How to Find Z Score

Z-score is a very useful tool in statistics. It is used in data analysis and deriving meaningful results by calculating the probability of z-scores.

If you’re having trouble understanding and calculating the z-score, you’re in the correct place!

Continue reading to gain a strong grasp on the following topics:

  • What is a z-score and its formula?
  • How to Find Z-score?
  • How to Find Z- Score in Excel?
  • How to Find Probability for a Z-Score?

Let’s get started with basics!

Part 1: Understanding Z-Score

1.1: What is a Z-score?

A z-score is a Standardized Normal z-score. 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 z-score.

Z- score ranges from -3 standard deviation up to +3 standard deviations. A negative z-score indicates that z-score lies below the mean value. Whereas, a positive z-score means that the z-score lies above the mean value.

Z-score 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 Z-Score 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 ‘z-score’.

Z-scores are used to analyze the deviation of a specific value ‘X’ from the average value. Hence, the calculation of z-score value needs the mean of the population, μ, and the normal population deviation, σ.

The formula for z-score calculation is

Z-score = (data point – mean)/standard deviation

or

Z score = ( x – µ ) / σ

Part 2: How to Find Z-score?

In some cases, you will need to find the z-score 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 z-score.

2.1 Calculate z-score when Mean and Standard Deviation are given

Let’s take an example and see how we can find z-score 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  z-score

      • z= (x−μ)/σ
      • z=  (30-25)/3
      • z= 5/3
      • z= 1.67

Step 2: Look up z-table

  • As ‘1.67’ is a positive value so we will look up positive z-table.
  • Read the z-table and find corresponding value.
  • Z-score for z-value ‘1.67’ is 0.9525.

how to find z score

Step 3: Convert z-score 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 z-score 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  z-score

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

Step 2: Look up z-table

how to find z score 1

  • Z-score for z-value ‘-2’ is 0.0228.

Step 3: Convert z-score 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: 1-0.0228=0.9772; 0.9772*100=97.72)

2.2 Calculate z-score 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 z-score 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 = ?
  • Z-score for Rob’s height, z-score = ?

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
  • 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 z-score formula i.e (data point- mean)/ standard deviation
        • Z-score = (6.5-5.98)/0.39
        • Z-score = 0.52 / 0.39
        • Z-score = 1.33
  • Step 5: Find value for z-score
    • The corresponding value for z-score = 0.9082

how to find z score

Part 3: How to Find Z- Score in Excel?

3.1 Calculate Z-Score 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 z-score 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 ‘Z-score’ in cell D1.
    • Enter the formula ‘=(C2-A2)/B2’ into an empty cell of D2.
  • Step 5
    • Press ‘Enter’.
    • The z-score will appear in cell D2 i.e.’-0.32′.

That’s it! You’ve found a z-score in Excel.

3.2 Create a Z- Calculator in Excel

Let’s quickly create a Z-Score calculator in Excel.

  • Merge the first two cells of Row 1 and Row 2.
  • Enter the title ‘Z-Score 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 ‘=(B3-B4)/B5’
    • Where, B3 contains value of data point
    • B4 contains value of population mean
    • B5 contains standard deviation.

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Quickly check the working of our calculator by typing in the values for the same problem

how to find z score 4

Let’s do some formatting of our calculator.

  • Select cell B3
  • Go to ‘Home’ > Borders > ‘Thick Box Border’.how to find z score 5
  • 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.

how to find z score 6

how to find z score 7

Part 4: How to Find Probability for a Z-Score?

Knowing z-score 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 z-score using either of the following methods.

Check Z Table PPT here >>

4.1 Find Probability using Z-Table

Finding probability by looking up z-table is a commonly used method. There are two z-tables for serving this purpose.

A positive z-table is used to find the probability of data points lying above the mean value. When a z-score is positive, the positive z-table is referred to.

A negative z-table is used to find the probability of data points lying below the mean value. The negative z-table is looked up when a z-score is a negative value.

Reading a z-score from z-table is easy. Follow the steps below:

  • See the ones digit and the tenths digit after a z-value 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 z-score 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
  • Z-score , z = – 0.32

Solution

  • Step 1: Find Probability
      • The ones and tenth digits of z-score is -0.3.
      • Go to row ‘-0.3’ and mark it.
      • The hundredths digit after decimal point of z-score 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 z-score is 0.3745
  • Step 2: Find Percentage
    • Multiply the obtained value by 100.
    • p =0.3745 *100 =37.45%

how to find z score 8

  • 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 z-score from 1.
      • 1-0.3745=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

Result

  • Helen performed better than 56 students.
  • 94 students performed better than Helen.

4.2 Find Probability using an online calculator

There are several p-value calculators available. If you are feeling lazy, you can quickly look up for the probability of a z-score 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 z-score 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.

how to find z score 9