Zscore is a very useful tool in statistics. It is used in data analysis and deriving meaningful results by calculating the probability of zscores.
If you’re having trouble understanding and calculating the zscore, you’re in the correct place!
Continue reading to gain a strong grasp on the following topics:
 What is a zscore and its formula?
 How to Find Zscore?
 How to Find Z Score in Excel?
 How to Find Probability for a ZScore?
Let’s get started with basics!
Part 1: Understanding ZScore
1.1: What is a Zscore?
A zscore is a Standardized Normal zscore. It is the number of deviations from the mean point in a normal distribution. You must know the mean and population standard deviation to calculate the zscore.
Z score ranges from 3 standard deviation up to +3 standard deviations. A negative zscore indicates that zscore lies below the mean value. Whereas, a positive zscore means that the zscore lies above the mean value.
Zscore is helpful in finding the probability of the occurrence of an event in the given data set. It is a useful tool for testing of hypothesis and analysis of data.
The data obtained from the survey results or tests conducted in a controlled environment have a huge variation in the possible values. These values may seem meaningless if we look at them independently. The values become a useful piece of information when compared with a reference value ‘mean’.
The comparison between a specific value and the average value gives a clear picture of the performance or success metric of the specific value.
1.2 What is the ZScore Formula?
It is possible to sample data sets on various normal distribution curves. The values are ‘Standardized’ from normal distribution to a standard normal distribution for ease of calculation. The converted and standardized value is called ‘zscore’.
Zscores are used to analyze the deviation of a specific value ‘X’ from the average value. Hence, the calculation of zscore value needs the mean of the population, μ, and the normal population deviation, σ.
The formula for zscore calculation is
Zscore = (data point – mean)/standard deviation
or
Z score = ( x – µ ) / σ
Part 2: How to Find Zscore?
In some cases, you will need to find the zscore when you already have the values for mean and standard deviation. In other cases, you may have the data set and the specific value for which you need to calculate the zscore.
2.1 Calculate zscore when Mean and Standard Deviation are given
Let’s take an example and see how we can find zscore when the mean and standard deviation is given.
Problem Statement # 1
The average score on the Statistics Grade Test is 25 and the standard deviation is 3. Joseph scored 30. Joseph scored higher than what percent of his class?
Data
 Population mean, μ = 25
 Standard deviation,σ=3
 Joseph Score, x=30
Solution
Step 1: Calculate zscore


 z= (x−μ)/σ
 z= (3025)/3
 z= 5/3
 z= 1.67

Step 2: Look up ztable
 As ‘1.67’ is a positive value so we will look up positive ztable.
 Read the ztable and find corresponding value.
 Zscore for zvalue ‘1.67’ is 0.9525.
Step 3: Convert zscore into the percentage

 0.9525*100=95.25%
Result
 Joseph performed better than 95.25% of the class.
Let’s take the same problem statement and find out the zscore for Alice who scored 19 in the Statistics Grade Test.
 Problem Statement # 2
The average score on the Statistics Grade Test is 25 and the standard deviation is 3. Alice scored 19. Alice scored higher than what percent of his class?
Data
 Population mean, μ = 25
 Standard deviation,σ = 3
 Alice Score, x = 19
Solution
Step 1: Calculate zscore


 z= ( x − μ ) / σ
 z= ( 19 – 25 ) / 3
 z= – 6 / 3
 z= 2

Step 2: Look up ztable
 As ‘2’ is a negative value so we will look up negative ztable.
 Read the ztable and find corresponding value.
 Zscore for zvalue ‘2’ is 0.0228.
Step 3: Convert zscore into the percentage

