Z-Score Table: Formula, Format and Application

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Z-score is the number of standard deviations from the mean. A data point represents the number of standard deviations above or below the mean.

A standard score for a raw score expressed as a Z-score is another name for it, and it can be plotted on a normal distribution curve.
The range of Z scores is -3 to +3 standard deviations.
We can use a Z-score to calculate the distance or difference between a value and the mean value.
When a variable is "standardised" , its mean and standard deviation are both changed to zero.

Table of Content

Z score formula
Z score table
Z score format
How to interpret Z score?
Z score and standard deviation
Application of Z score
Things to Remember
Previous Year Questions
Sample Questions

Key Terms: Variance, Standard Deviation, Probability, Z Score,Random, Variable, Mean, Zero,

Z Score Formula

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It is a method for contrasting test results with those of a "normal" population.

If X is a random variable with a mean (μ) and a standard deviation (σ), it is possible to compute its Z-score by deducting the mean from X and dividing the result by the standard deviation.

z = (x – μ) / σ

here, x = test value

μ = mean

and σ = standard deviation

Z Score Table

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Finding the data value's z-score is the first step in locating a specific area under a normal curve.

The area may then be found using a Z-score table.
A Z-Score Table is a chart that displays the percentage of values (or area percentage) on a standard normal distribution that are to the left of a specific z-score.

Positive Z score table: An elevated observed value above the mean of all values is indicated by a positive Z-score.
Negative Z-scores Table: An observed value that is lower than the mean of all values has a negative Z-score value.

Right Z table

The region on the curve's right side can be seen in this z-table (normal distribution table). The region between z=0 and any positive value can be found using these values. Look at this left-tail z-table instead if you're looking for a region in a left tail.

