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Home → Statistics → Z-score calculator Saved Solutions Z - score calculator

This calculator solves three types of problems related to z-scores.

Tutorial

What is z – score?

The z-score measures how many standard deviations the observed value is above or below the population mean.

Z scores were created so that we could easily compare some data to the average value.

For example, knowing that someone is 175 cm tall doesn't tell us anything about how that person compares to the rest of the population. Is this person higher or smaller than average? On the other hand, if person's height has a z-score of 0.15, it means he is somewhat higher than the average; furthermore, we can calculate that he is taller than 55% of the population.

How to use this calculator?

The calculator solves three types of problems.

Type 1. Find the area (probability) to the left, right, or between two Z-scores. Type 2. Find the z-score if the cumulative probability level (p-value) is given. Type 3. Find the z-score based on the raw value, mean and standard deviation of a population.

Area to the left

Example:

Find probability of selecting the z-score less than 0.46

Step 1: Draw the image:

Step 2: Use z-score table to read the result.

How to calculate z - score?

Case 1: If we know the raw score (x), mean (μ) and standard deviation (σ), we calculate the z-score using the following formula.

Z = (x - μ)/σ

Example: Given x = 72, μ = 65, and σ = 5, the z-score would be

Z = (x - μ)/σ = (72-65)/5 = 1.4

Thus, the number 72 is 1.4 standard deviations above the population mean.

Case 2: If we know the p-value, we can calculate the z-score using the standard normal table.

For example if p = 0.742 than, using the standard normal table we can find that the z-score is 0.9920.

Case 3: If we are given a dataset, then we need to apply the following steps.

1. Calculate the mean μ using the formula μ = Σx/n,

2. Calculate the standard deviation using σ formula σ2 = Σ(x - μ)2 / (n-1),

3. Apply the formula Z = (x - μ)/σ.

Example: The dataset of exam scores is 45, 51, 67 and 55. Find the z-score for x = 60.

Step1: Find the mean.

μ = (45+61+67+55)/4 = 57

Step2: Find the standard deviation.

σ2 = [(45-57)2+(61-57)2+(67-57)2+(55-57)2]/(4-1) σ2 = [(144+16+100+4]/3 σ2 = 88 σ = 9.38

Step3: Calculate z

z = (60 - 57)/9.38 = 0.319

How to interpret z - score?

The z score shows how many standard deviations you are above or below the population mean.

Positive z-score indicates that the observation is above average.

Negative z-score indicates that the observation is below average.

Approximately 68% of the data have a z-score of -1 to 1. This means that 68% of observations are less than one standard deviation away from the mean. If your z score on the exam is 1.5, you are far above the average. On the other side, a z-score of -0.25 indicates that you are slightly below average.

Z-score table

The p-value is shown by the green number, and the z-score is presented by the orange numbers. Every value in the table represents the area between Z=0 and a Z-score.

z	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.5000	0.5040	0.5080	0.5120	0.5159	0.5199	0.5239	0.5279	0.5318	0.5358
0.1	0.5398	0.5438	0.5479	0.5517	0.5557	0.5596	0.5635	0.5674	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.5909	0.5948	0.5987	0.6025	0.6064	0.6102	0.6140
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6405	0.6443	0.6480	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6843	0.6879
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7122	0.7156	0.7190	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7421	0.7453	0.7485	0.7517	0.7549
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7733	0.7763	0.7793	0.7823	0.7852
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.851	0.8078	0.8105	0.8132
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8314	0.8339	0.8364	0.8389
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8576	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8769	0.8790	0.8810	0.8829
1.2	0.8849	0.8869	0.8888	0.8906	0.8925	0.8943	0.8961	0.8979	0.8997	0.9014
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9114	0.9130	0.9146	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9264	0.9278	0.9292	0.9305	0.9318
1.5	0.9332	0.9345	0.9358	0.9370	0.9382	0.9394	0.9406	0.9417	0.9429	0.9440
1.6	0.9452	0.9463	0.9474	0.9485	0.9495	0.9505	0.9515	0.9525	0.9535	0.9544
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9624	0.9632
1.8	0.9641	0.9648	0.9656	0.9664	0.9671	0.9678	0.9685	0.9692	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9755	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9807	0.9812	0.9816
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9853	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9877	0.9880	0.9884	0.9887	0.9889
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9908	0.9911	0.9913	0.9915
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9928	0.9930	0.9932	0.9934	0.9936
2.5	0.9938	0.9939	0.9941	0.9943	0.9945	0.9946	0.9947	0.9949	0.9950	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9958	0.9959	0.9960	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9972	0.9973
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9978	0.9979	0.9980	0.9980
2.9	0.9981	0.9982	0.9983	0.9983	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986
3.0	0.9986	0.9987	0.9987	0.9988	0.9988	0.9988	0.9988	0.9989	0.9989	0.9990

Resources

1. Z - score - solved word porblems

2. Z - score - definition, formula, calculation and interpretation

P from Z Z-score from P Find Z score Z-score calculator P from Z => find probability to the left, right or between two Z-scores. help ↓↓ examples ↓↓ INSTRUCTIONS: 1 . Enter cutoff points to find the area under normal curve. 2 . You can use integers (10), decimal numbers (10.2) and fractions (10/3). 0123456789–.C ✖

Example problem: Your z-score on the math exam is 0.542. What percentage of students had fewer points than you?

P ( < Z )

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Z-score calculator Z from P => find the z-score using the cumulative probability. help ↓↓ examples ↓↓ INSTRUCTIONS: 1 . Find the z-score for a given probability 2 . You can use integers (10), decimal numbers (10.2) and fractions (10/3). 0123456789–.C ✖

Example problem:You scored better than 73.5% of the students. What's your z-score?

P value = Tails:

Left tailed

Right tailed

Two tailed

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Z-score calculator Compute z-score based on a raw score help ↓↓ examples ↓↓ INSTRUCTIONS: 1 . This calculator finds z-score based on raw values. To use this calculator you will need population mean and standard deviation. 2 . You can use integers (10), decimal numbers (10.2) and fractions (10/3). 0123456789–.C ✖

Example problem:The average math test score is 72, with a standard deviation of 8. What is the z-score for a student who scored 64?

Raw score = Population Mean μ = Standard Deviation σ = Also, find the probability of observing data that is less than the raw score.

OK

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Examples ex 1: An average grammar test score is 65, with a standard deviation of 8. For a student who received a score of 75 on the test, what is the z-score? ex 2: Find probability P(Z < 0.23) ex 3: Find probability P(-2.1 < Z < 0.1396) ex 4: Jim scored better than 81.2% of the students. What's Jim's z-score? Related calculators 1: Normal distribution 2: Descriptive statistics 3: Standard deviation 4: Probability distributions 5: Correlation and Regression ×

Z - score calculator – Widget Code

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452 861 664 solved problems working... × ans: syntax error C DEL ANS ± ( ) ÷ × 7 8 9 – 4 5 6 + 1 2 3 = 0 . About the Author

Welcome to MathPortal. This website's owner is mathematician Miloš Petrović. I designed this website and wrote all the calculators, lessons, and formulas.

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