 0.0228*100=2.28%
Result
 Alice performed better than only 2.28% of the class. In other words, 97.72% of the test takers performed better than Alice.
 (Calculation Hint: 10.0228=0.9772; 0.9772*100=97.72)
2.2 Calculate zscore when Mean and Standard Deviation are not given
Consider the other case, now. You have data set and the particular value which needs to be compared with the remaining data set. The mean of the population and the standard deviation is not given.
Problem Statement
The height of five friends Chris, Tyler, Kristine, Sophia, and Rob are 6.2, 6.0, 5.5,5.7,6.5 respectively. What is the zscore corresponding to Rob’s height?
Data
 Height of Chris = 6.2
 Height of Tyler = 6.0
 Height of Kristine = 5.5
 Height of Sophia = 5.7
 Height of Rob = 6.5
Required
 Mean, μ = ?
 Standard Deviation, σ = ?
 Zscore for Rob’s height, zscore = ?
Solution
 Step 1: Calculate Population Mean


 The mean is the average of all data points in the data set.
 Calculate the sum of values of all data points i.e. 6.2+6.0+5.5+5.7+6.5 =29.9
 Count the number of data points in the data set i.e. n=5
 Use formula for calculating mean = ( sum of all values)/ sample size
 Calculate mean = 29.9/5
 Hence, the population mean, μ= 5.98

 Step 2: Find Variance


 Subtract mean from each data point
 6.2 – 5.98 = 0.22
 6.0 – 5.98 = 0.02
 5.5 – 5.98 = 0.46
 5.7 – 5.98 = 0.28
 6.5 – 5.98 = 0.52
 Square each result
 0.22 ^ 2 = 0.0484
 0.02 ^ 2 = 0.0004
 0.46 ^ 2 = 0.2116
 0.28 ^ 2 = 0.0784
 0.52 ^ 2 = 0.2704
 Find the sum of the squared values
 0.0484+0.0004+0.2116+0.0784+0.2704=0.6092
 Divide by n – 1, where n is the number of data points.
 0.6092/4 = 0.1523
 Hence variance = 0.1523
 Subtract mean from each data point

 Step 3: Calculate Standard Deviation


 The standard deviation of a data set is calculated by taking under root of the variance.
 Standard deviation, σ = √0.1523
 Standard deviation, σ =0.39

 Step 4: Calculate z score


 Insert values in the zscore formula i.e (data point mean)/ standard deviation
 Zscore = (6.55.98)/0.39
 Zscore = 0.52 / 0.39
 Zscore = 1.33
 Insert values in the zscore formula i.e (data point mean)/ standard deviation

