Measurement and Statistical Concepts

# Numbers and Measurement

**Measurement**: the process of assigning numerals or numbers to observationsWe enhance our ability to measure attributes by defining the basic size of the unit of measurement for a test -- the use of real numbers

Real numbers are also continuous because they represent any quantity along a number line

# Properties of Measurement in Relation to Numbers

The score from a psychological test is based on the properties of numbers and how we treat 0

The level of measurement of a particular score can be determined by the presence or absence of 4 properties

## Nominal

number of classes (classification), mode

**Qualitative**: sex or hair color, distinguishing labels or categories

## Ordinal

median, percentiles, order statistics

**Quantitative**: class rank, hardness of minerals, order finish in a competitive running race

## Interval

equality of intervals of scores along the score continuum

**Quantitative**: temperature (Celsius), standardized test scores

## Ratio

equality of ratio

**Quantitative**: temperature (Kelvin)

# Scaling

the mathematical techniques used for determining what numbers should be used to represent different amounts of a property or attribute being measured

# Statistical Foundations for Psychometrics

## Descriptive Statistical Techniques

used to organize and describe data

order and group scores into distributions and describe the scores

calculate a single number that summarizes a set of scores

represent scores graphically

applied to samples and populations

## Inferential Statistical Techniques

used to help make educated guesses about populations based on the samples

# Variables, Frequency Distributions, and Scores

measurements acquired on a variable/s are part of the data collection process -- refers to a property whereby members of a group differ from one another (i.e., measurements change from one person to another)

## Constant

refers to property whereby members of a group do not differ from one another (e.g., all persons in a study or taking an examination are female; thus, biological sex is constant)

variables can be qualitative or quantitative; quantitative variables can be discrete or continuous

### Discrete

values can only be whole numbers

### Continuous

can assume any value

## Frequency Distribution

a tabulation of the number of occurrences of each score value

# Shape, Central Tendency, and Variability of Score Distributions

The shape of a distribution is defined as symmetric, positively skewed, and negatively skewed; can also be unimodal or bimodal

## Central Tendency

the score value at the center (position at the center of the x-axis) that marks the center of the distribution of scores -- mean, median, and mode

Since a measure of central tendency is a single number, it is a concise way to provide an initial view of the set of scores.

Measures of central tendency can quickly and easily be compared.

Many inferential statistical techniques use a measure of central tendency to test hypotheses of various types.

Measures of variability for a set of scores or measurements provide a value of how spread or dispersed the individual scores are in a distribution -- variance and standard deviation

### Variance

the degree to which measurements or scores differ from the mean of the population or sample

### Standard Deviation

the square root of the variance

## Percentiles

used to provide an index of the relative standing for a person with a particular score relative to the other scores (persons) in a distribution

## Z-Score

A way to rescale or standardize the value of 0 so that it means the same thing in every distribution. We can then directly compare scores from different tests with different distributions.

## T-Score

a mean of 50 and a standard deviation of 10

## Stanine Cycle

All raw scores are converted to a single-digit system of scores ranging from 1 to 9. The mean of stanine scores is always 5 and the standard deviation is 2.

# Measurement and Statistical Concepts

# Numbers and Measurement

**Measurement**: the process of assigning numerals or numbers to observationsWe enhance our ability to measure attributes by defining the basic size of the unit of measurement for a test -- the use of real numbers

Real numbers are also continuous because they represent any quantity along a number line

# Properties of Measurement in Relation to Numbers

The score from a psychological test is based on the properties of numbers and how we treat 0

The level of measurement of a particular score can be determined by the presence or absence of 4 properties

## Nominal

number of classes (classification), mode

**Qualitative**: sex or hair color, distinguishing labels or categories

## Ordinal

median, percentiles, order statistics

**Quantitative**: class rank, hardness of minerals, order finish in a competitive running race

## Interval

equality of intervals of scores along the score continuum

**Quantitative**: temperature (Celsius), standardized test scores

## Ratio

equality of ratio

**Quantitative**: temperature (Kelvin)

# Scaling

the mathematical techniques used for determining what numbers should be used to represent different amounts of a property or attribute being measured

# Statistical Foundations for Psychometrics

## Descriptive Statistical Techniques

used to organize and describe data

order and group scores into distributions and describe the scores

calculate a single number that summarizes a set of scores

represent scores graphically

applied to samples and populations

## Inferential Statistical Techniques

used to help make educated guesses about populations based on the samples

# Variables, Frequency Distributions, and Scores

measurements acquired on a variable/s are part of the data collection process -- refers to a property whereby members of a group differ from one another (i.e., measurements change from one person to another)

## Constant

refers to property whereby members of a group do not differ from one another (e.g., all persons in a study or taking an examination are female; thus, biological sex is constant)

variables can be qualitative or quantitative; quantitative variables can be discrete or continuous

### Discrete

values can only be whole numbers

### Continuous

can assume any value

## Frequency Distribution

a tabulation of the number of occurrences of each score value

# Shape, Central Tendency, and Variability of Score Distributions

The shape of a distribution is defined as symmetric, positively skewed, and negatively skewed; can also be unimodal or bimodal

## Central Tendency

the score value at the center (position at the center of the x-axis) that marks the center of the distribution of scores -- mean, median, and mode

Since a measure of central tendency is a single number, it is a concise way to provide an initial view of the set of scores.

Measures of central tendency can quickly and easily be compared.

Many inferential statistical techniques use a measure of central tendency to test hypotheses of various types.

Measures of variability for a set of scores or measurements provide a value of how spread or dispersed the individual scores are in a distribution -- variance and standard deviation

### Variance

the degree to which measurements or scores differ from the mean of the population or sample

### Standard Deviation

the square root of the variance

## Percentiles

used to provide an index of the relative standing for a person with a particular score relative to the other scores (persons) in a distribution

## Z-Score

A way to rescale or standardize the value of 0 so that it means the same thing in every distribution. We can then directly compare scores from different tests with different distributions.

## T-Score

a mean of 50 and a standard deviation of 10

## Stanine Cycle

All raw scores are converted to a single-digit system of scores ranging from 1 to 9. The mean of stanine scores is always 5 and the standard deviation is 2.