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# Z - score calculator

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

Z-score calculator
P from Z => find probability to the left, right or between two Z-scores.
help ↓↓ examples ↓↓ tutorial ↓↓

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

Z-score calculator
Z from P => find the z-score using the cumulative probability.
help ↓↓ examples ↓↓ tutorial ↓↓

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

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Z-score calculator
Compute z-score based on a raw score
help ↓↓ examples ↓↓ tutorial ↓↓

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?

Also, find the probability of observing data that is less than the raw score.
working...
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?
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TUTORIAL

In simple terms, 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, if a 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 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

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

For example if p = 0.345 than, using the standard normal table we can find that the z-score is -0.399.

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 same formula we used in Case 1: 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?

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

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