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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.

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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?

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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.

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 σ^{2} = 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.

σ^{2} = [(45-57)^{2}+(61-57)^{2}+(67-57)^{2}+(55-57)^{2}]/(4-1)

σ^{2} = [(144+16+100+4]/3

σ^{2} = 88

σ = 9.38

Step3: Calculate z

z = (60 - 57)/9.38 = 0.319

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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