Z Score Table and Chart (How to use Z-Score) (2024)

Z table or the Z-score table also called the standard normal table is a mathematical table used to find the probability that the z-score value is below, above, or between the values on the standard normal distribution.

Standard scores are most commonly called z-scores. In this post, equivalent terms like z-values, standardized variables, and normal scores are used.

The standard normal distribution table is the normal distribution having a mean value of 0 and a standard deviation of 1.

The standard score value gives you how far the raw score value is from the mean.

The Z-score value either be Positive or Negative depending on raw score value is greater than the mean or less than the mean.

The positive Z score value represents the raw score value is higher than the mean of the distribution. Use the Z score table pdf file to determine the corresponding area or probability to a positive z score value.

For example, the z-score value of 2 indicates it is 2 standard deviations greater than the mean.

A negative Z score value indicates the raw score value is below the mean of the distribution. Use the negative z table pdf file to determine the probability of a negative z score value.

Positive Z Score Table

Use the standard normal distribution table or z table chart to find values on the right of the mean distribution.

Corresponding values are higher than the mean of the distribution.

The positive z score in the z-table represents the area under the bell curve to the right of z.

Z Score Table and Chart (How to use Z-Score) (1)

Negative Z Score Table

Use the Negative Z score table or Z table chart to find values on the left of the mean distribution in the following z table.

Corresponding values are less than the mean of the distribution. The negative z score in the z-table represents the area under the bell curve to the left of z.

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How to read Z Table

Z table is used to find the z score value lies on the left of the mean or right of the mean distribution.

To use the Z table, you should have a Z score value.

You can refer Z score calculator article which explains how to calculate the Z score using the z score formula below

z = (x -μ )/σ

Where:μ = is the population mean for the unstandardized valueσ = is the population standard deviation for the unstandardized valuex = is the raw score value

z = is the calculated z-score value.

If the z score value is positive, use the positive z table to find the area on the right of the mean of the distribution.

If the z score value is negative, use the negative z table to find an area on the left of the mean of the distribution.

Let’s consider an example to find the z score value and corresponding probability.

We have below the inputs parameter

The mean of the population for the unstandardized value = 75

The population standard deviation for the unstandardized value = 15

Raw Score Value: 83

Using the above input values as per the Z score formula, it calculates the Z score value below

Z score = (83 – 75)/15Z = 0.54

Now as we have the Z score value, use the z table chart.

The z score value is positive hence we will use the positive z table chart to find an area on the right of the mean.

To find a z-score using the table, look up the z-score value on the positive z-score table.

Find the first two digits of the z score to map it on the Y-axis of the normal distribution table, in our case first, two digits are 0.5

Find the value of digits at the second decimal position in z score value, it is 0.04

In the below z table, the yellow color box indicates the mapping of the z score value on the Y-axis and X-axis to find the corresponding area as marked in red color

Z Score Table and Chart (How to use Z-Score) (3)

The corresponding area is 0.7054 which means 70.54%

For the normal standard curve, what percent of the data values are below the third quartile denoted by Q3, and also find the z-score corresponding to Q3?

We know the Q3 Quartile represents the 75th percentile in the standard normal distribution.

It means there 75% of the data values lies below the third Quartile in the standard normal distribution.

Let’s first draw the normal distribution curve for percentile = 0.75

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To find the z-score for the 75th percentile, we will follow the below steps

Step-1

Go to the z score chart and check the probability closest to 0.75 in the values inside the table.

Sometimes the exact values do not exist, in that case, we will consider the best closest value.

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The closest value in the Z-table is 0.7517

Step-2

Find the z-score corresponding to this value i.e in this case its corresponding row value is 0.6 and its corresponding column value is 0.08

Step-3

Combine these numbers as 0.6+0.08 = 0.68

Conclusion

The Z-score for the Q3 Quartile is 0.68 which means 75% of the data point of the normal distribution is below 0.68 z-score.

Z-Score Percentile Table for Normal Distribution

The below table displays the z-score percentile table for the standard normal table

PercentileZ-Score
5-1.645
10-1.282
20-0.842
40-0.253
500
600.253
800.842
851.036
901.282
951.645
971.881
982.054
992.326

Z Score Percentile Table for Normal Distribution

Z Value Table for Confidence Intervals

To find the z score for the confidence intervals, use the below table.

Confidence Interval ((1–α) * 100%)Significance level (α)Critical Z-Value (Z-score)
80%0.201.282
85%0.151.439
90%0.101.645
95%0.051.960
97%0.032.17
98%0.022.326
99%0.012.576
99.5%0.0052.807
99.8%0.0023.090
99.9%0.0013.291

What’s the difference between Z-Score vs Percentile?

