English short essay

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Unit5_Statistics...pdf

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Statistics for Score Interpretation:

The Basic Mathematics of

Measurement

One does not need to be a statistical

wizard to grasp the basic mathematical

concepts needed to understand major

measurement issues.

Measurement

• Measurement is defined as a set of rules for

assigning numbers to represent objects,

traits, attributes, or behaviors.

Scales of Measurement

• Nominal Scales: qualitative system for categorizing

objects or people. Examples: Gender - Female =1, Male =

2; Eye Color - Brown =1, Blue =2, Green = 3.

• Ordinal Scales: allows you to rank people or objects

according to the quantity of a characteristic. Example:

Graduation Class Rank - 1 = Valedictorian, 2 =

Salutatorian, 3 = 3rd Rank, etc..

Scales of Measurement

• Interval Scales: allows ranking on a scale with equal

units. Examples: IQs, GRE scores

• Ratio Scales: properties of interval scales with a true zero

point. Examples: Height in inches, Weight in pounds

Why “Scale” matters

• There is a hierarchy among the scales with

nominal scales being the least sophisticated and

providing the least information and ratio scales

being the most sophisticated and providing the

most information.

• Interval and ratio level data allow the use of the

more powerful parametric statistical procedures.

Distributions

• Distribution: a set of scores.

• Raw Score Distributions

• Frequency Distributions

Ungrouped Frequency Distribution

Grouped Frequency Distribution

• Frequency Graphs

4 5 6 7 8 9 10

Homework Score

Shapes of Distributions

• Symmetric Distributions

• Normal Distribution (Bell-Shaped Curve)

Special symmetric distribution that is

unimodal with mode = median = mean

• Skewed Distributions Positive Skew

Negative Skew

Descriptive Statistics

• Measures of Central Tendency

* Mean

* Median

* Mode

Descriptive Statistics

• Measures of Variability

* Range

* Variance

Standard Deviation

Correlation Coefficients

• A correlation coefficient is a mathematical

measure of the relationship between two

variables.

• The correlation coefficient was developed

by Karl Pearson and is designated by the

letter r.

Correlation (r)

• Correlations range from -1.0 to +1.0

• Correlations differ on two parameters: size and

sign.

• Sign - can be positive or negative. Indicates the

pattern of the relationship.

• Size - a correlation of 0.0 indicates the absence of

a relationship; the closer the correlation gets to

1.0, the stronger the relationship; a 1.0 indicates a

perfect relationship.

Scatterplots

• Scatterplots: graph depicting the

relationship between two variables (X & Y).

Each mark in the scatterplot actually

represents two scores, an individual’s scores

on the X and the Y variable.

Major Types of Correlations

• Pearson Product-Moment Correlation: both

variables continuous and on an Interval or

Ratio scale.

• Spearman Rank-Difference Correlation:

both variables on an Ordinal scale.

Major Types of Correlations

• Point-Biserial Correlation: one variable

continuous and on Interval/Ratio scale, the

other a genuine dichotomy (e.g., true/false).

• Biserial Correlation: both variables

continuous and on Interval/Ratio scale, but

one is reduced to two categories (i.e.,

dichotomized).

Factors that Effect Correlations

• Most correlations assume a linear

relationship (falling on a straight line). If

another type of relationship exists,

traditional correlations may underestimate

the correlation.

• If there is a restriction of range in either

variable, the magnitude of the correlation

will be reduced.

Qualitative Interpretation of

Correlations

• General Guidelines:

• < 0.30 Weak

• 0.30 - 0.70 Moderate

• > 0.70 Strong

• These are not universally accepted and you

might see other guidelines.

Statistical Significance of

Correlations

• Statistical significance is determined both

by the size of the correlation coefficient and

the size of the sample.

• This and related topics are covered in most

introductory statistics texts and courses.

Quantitative Interpretation of

Correlations

• Coefficient of Determination (r2): the

proportion of variance on one variable that

is determined or predictable from the other

variable.

• Coefficient of Nondetermination (1-r2): the

proportion of variance in one variable that is

not determined or predictable from the other

variable.

Correlation & Prediction

• When variables are correlated, particularly

when there is a strong correlation,

knowledge about performance on one

variable provides information that can help

predict performance on the other variable.

Linear Regression

• A statistical technique for predicting scores

on one variable (criterion or Y) given a

score on another (predictor or X).

• Predicts criterion scores based on a perfect

linear relationship.

• Strong correlations result in accurate

predictions; weak correlations result in less

accurate predictions.

Correlation & Causality

• It is a common misconception that if two

variables are correlated one is causing the

other.

• This is not the case!