z	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0	0	0.004	0.008	0.012	0.016	0.0199	0.0239	0.0279	0.0319	0.0359
0.1	0.0398	0.0438	0.0478	0.0517	0.0557	0.0596	0.0636	0.0675	0.0714	0.0753
0.2	0.0793	0.0832	0.0871	0.091	0.0948	0.0987	0.1026	0.1064	0.1103	0.1141
0.3	0.1179	0.1217	0.1255	0.1293	0.1331	0.1368	0.1406	0.1443	0.148	0.1517
0.4	0.1554	0.1591	0.1628	0.1664	0.17	0.1736	0.1772	0.1808	0.1844	0.1879
0.5	0.1915	0.195	0.1985	0.2019	0.2054	0.2088	0.2123	0.2157	0.219	0.2224
0.6	0.2257	0.2291	0.2324	0.2357	0.2389	0.2422	0.2454	0.2486	0.2517	0.2549
0.7	0.258	0.2611	0.2642	0.2673	0.2704	0.2734	0.2764	0.2794	0.2823	0.2852
0.8	0.2881	0.291	0.2939	0.2967	0.2995	0.3023	0.3051	0.3078	0.3106	0.3133
0.9	0.3159	0.3186	0.3212	0.3238	0.3264	0.3289	0.3315	0.334	0.3365	0.3389
1	0.3413	0.3438	0.3461	0.3485	0.3508	0.3531	0.3554	0.3577	0.3599	0.3621
1.1	0.3643	0.3665	0.3686	0.3708	0.3729	0.3749	0.377	0.379	0.381	0.383
1.2	0.3849	0.3869	0.3888	0.3907	0.3925	0.3944	0.3962	0.398	0.3997	0.4015
1.3	0.4032	0.4049	0.4066	0.4082	0.4099	0.4115	0.4131	0.4147	0.4162	0.4177
1.4	0.4192	0.4207	0.4222	0.4236	0.4251	0.4265	0.4279	0.4292	0.4306	0.4319
1.5	0.4332	0.4345	0.4357	0.437	0.4382	0.4394	0.4406	0.4418	0.4429	0.4441
1.6	0.4452	0.4463	0.4474	0.4484	0.4495	0.4505	0.4515	0.4525	0.4535	0.4545
1.7	0.4554	0.4564	0.4573	0.4582	0.4591	0.4599	0.4608	0.4616	0.4625	0.4633
1.8	0.4641	0.4649	0.4656	0.4664	0.4671	0.4678	0.4686	0.4693	0.4699	0.4706
1.9	0.4713	0.4719	0.4726	0.4732	0.4738	0.4744	0.475	0.4756	0.4761	0.4767
2	0.4772	0.4778	0.4783	0.4788	0.4793	0.4798	0.4803	0.4808	0.4812	0.4817
2.1	0.4821	0.4826	0.483	0.4834	0.4838	0.4842	0.4846	0.485	0.4854	0.4857
2.2	0.4861	0.4864	0.4868	0.4871	0.4875	0.4878	0.4881	0.4884	0.4887	0.489
2.3	0.4893	0.4896	0.4898	0.4901	0.4904	0.4906	0.4909	0.4911	0.4913	0.4916
2.4	0.4918	0.492	0.4922	0.4925	0.4927	0.4929	0.4931	0.4932	0.4934	0.4936
2.5	0.4938	0.494	0.4941	0.4943	0.4945	0.4946	0.4948	0.4949	0.4951	0.4952
2.6	0.4953	0.4955	0.4956	0.4957	0.4959	0.496	0.4961	0.4962	0.4963	0.4964
2.7	0.4965	0.4966	0.4967	0.4968	0.4969	0.497	0.4971	0.4972	0.4973	0.4974
2.8	0.4974	0.4975	0.4976	0.4977	0.4977	0.4978	0.4979	0.4979	0.498	0.4981
2.9	0.4981	0.4982	0.4982	0.4983	0.4984	0.4984	0.4985	0.4985	0.4986	0.4986
3	0.4987	0.4987	0.4987	0.4988	0.4988	0.4989	0.4989	0.4989	0.499	0.499
3.1	0.499	0.4991	0.4991	0.4991	0.4992	0.4992	0.4992	0.4992	0.4993	0.4993
3.2	0.4993	0.4993	0.4994	0.4994	0.4994	0.4994	0.4994	0.4995	0.4995	0.4995
3.3	0.4995	0.4995	0.4995	0.4996	0.4996	0.4996	0.4996	0.4996	0.4996	0.4997
3.4	0.4997	0.4997	0.4997	0.4997	0.4997	0.4997	0.4997	0.4997	0.4997	0.4998
3.5	0.4998	0.4998	0.4998	0.4998	0.4998	0.4998	0.4998	0.4998	0.4998	0.4998
3.6	0.4998	0.4998	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999
3.7	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999
3.8	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999	0.4999

Left Z table

The region to Z's left can be seen in this table. Simply put, the region of a left hand tail. Use the right-hand z table to determine a value between z=0 and a positive number.

Z	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0	0.5	0.504	0.508	0.512	0.516	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.591	0.5948	0.5987	0.6064	0.1064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.648	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.67	0.6736	0.6772	0.6808	0.6844	0.6879
0.5	0.6915	0.695	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.719	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.758	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.791	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.834	0.8365	0.8389
1	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.877	0.879	0.881	0.883
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.898	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
1.5	0.9332	0.9345	0.9357	0.937	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.975	0.9756	0.9761	0.9767
2	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.983	0.9834	0.9838	0.9842	0.9846	0.985	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.989
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.992	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
2.5	0.9938	0.994	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.996	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.997	0.9971	0.9972	0.9973	0.9974
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.998	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
3	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.999	0.999

Z Score Format

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There are several formats available for the z-score tables. Here are the two most widely used z-score formats:

The probability or yielding area.
The first format aids in calculating the probability or the area.

Starting at the mean, move rightward until you reach the necessary z-score. These charts are frequently referred to as "cumulative from the mean."

The user has ensured that they take this into mind and make the appropriate modifications while using the table because the table only uses half of the area under the normal curve. The table in question only contains positive z-scores sequential from the left.

