# Z-Score Calculator

We will apply the z-score formula below:

$z=\frac{x-\mu }{\sigma }$

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## What is a Z score ?

Z-scores are a technique of comparing outcomes to a "normal" population.

When measured in standard deviation units, a z-score represents the location of a raw score in terms of its distance from the mean.

If the value is above the mean, the z-score is positive; if it is below the mean, the z-score is negative.

It's also known as a standard score since normalizing the distribution enables for comparison of results on various types of variables.

Test or survey results can have millions of alternative outcomes and units, making them appear meaningless.

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## What Z score formula should I use?

The Z-score formula you use depends on the how the question is asked

Option 1

If you have a question on calculating z-score given the mean, standard deviation and a data point, you ought to select the first option on “Data Point (s).”

The Z score formula in this case is.  $z=\frac{x-\mu }{\sigma }$

A sample question suitable for this option could look like this:

A random variable x has a normal distribution with a mean of 8 and a standard deviation of 4. Find the Z score given x=12.

Option 2

If you have a question on calculating z-score given the Sample mean, Sample Mean, Sample Size, Population Mean and Population Standard Deviation you ought to select the second option on “Sample mean and size.”

A sample question suitable for this option could look like this:

In a Psychology class, the mean weight of students is 60 and has a standard deviation of 5. What is the probability of finding a random sample of 4 students with a mean weight of 65 Kgs, assuming that the weights are normally distributed?

Option 3

If you have a question on calculating z-score given the A Data Sample, Population Mean, and Standard Deviation you ought to select the third option on “A Data Sample.”

A sample question suitable for this option could look like this:

In a hospital, the mean height of patients in a single day  is 165cm and has a standard deviation of 5. Find the z-score of individual attending the hospital if the following height data was collected in that particular day assuming that the heights are normally distributed? The data; 157.8cm, 170cm, 145 cm, 169cm,175cm.

Note: Key in this data into the calculator without units as

157.8,170,145,169,175

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## How to interpret z score

The z-score value indicates how many standard deviations you are from the mean.

A z-score of 0 indicates that the data is average. The raw score is higher than the mean average if the z-score is positive. The raw score is below the mean average when the z-score is negative.

For instance, if a z-score is equal to +2, it posits that it is 2 standard deviations above the mean. Conversely, a negative z-score depicts that the raw score is below the mean average. For example, if a z-score is equal to -1, it is 1 standard deviation below the mean.

It enables researchers to determine the likelihood of a score falling within a conventional normal distribution. It also allows us to compare two scores from distinct samples (which may have different means and standard deviations).

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