z-Scores: Location in Distributions

Agenda Prayer A bit more about standard deviations z-Scores: the basics Standardizing distributions Tuesday:

• More on standardized distributions /T-scores

• Using R• The STORY in your data

Descriptive Statistics in Research

LuAnn, S., J. Walter and D. Antosh. (2007) Dieting behaviors of young women post-college graduation. College Student Journal 41:4.

CNN-Money uses Standard Deviation

Make fear and greed work for youWall Street constantly swings between these two emotions. You can either get caught in the frenzy - or profit from it.By Janice Revell, Money Magazine senior writerLast Updated: July 21, 2009: 10:56 AM ET

“Making matters worse, the big stock bet would be far riskier on a year-to-year basis than other strategies. The most common measure of portfolio risk is standard deviation, which tells you how much an investment's short-term returns bounce around its long-term average. Since 1926 stocks have returned average gains of 9.6% a year, with a standard deviation of 21.5 percentage points, according to Ibbotson Associates. That means that about two-thirds of the time, the annual return on stocks landed 21.5 percentage points below or above the average - that is, in any given year, your results would range from a 12% loss to a 31% gain. You'd need either an iron stomach or a steady supply of Zantac to stay the course. And if you happened to be at or near retirement when one of those really bad years hit, you might have to rethink your plans.”

http://money.cnn.com/2009/07/20/pf/funds/fear_greed.moneymag/

When you finish studying z-scores you should be able to …

Explain how z-scores provide a description of a location in a distribution Transform an X score into a z-score Transform z-scores back into X scores, when the mean and standard

deviation are given. Use z-scores to make comparisons across variables and individuals. Describe the effects when an entire data set is standardized by

transforming all the scores to z-scores, including the impact on the shape, mean and standard deviation, and its comparability to other standardized distributions.

Use z-scores to transform a distribution into a standardized distribution. Use SPSS to create standardized scores for a distribution.

Two distributions of exam scores

Locations and Distributions

Exact location is described by z-score• Sign tells whether score is located

above or below the mean

• Number tells distance between score and mean in standard deviation units

Two distributions of exam scores

Relationship of z-scores and locations

64 67 70 73 7646 58 70 82 94

Learning Check

• A z-score of z = +1.00 indicates a position in a distribution ____

• Above the mean by 1 pointA• Above the mean by a distance equal to 1

standard deviationB

• Below the mean by 1 pointC• Below the mean by a distance equal to 1

standard deviation D

Learning Check - Answer

• A z-score of z = +1.00 indicates a position in a distribution ____

• Above the mean by 1 pointA• Above the mean by a

distance equal to 1 standard deviation

B• Below the mean by 1 pointC• Below the mean by a distance

equal to 1 standard deviation D

Learning Check

• Decide if each of the following statements is True or False.

• A negative z-score always indicates a location below the mean

T/F• A score close to the mean

has a z-score close to 1.00

T/F

Answer

• Sign indicates that score is below the meanTrue

• Scores close to 0 have z-scores close to 0.00False

COMPUTING Z-SCORES FROM XSTANDARD SCORES FROM Z-SCORES

Equation for z-score

Numerator is a deviation score

Denominator expresses deviation in standard deviation units

XX

z

Determining raw score from z-score

Numerator is a deviation score

Denominator expresses deviation in standard deviation units

zXXX

z

X so

Example of converting a score

Learning Check

• For a population with μ = 50 and σ = 10, what is the X value corresponding to z=0.4?

• 50.4A• 10B• 54C• 10.4D

Learning Check - Answer

• For a population with μ = 50 and σ = 10, what is the X value corresponding to z=0.4?

• 50.4A• 10B• 54C• 10.4D

Learning Check

• Decide if each of the following statements is True or False.

• If μ = 40 and X = 50 corresponds to z=+2.00, then σ = 5 points

T/F• If σ = 20, a score above the

mean by 10 points will have z = 1.00

T/F

Answer

• If 2σ = 10 then σ = 5 True

• Why?False

z-Scores for Comparisons

All z-scores are comparable to each other Scores from different distributions can be

converted to z-scores The z-scores (standardized scores) allow

the comparison of scores from two different distributions along

5.3 Standardizing a Distribution

• Every X value can be transformed to a z-score

• Characteristics of z-score transformation– Same shape as original distribution– Mean of z-score distribution is always 0.– Standard deviation is always 1.00

• A z-score distribution is called a standardized distribution

Transformation of a Population of Scores

Axis Re-labeling

Shape of Distribution after z-Score Transformation

Creating a Standardized Distribution

Learning Check• Last week Andi had exams in Chemistry and in Spanish.

On the chemistry exam, the mean was µ = 30 with σ = 5, and Andi had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 6 and Andi had a score of X = 65. For which class should Andi expect the better grade?

• ChemistryA

• SpanishB

• There is not enough information to knowC

Learning Check - Answer• Last week Andi had exams in Chemistry and in Spanish.

On the chemistry exam, the mean was µ = 30 with σ = 5, and Andi had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 6 and Andi had a score of X = 65. For which class should Andi expect the better grade?

• ChemistryA

• SpanishB

• There is not enough information to knowC

Concepts

Equations

Interpretation

WHAT ARE YOUR

QUESTIONS?

z-Scores for Comparisons

All z-scores are comparable to each other Scores from different distributions can be

converted to z-scores The z-scores (standardized scores) allow

the comparison of scores from two different distributions along

Other Standardized Distributions

Process of standardization is widely used• SAT has Mean = 500 and σ = 100• IQ has Mean = 100 and σ = 15 Point

Standardizing a distribution has two steps• Original raw scores transformed to z-scores• The z-scores are transformed to new X values

so that the specific μ and σ are attained.

Creating a Standardized Distribution

This form of standardized score, withM = 50 and = 10, is known as a T-score.

Looking to Inferential Statistics

Interpretation of research results depends on determining if (treated) sample is noticeably different from the population

One technique for defining noticeably different uses z-scores.

Diagram of Research Study

Distributions of weights

Learning Check• Last week Andi had exams in Chemistry and in Spanish.

On the chemistry exam, the mean was µ = 30 with σ = 5, and Andi had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 6 and Andi had a score of X = 65. For which class should Andi expect the better grade?

• ChemistryA

• SpanishB

• There is not enough information to knowC

Learning Check - Answer• Last week Andi had exams in Chemistry and in Spanish.

On the chemistry exam, the mean was µ = 30 with σ = 5, and Andi had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 6 and Andi had a score of X = 65. For which class should Andi expect the better grade?

• ChemistryA• SpanishB• There is not enough

information to knowC

Learning Check TF

• Decide if each of the following statements is True or False.

• Transforming an entire distribution of scores into z-scores will not change the shape of the distribution.

T/F• If a sample of n = 10 scores is

transformed into z-scores, there will be five positive z-scores and five negative z-scores.

T/F

Concepts

Equations

Interpretation

WHAT ARE YOUR

QUESTIONS?

z-Scores: Location in Distributions