Z-Score:

  • Z-score measures how much a z-score deviates from the mean of the distribution in terms of standard deviation.
  • It tells us about the position of the data value in the normal distribution.
  • It describes the distance between two extreme data points in the distribution very accurately.
  • Z-score ranges from −∞ to ∞
  • It can be used only in the case of normal distribution.

Percentile:

  • The Percentile indicates the percentage of data values lies below the certain z-value in the distribution.
  • It only tells us about the fraction of data values lies below the data value but does not specify their positions in the distribution.

Z Score Table Pdf

You can download the z score table pdf file having the positive z score chart and negative z score chart in pdf file from given below link.

Z Score Table and Chart (How to use Z-Score) (2024)

FAQs

How do you write an z-score answer? ›

The formula for calculating a z-score is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation.

How to find the z-score step by step? ›

Use the following format to find a z-score: z = X - μ / σ. This formula allows you to calculate a z-score for any data point in your sample. Remember, a z-score is a measure of how many standard deviations a data point is away from the mean.

How do you use AZ table with a negative z-score? ›

A Z-score table shows the percentage of values (usually a decimal figure) to the left of a given Z-score on a standard normal distribution. For negative Z-scores, look up the positive version on this table, and subtract it from 1.

What does the z-score tell you? ›

A z-score measures exactly how many standard deviations above or below the mean a data point is. Here are some important facts about z-scores: A positive z-score says the data point is above average. A negative z-score says the data point is below average.

What is the z-score for dummies? ›

Z-score is a result of standardizing an individual data point. Simply put, a z-score gives us an idea of how far the data point is from the mean measured in terms of standard deviation(σ). For instance, a z-score of 2.5 indicates that the value is between 2 to 3 standard deviations from the mean and is not so common.

How do you guide z-scores? ›

A z-score of 0 indicates that the value is equal to the mean. A z-score of 1 indicates that the value is one standard deviation above the mean, and a z-score of -1 indicates that the value is one standard deviation below the mean. Z-scores can identify projects or programs performing better or worse than expected.

How to interpret z-score growth chart? ›

Z-score equal to 0 means an average value, while a z-score of +1 means the value is one SD above the mean value of the population. Z-score charts (also known as centile growth charts) are used in paediatric growth follow-up and to compare anthropometrical variables to detect the presence of malnutrition or disease [3].

What is the fastest way to find the z-score? ›

Calculate the z-score using the formula z = (x - mean) / standard deviation .

How to solve z-score problems? ›

By using the z-score formula: z = (x - μ) / σ we can convert any distribution to the standard normal distribution. Here the Greek letter μ the mean and σ is the standard deviation. The standard normal distribution is a special normal distribution. It has a mean of 0 and its standard deviation is equal to 1.

How is the z-score formula written? ›

Z Score = (x − x̅ )/σ

x = Standardized random variable. x̅ = Mean. σ = Standard deviation.

How do you use z-score in a table? ›

To use a z-table, first turn your data into a normal distribution and calculate the z-score for a given value. Then, find the matching z-score on the left side of the z-table and align it with the z-score at the top of the z-table. The result gives you the probability.

How to use z-table to find critical value? ›

The z critical value can be calculated as follows:
  1. Find the alpha level.
  2. Subtract the alpha level from 1 for a two-tailed test. For a one-tailed test subtract the alpha level from 0.5.
  3. Look up the area from the z distribution table to obtain the z critical value.

What is the z-score on a graph? ›

A z-score of 0 indicates that the given point is identical to the mean. On the graph of the standard normal distribution, z = 0 is therefore the center of the curve. A positive z-value indicates that the point lies to the right of the mean, and a negative z-value indicates that the point lies left of the mean.

How do you find the z test value from a table? ›

To use a z-table, first turn your data into a normal distribution and calculate the z-score for a given value. Then, find the matching z-score on the left side of the z-table and align it with the z-score at the top of the z-table. The result gives you the probability.

How do you find the z-score of a number in a data set? ›

You would use the following formula: Z-score = (the initial data point – mean)/standard deviation. Brunner uses an example of finding out how a student's test score of 90 compared with the scores his peers received, which are 75, 80, 85, 90, and 95. First, we find out the mean of this data set, which is 85.

How to use z-table to find p value? ›

Finding the p value from a z value, using the table with standard normal probabilities
  1. Find the row corresponding to the z value you found up to the first decimal, and the column corresponding to the second decimal. ...
  2. The two sided p value is 2×(1−pleft)

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