This table format helps in determining the region or probability starting from the last left value, which is negative infinity, and moving right above the necessary z-score.
So, "cumulative from the left" is how these tables are referred to.
In contrast to the first choice, the table operates with the entire region under the normal curve and requires little customization.
This style allows for the use of both positive and negative z-score values.

Read More: Probability density function

How to Interpret Z score?

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Z score interpretation is as follows:

An element that is below the mean is indicated by a z-score that is less than 0.
An element bigger than the mean is indicated by a z-score greater than 0.
An element that is equal to the mean is represented by a z-score of 0.
A z-score of 1 denotes an element that is 1 standard deviation above the mean, a z-score of 2 denotes an element that is 2 standard deviations above the mean, etc.
A z-score of -1 denotes an element that is 1 standard deviation from the mean; a z-score of -2 denotes an element that is 2 standard deviations from the mean; etc.
Approximately 68% of the items in a set with many elements have a z-score between -1 and 1, 95% have a z-score between -2 and 2, and 99% have a z-score between -3 and 3.

Read More: Conditional Probability

Z score and Standard Deviation

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A z-score simply represents the number of standard deviations from the mean value of the reference population, which is a population for which known data have been documented, such as in the charts the CDC collects regarding people's weights. Consider this:

In the case of a z-score of 1, the standard deviation is present above the mean.
The standard deviation can be found below the mean with a score of -2.
With a score of 1.8, the standard deviation is present above the mean and the score is 1.8.
A z-score indicates precisely where the score falls on a normal distribution curve.
Z-scores range from zero, which indicates that the values are exactly average, to +3, which indicates that the values are well above average.

Application of Z Score

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The mean and standard deviation serve as the distribution's centre and measure of the variable's level of variability, respectively, in a normal distribution.

To figure out the likelihood of a certain value in order to figure out the area under any normal distribution, use the z-table to get the areas for a computed z-score.
The likelihood that a value will occur can be determined using this.
Not all z-score tables are created equal.
The mean and standard deviation of the entire population must be correctly known in order to calculate the z score.
Many different fields make extensive use of Darar and other z-score technologies.

This can be enlisted in the manners specified below:

Z test

The Z- test is frequently employed in the standardisation of testing, and it can also be compared to a student's t-test.

The t-test is considerably more frequently employed since it is more accurate
Because it is frequently quite challenging to compute the complete population.

Prediction Intervals

Let's take into consideration a lower endpoint and an upper endpoint in order to generate the Z score based on prediction intervals. These two specific intervals support more future observations that are likely to occur farther in the interval.

Process Constant

One may quickly determine how off-target a process is functioning with the aid of the programme, which is referred to as a process constant.

Analysis by Principal Components

The variables measured in this kind of application are only taken on various scales.

Despite the fact that the variables are usually assessed on a comparable scale
The outcome that is frequently produced in both situations is standardised and often has quite distinct ranges.

Read More:

Relevant Concepts
Set Theory	Binary Operations	Types Of Relations
Quotient Rule	Maxima And Minima	Area Under Curve

Things to Remember

The percentage of values that fall below a given z score is shown in a mathematical table known as a z score table.
Z score tables come in both positive and negative varieties.
The area under a typical normal distribution curve to the left of the z score is provided in the z table.
The Z score formula: z = (x – μ) / σ here, x = test value, μ = mean
A z-score simply represents the number of standard deviations from the mean value of the reference population
The fact that standard scores always presume that all distributions are normal distributions is one of their main drawbacks.
When a variable is "standardised" , its mean and standard deviation are both changed to zero.

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Sample Questions

Ques. In the Z score, what does the Z stand for? (3 marks)

Ans. The term "standard deviation" describes a line along which a certain data point falls. The amount of standard deviations on a specific data point, either above or below the mean value, is what is referred to as the Z score or standard score. The Z value in the Z Score represents how far a given value deviates from the standard deviation.

