Applied statistics - Chapter 4: Transformation and Z-scores

Linear transformation

Involves the use of a constant

Linear transformation (con’t)

Affects mean and/or standard deviation, but not the shape

What happens when a constant is added or subtracted from?

The mean gets changed

What happens when a constant is multiplied or divided from?

Both the mean and standard deviation get changed

Non-Linear data transformation

Change the shape, mean, and the standard deviation of the distribution

Why’s the shape of the distribution changed in a non-linear data transformation?

Because the transformations change individual scores NOT in a uniform manner

Why do we learn about transformations?

To standardize scores and statistics

How are z-scores useful?

A z-score will describe the exact location of a score within a single distribution

e.g. If your score of 70% on a test is z-transformed to give z = +2, you’d immediately know that you have done very well compared with the rest of the class

How are z-scores useful? (con’t)

You can compare a z-score in one distribution to a z-score in another distribution

e.g.

• On a math test, you scored 78. M=70, s=4 (z = +2),

• On an English test, you scored 90. M=80, s=10 (z = +1)

  • You performed better on the math test in comparison to the rest of the class

Important Characteristics of Z-score Distribution

1. Mean Z-scores = 0

2. Variance and Standard Deviation of Z-scores = 1

3. Converting observed scores (X scores) to Z-scores doesn’t change the shape of the distribution

Important Characteristics of Z-score Distribution (con’t)

• If the distribution of X scores is NOT normal, the distribution of Z scores will NOT be normal

• If the distribution of X scores is normal, the distribution of Z scores will be normal

The Standard Normal Distribution

Transforming a normal distribution of scores into z-scores, which allows us to place our z-scores on the standard normal distribution

The Standard Normal Distribution (con’t)

A standard normal distribution is a normal (bell-shaped) distribution with mean 0 and standard deviation 1

The Standard Normal Distribution (III)

Allows us to determine the proportion of area within specific areas of the normal distribution

i.e. Areas within the standard normal distribution are divided by standard deviation units above and below the mean