 Step 5: Find value for zscore

 The corresponding value for zscore = 0.9082
Part 3: How to Find Z Score in Excel?
3.1 Calculate ZScore in Excel
Problem Statement
150 students take Statistics exams. The population mean (μ) is 95 and the standard deviation (σ) is 19. Helen scored 89 in the exam. What is the zscore corresponding to Helen’s score?
Data
 Total number of students, n = 150
 Population Mean, μ = 95
 Standard Deviation, σ = 19
 Data Point , x = 89
Solution
 Step 1
 Label the column header as ‘Mean’ in cell A1.
 Type the population mean ’95’ into a blank cell, A2.
 Step 2
 Label the column header as ‘Population Standard Deviation’ in cell B1.
 Type the standard deviation ’19’ into a blank cell, B2.
 Step 3
 Label the column header as ‘Data Point’ in cell C1.
 Type the data value ’89’ into a blank cell, C2.
 Step 4
 Label the column header as ‘Zscore’ in cell D1.
 Enter the formula ‘=(C2A2)/B2’ into an empty cell of D2.
 Step 5
 Press ‘Enter’.
 The zscore will appear in cell D2 i.e.’0.32′.
That’s it! You’ve found a zscore in Excel.
3.2 Create a Z Calculator in Excel
Let’s quickly create a ZScore calculator in Excel.
 Merge the first two cells of Row 1 and Row 2.
 Enter the title ‘ZScore Calculator’.
 Label the first cell of Row 3 as ‘Data Point’.
 Label the first cell of Row 4 as ‘Population Mean’.
 Label the first cell of Row 5 as ‘Standard Deviation’.
 In the next row, enter label ‘Z Score’.
 In the next cell, type in the formula ‘=(B3B4)/B5’
 Where, B3 contains value of data point
 B4 contains value of population mean
 B5 contains standard deviation.
Quickly check the working of our calculator by typing in the values for the same problem
Let’s do some formatting of our calculator.
 Select cell B3
 Go to ‘Home’ > Borders > ‘Thick Box Border’.
 Repeat this cell B3, B4,B5 and B7.
 Go to View tab> Uncheck Gridlines.
 This will disappear the grids in your excel calculator and the look will be pleasant.
Part 4: How to Find Probability for a ZScore?
Knowing zscore for a data point alone does not make much sense. The positive and negative sign can only tell you if the specific value lies above or below the mean.
In order to get more useful information and gain insight of data, we need to calculate the probability for the occurrence of an event. Also, we should be able to compare data point with a standard and derive the conclusive result.
This can be achieved by calculating the probability for a zscore using either of the following methods.
Check Z Table PPT here >>
4.1 Find Probability using ZTable
Finding probability by looking up ztable is a commonly used method. There are two ztables for serving this purpose.
A positive ztable is used to find the probability of data points lying above the mean value. When a zscore is positive, the positive ztable is referred to.
A negative ztable is used to find the probability of data points lying below the mean value. The negative ztable is looked up when a zscore is a negative value.
Reading a zscore from ztable is easy. Follow the steps below:
 See the ones digit and the tenths digit after a zvalue decimal (It is the first digit after the decimal point).
 Go to the row corresponding to the value and mark it.
 Cross the column now and locate the hundredth digit after the decimal value (it is the second digit after the decimal point).
 Note the value present at the intersection of the row and column.
 To understand the percentage, multiply the acquired value by hundred.
Let’s practice an example by continuing with our problem statement.
Problem Statement
150 students take Statistics exams. The population mean (μ) is 95 and the standard deviation (σ) is 19. Helen scored 89 in the exam and the zscore is ‘0.32’. Find out (i) Helen performed better than how many students (ii) How any students performed better than Helen?
Data
 Total number of students, n = 150
 Population Mean, μ = 95
 Standard Deviation, σ = 19
 Data Point , x = 89
 Zscore , z = – 0.32
Solution
 Step 1: Find Probability


 The ones and tenth digits of zscore is 0.3.
 Go to row ‘0.3’ and mark it.
 The hundredths digit after decimal point of zscore is 0.02
 Go to column 0.02 and mark it.
 Now, circle the value at the intersection of marked row and column.
 Note the value.
 The probability corresponding to Helen’s zscore is 0.3745

 Step 2: Find Percentage

 Multiply the obtained value by 100.
 p =0.3745 *100 =37.45%
 Step 3: Find Helen performed better than how many students i.e p ( Z < z )


 ‘P’ indicates that Helen performed better than 37.45 % of the class.
 Use percentage to calculate the number of students i.e.(37.45*150)/100 = 56.175.
 Since human beings can not be a partial value so we round off the figure to 56.

 Step 4: Find how many students performed better than Helen i.e. p (Z > z)

 Calculate the percentage of remaining students by subtracting p of zscore from 1.
 10.3745=0.6255
 0.6255*100=62.55%
 Use percentage to calculate the number of students i.e. (62.55*150)/100 =93.8255
 Since human beings can not be a partial value so we round off the figure to 94.
 Hint: You can simply subtract the number of students obtained in step 3 from the total number of test takers i.e. 150 – 56 = 94
 Calculate the percentage of remaining students by subtracting p of zscore from 1.
Result
 Helen performed better than 56 students.
 94 students performed better than Helen.
4.2 Find Probability using an online calculator
There are several pvalue calculators available. If you are feeling lazy, you can quickly look up for the probability of a zscore using online calculators.
We recommend the Omni Calculator because it gives quick answer to p (z < Z) and p (z > Z).
We have typed in the zscore of our aforementioned example and the p values are automatically calculated. Notice that both results match. You can use p value calculator by clicking here.