Ques. How is Z Score used in real existence? (3 marks)

Ans. In accounting for finances, Z Score is employed. It gauges the variability of an observation. Businesses can use it to assess financial positioning, market volatility, and other factors. In general, a Z Score of less than 1.8 implies significant losses for a corporation or that it may file for bankruptcy. A high ranking (nearer to 3) attests to a company's solid financial status. In order to make better comparisons, analysts might adapt data scores from diverse data sets.

Ques. What Kinds of Z Score Tables Exist? (2 marks)

Ans. There are two kinds of z tables: negative and positive. For negative z score values and positive ones, respectively, the negative z table is employed.

Ques. How do z-scores and the normal distribution relate to one another? (3 marks)

Ans. Z-scores are used to standardise and compare data across various datasets in conjunction with the normal distribution. A probability distribution known as the normal distribution is frequently used to simulate real-world events, and z-scores enable us to transform any normal distribution into a typical normal distribution with a mean of zero and an SD (standard deviation) of one.

Ques. Use the z table to calculate the value if the raw score is 250, the mean is 150, and the standard deviation is 86. (3 marks)

Ans. The formula for the z score is given as:

z = (x – μ) / σ

here, x = 250,

μ = 150 and

σ = 86

z = 1.16

The value obtained from the positive z table is 0.8770.

Ques. How can one find a Value on the Z Score Table? (3 marks)

Ans. The following are the steps to find a value on the z score table:

Utilise the indication to locate the right table.
Find the necessary row using the first two numbers of the score.
Locate the necessary column by using the second digit following the decimal.
The necessary value will be revealed by finding the intersection of the row and the column.

Ques. According to a poll of 250 people, the participants' average annual income was $50,000, with a $10,000 standard deviation. What is the participant's z-score if they make $70,000? (3 marks)

Ans. The formula z = (x - mean) / standard deviation is used to calculate the z-score. Once the data are plugged in, we obtain: z = (70,000 - 50,000) / 10,000 = 2.

A participant earning $70,000 has a z-score of 2, which indicates that their income is 2 standard deviations above the mean of the group.

Ques. The average test score for a class of pupils is 70, while the standard deviation is 12. What percentage of students actually scored above 85? (3 marks)

Ans. For the provided data, the z score is,

z= (85-70)/12=1.25

The percentage of the data contained in this score, according to the z score table, is 0.8944.

This indicates that 89.44% of the students fall within the range of test scores of 85, making the proportion of students with test scores exceeding 85 equal to (100 – 89.44%)% = 10.56%.

Ques. How does Z Score normalisation work? (3 marks)

Ans. When analysing data, one should keep in mind that the score is a crucial factor. This is a statistical analysis of the results and information. It performs the role of a data administrator by assisting in the prediction of the likelihood that a score will be present within the normal distribution of data.

Ques. Find the value of P(z 1.26) using the z table. (2 marks)

Ans. P(z ≥ 1.26) = 1 - P(z < 1.26)

The z-score table provides us with

P(z < 1.26) = 0.8962

= 1 - 0.896

= 0.1038

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Z-Score Table: Formula, Format and Application

Z Score Formula

Z Score Table

Right Z table

Left Z table

Z Score Format

How to Interpret Z score?

Z score and Standard Deviation

Application of Z Score

Z test

Prediction Intervals

Process Constant

Analysis by Principal Components

Things to Remember

Previous Year Questions

Sample Questions

CBSE CLASS XII Related Questions

1.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

2.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

3.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

4.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

6.
A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

Comments

SUBSCRIBE TO OUR NEWS LETTER

Z-Score Table: Formula, Format and Application

Z Score Formula

Z Score Table

Right Z table

Left Z table

Z Score Format

How to Interpret Z score?

Z score and Standard Deviation

Application of Z Score

Z test

Prediction Intervals

Process Constant

Analysis by Principal Components

Things to Remember

Previous Year Questions

Sample Questions

CBSE CLASS XII Related Questions

1.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

2.If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

3.Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

4. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

6.A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

2.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

3.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

4.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

6.
A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is