Normal Distributions and Standard (z) Scores

Normal Distributions and Standard (z) Scores

• The normal distribution is a continuous probability distribution that is symmetrical on both sides of the mean, so the right side of the center is a mirror image of the left side. The area under the normal distribution curve represents probability and the total area under the curve sums to one.

• The normal distribution is often called the bell curve because the graph of its probability density looks like a bell. It is also known as called Gaussian distribution, after the German mathematician Carl Gauss who first described it.

• Fig. 2.9.1 shows normal curve.

• A normal distribution is determined by two parameters the mean and the variance. A normal distribution with a mean of 0 and a standard deviation of 1 is called a standard normal distribution.

z Scores

• The Z-score or standard score, is a fractional representation of standard deviations from the mean value. Accordingly, z-scores often have a distribution with no average and standard deviation of 1. Formally, the z-score is defined as :

            Z = X-μ / σ

where μ is  mean, X is score and σ is standard deviation

• The z-score works by taking a sample score and subtracting the mean score, before then dividing by the standard deviation of the total population. The z-score is positive if the value lies above the mean and negative if it lies below the mean.

• A z score consists of two parts:

a) Positive or negative sign indicating whether it's above or below the mean; and

b) Number indicating the size of its deviation from the mean in standard deviation units

 • Why are z-scores important?

• It is useful to standardized the values (raw scores) of a normal distribution by converting them into z-scores because:

 (a) It allows researchers to calculate the probability of a score occurring within a standard normal distribution;

(b) And enables us to compare two scores that are from different samples (which may have different means and standard deviations).

• Using the z-score technique, one can now compare two different test results based on relative performance, not individual grading scale.

Example 2.9.1: A class of 50 students who have written the science test last week. Rakshita student scored 93 in the test while the average score of the class was 68. Determine the z-score for Rakshita's test mark if the standard deviation is 13.

Solution: Given,

Rakshita's test score, x = 93, Mean (u) = 68, Standard deviation (σ) = 13 The z-score for Rakshita's test score can be calculated using formula as,

Ꮓ = X- μ / σ = 93-68 / 13 = 1.923

Example 2.9.2: Express each of the following scores as a z score:

(a) Margaret's IQ of 135, given a mean of 100 and a standard deviation of 15

(b) A score of 470 on the SAT math test, given a mean of 500 and a standard deviation of 100.

Solution :

a) Margaret's IQ of 135, given a mean of 100 and a standard deviation of 15

Given, Margaret's IQ (X) = 135, Mean (u) = 100, Standard deviation (o) = 15

The z-score for Margaret's calculated using formula as,

Z = X- μ / σ = 135-100 / 15 =2.33

b) A score of 470 on the SAT math test, given a mean of 500 and a standard deviation of 100

Given,

Score (X) = 470, Mean (u) = 500, Standard deviation (6)= 100

The z-score for Margaret's calculated using formula as,

Z = X-μ / σ = 470-500 /100 = 0.33

Standard Normal Curve

• If the original distribution approximates a normal curve, then the shift to standard or z-scores will always produce a new distribution that approximates the standard normal curve.

• Although there is an infinite number of different normal curves, each with its own mean and standard deviation, there is only one standard normal curve, with a mean of 0 and a standard deviation of 1.

Example 2.9.3: Suppose a random variable is normally distributed with a mean of 400 and a standard deviation 100. Draw a normal curve with parameter label.

Solution: