area under the curve into six sections. The width of each section is one standard deviation. The curve is symmetric on both sides of the mean.

The total area under the curve is probability 1 or 100%.

We can tell the probability of a given value ‘X’ at the subdivision of -3, -2,-1,0,1,2,3 by looking at the graph. However, we can not tell the same for any point lying anywhere in the curve other than these points. Hence, z-score is needed to know the probability at a given instant in the standard normal distribution curve.

What is z- table?

Z-table is used to calculate the probability of a random variable ‘Z’ corresponding to any z-value. Each value in the table indicates a cumulative area under the curve. The z-table tells us how well a particular metrics has performed in comparison with its competitors in the form of a percentage.

The z-table is divided into two sections:

• The ‘row’ indicates the integer part and first decimal of the z-score.
• The ‘column’ indicates the second decimal of the z-score.
• The value at the intersection of row and column is the probability.

There are two z-tables: Positive z-score table and a negative z-score table. You will need both z-tables to calculate the probability of z-scores. The two tables prevent the statistician from being overwhelmed by the data.

Negative z-score table

The negative z score table is used to find values to the left of the mean. The table entries represent the area under the bell curve to the left of the ‘z-value’ where z-value occurs before the mean. In simple words, this area contains the values which are less than the mean value.

Positive z-score table

The positive z score table is used to find values to the right of the mean. The table entries represent the area under the bell curve to the left of the ‘z-value’ where z-value occurs after the mean. In simple words, this area contains the values which are greater than the mean value.

How a value ‘X’ from a normal distribution is converted into z-score?

Z-score applies to a number of situations in real-life and can be used to measure the performance of ‘X’ in comparison to the competitors. In order to convert a value ‘X’ to a z-score, the population mean (µ) and the standard deviation (σ) of the data sample must be known.

Z-score for any value from the data set can be calculated as:

z= (x- µ)/σ

Example

Consider a group of 150 applicants who took a Statistics test. Jhona was among the test takers and secured only 400 out of 1000. The average score is 750 (µ) and the standard deviation was 150 (σ). Now, we want to examine the performance of Johna in comparison with her peers.

Let’s calculate the z-score corresponding to Jhona’s actual test score.

i.e.

z=(400-750)/150

z= -2.33

Hence, the z-score for Johna’s test score is -2.33. Since the score is negative, so we can easily tell that she underperformed as compared to her peers.

To know the exact percentage of deviation from the average score, you need to use the negative z-table.

How to use z-tables to calculate z-score?

You can find the probability of any value belonging to a statistical sample with a standard normal distribution (Z) by the following steps:

• See the sign ‘+’ or ‘-‘ of your z-value.
• If it is a positive value, use positive z-table.
• If it is a negative value, use negative z-table.
• See the ones digit and the tenths digit after a decimal of z-value (It is the first digit after the decimal point).
• Go to the row corresponding to the value and mark it.
• Now, traverse through the column and find the hundredths digit after decimal value (It is the second digit after the decimal point).
• Circle the value present at the intersection of the row and column.
• Multiply the obtained value with hundred to know the percentage.

Let’s practice out using the same example to see how it works.

The z-score for Johna is ‘-2.33’.

Step 1: The first step is to go to the row corresponding to ‘-2.3’.

Step 2: The next step is to go to the column  of ‘0.03’

Step 3: See the value at the intersection point of row and column.

Step 4: The value is 0.0099

Step 5: Multiply the value with 100 to get the percentage

1. i.e.  0.0099 *100=0.99
2. The percentage is 0.99%.

Understanding the result

This result indicates that Johna’s performed better than only 0.01% of the students. 0.01% of 150 students is only 1.5. Since human beings cannot be part, so we will round it to 1. Jhona performed better than 1 student only.

Practice Questions

Question 1: What is p(Z<2.16)?

Steps

• Use positive z-table.
• Go to row ‘2.1’ and column ‘0.06’.
• See the value at the intersection point.
• The value is 0.9846.

Answer: The probability of p(Z<2.16) is 0.9846 or 98.46%.

Question 2: What is p(Z<-0.06)?

Steps

• Use negative z-table.
• Go to row ‘-0.0’ and column ‘0.06’.
• See the value at the intersection point.
• The value is 0.4761.

Answer: The probability of p(Z<0.06) is 0.4761 or 47.61%.

Question 3: What is p(Z>1.03)?

Explanation

The standard normal curve tells us only about the area less than a given point. In other words, we can only know the probability of a given value which is less than ‘z’.

∵ p(Z<1.03) + p(Z>1.03) =1

∴ P(Z>1.03)=1- p(Z<1.03)

If we want to know the probability of a value greater than ‘z’, we need to perform additional steps. First, we calculate the probability of value less than ‘z’. The next step is to subtract this probability form the total of 1.

By doing so, we get the probability of occurrence of a random variable Z  greater than ‘z’.

Steps

• Use positive z-table.
• Go to row ‘1.0’ and column ‘0.03’.
• See the value at the intersection point.
• The value is 0.8485.
• The probability of p(Z<1.03) is 0.8485.
• Subtract the value from 1. i.e. 1-0.8485=0.1515

Answer: The probability of p(Z>1.03) is 0.1515 or 15.15%.

Question 4: What is p (Z>-1.50)?

Steps

• Use negative z-table.
• Go to row ‘-1.5’ and column ‘0.00’.
• See the value at the intersection point.
• The value is 0.8485.
• The probability of p(Z<1.03) is 0.0668.
• Subtract the value from 1.
• .1-0.8485=0.9332

Answer: The probability of p(Z>–1.05) is 0.9332 or 93.32%.

Interesting Facts

The standard normal curve is symmetric. The tail of curve below any value -z looks exactly like the tail above +z.

i.e. p(Z<-z) = p(Z>+z)

• Let’s say, z =-1.3
• Using tables we calculate,
• p( Z<-1.3) = (0.0968)
• p (Z>+1.3) = 1- (0.9032)  =0.0968
• Hence verified that
• p (Z<-1.3) = p (Z>+1.3)

Also, the probabilities are not at a single point but a cumulative area so any p(Z<z)=p(Z≤z).