Standard Scores

Definition

The standard score, or the z-score, is the number of standard deviations a raw score (observed value or point) is above or below the mean value of what is being measured.

Mathematically, it is defined as:

where:

  • is some data point
  • is the mean
  • is the standard deviation

Explain how the z-score table does the integration for you

[!example]
Given a standard normal distribution, calculate:


  1. , so

  2. Since the normal distribution is symmetrical, we can use positive values:


Example

A set of data is normally distributed as follows:

what percent of data is less than ?

solution
We know that and . Calculating our z-score:

From a z-score table, we find that:

so of data is less than .

Example

The masses, in grams, of a set of organic apples at a local orchard are normally distributed. 37% of the apples are less than 100g, and 27% are greater than 200g. Find the mean and standard deviation of the weight of the apples.

solution
From a z-score table, we have:

so

and

Subtracting (2) from (1):

so . Substituting this back into (2) gives .

So the mean is and the STDEV is .

[!example]
Suppose that final grades in this course follow a normal distribution with and .

  1. Find the probability that a student has a final grade of 83 or more
  2. Calculate the 95th percentile

solution

P(X \ge 83) & = P\left( \frac{X - \mu}{\sigma} \ge \frac{83 - \mu}{\sigma} \right) \
& = P\left( Z \ge \frac{83 - 75}{8} \right) \
& = P(Z \ge 1) \
& = 1 - 0.83134 \
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