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A z-score is a statistical measure that can be interpreted with the use of a table to find the percentage of data between the given data and the mean in a normal distribution. Beware, however, that some tables are set up to find what percent of ALL data is less than the given data value. It is always important to read the table’s explanation before using it.
To compute a z-score, use the formula: 𝑧 = 𝑥− 𝑥̅ which is the same as z = data
– mean 𝜎
std. dev.
A z-score of 0 only occurs when the given data equals the mean.
A z-score that is negative will result when the given data is less than the mean.
A z-score that is positive will result when the given data is greater than the mean.
Finding a z-score
The formula is not complicated, but you must either already have or be able to find the mean and standard deviation for the data and have a particular piece of data given. Consider the following:
Compute the z-score if the mean is 75, the standard deviation is 8.2, and the given piece of data is 83. 𝑧 = 𝑥− 𝑥̅
= 𝜎
83−75
=
8.2
8
8.2
= 0.98 (Round answers to two decimal places.)
Questions relating to z-scores could ask the following:

Find percent of data between the given and the mean.

Find percent of data that is less than the given. (also known as percentile)

Find percent of data that is greater than the given.

Find percent of data that is between two givens.
In most cases there will be a context to the question and not just a z-score that is given. You will need to compute the z-score and then interpret which of the four scenarios above is present in the problem.
Find percent of data between the given z-score and the mean
Ex.: What percentage of data lies between a z-score of 1.35 and the mean?
At z = 1.35, the reported number is 41.15, so 41.15% of the data lies between the given value and the mean.
Ex.: What percentage of data lies between a zscore of −2.24 and the mean?
The z-score being negative is only relevant to the given piece of data being less than the mean. At z = 2.24, the reported number is 48.75%, so 48.75% of the data lies between the given value and the mean.

Find percent of data that is less than the given z-score (also known as percentile)
Ex.: What percentage of data lies below a z-score of 1.96?
At z = 1.96, the reported percentage is 47.50%, but this is only the percentage between the given z and the mean.
The 50% that is below the mean must be added to the
47.50% to see that 97.50% of the data lies below a z-score of 1.96.
Ex.: What percentage of data lies below a zscore of −1.44?
At z = −1.44, the reported percentage is 42.51%, but this is the percentage between the given z and the mean. Since the z-score is already to the left of the mean, you need to subtract this value from 50% and get 7.49% of the data lies below the z = −1.44.
Find percent of data that is greater than the given z-score
Ex.: What percentage of data is greater than a z-score of 0.25?
At z = 0.25, the reported percentage is 9.87%, but this is the percentage between the given z and the mean. Since you are to the right of the mean, subtract this percentage from
50% and get 40.13% of the data lies above a z-score of
0.25.
Ex.: What percentage of data is greater than a zscore of −1.06?
At z = −1.06, the reported percentage is 35.54%. This time you want what is greater than the given z. You still need the 50% to the right of the mean, so add 35.54 and 50 to get 85.54% of the data is greater.
Find percent of data that lies between two given z-scores
Ex.: What percentage of data lies between zscores of −0.91 and
1.04?
Since one z-score was negative and the other was positive, all you have to do is add the percentages found. Z = −0.91 gives 31.86%. Z = 1.04 gives 35.08%
31.86 + 35.08 = 66.94%
Ex.: What percentage of data lies between z-scores of 0.70 and
2.02?
Since both z-scores are positive, subtract the smaller percentage from the larger.
Z = 0.70 gives 25.80%, and z = 2.02 gives 47.83%.
47.83 – 25.80 = 22.03%
Ex.: What percentage of data lies between zscores of −1.83 and
−1.95?
Since both z-scores are negative, subtract the smaller percentage from the larger.
Z = −1.83 gives 46.64%, and z = −1.95 gives 47.44%.
47.74 – 46.64 = 1.1%