Chapter 5

Characteristics of Normal Curve

-Most test scores cluster or fall near the middle of the distributions, forming the average or central tendency. The farther right or left you move, the fewer the number of scores there are. -Most people will scores near the middle of the distributions, making the center of the distribution the highest point. -The curve can continue to infinity, and therefore the right and left tails will never touch the baseline.

Histogram

A bar graph used to represent frequency data in statistics.

Mini-Mental State Examination (MMSE)

A fairly short cognitive ability test often used to screen individuals for and estimate the severity of cognitive impairment. It is also used to measure cognitive changes over time - for example, to measure how an individual is responding to treatment.

Outliers

A few values that are significantly higher or lower than most of the values. When dealing with measures of central tendency having extreme outliers can alter the meaning of the mean and make the mode or median more preferable measures of central tendency because are they are more informative. (i.e. tooth fairy money)

Median

A measure of central tendency for a distribution, represented by the score that separates the upper half of the scores in a distribution from the lower half.

Norm Group

A previously tested group of individuals whose scores are used for comparison purposes.

Ratio Scales

A scale of measurement in which there is an absolute zero point, indicating an absence of the variable being measured. An implication is that ratios of numbers on the scale can be formed (generally, these are physical measures such as weight or timed measures such as duration or reaction time).

Class Intervals

A set of defined range limits in which the data are grouped. These categories should be of equal length and mutually exclusive and exhaustive, meaning that each data point should fit into only one class. Lyman suggests you should aim for approximatey 15 groups. Highest score - lowest score/15 + 1 if even so each interval will have midpoint.

Scale

A set of transformed scores that are often used to interpret a test. There are two types of transformations: -linear -area

Normal Curve

A symmetrical, bell-shaped curve that describes the distribution of many types of data; most scores fall near the mean and fewer and fewer near the extremes.

Linear Transformations - T Scores

Always have a mean of 50 and a standard deviation of 10. Formula = T = (z x 10) + 50 T scores are preferred because they are always positive and that makes it easier to understand because people generally think of raw scores as part of a 100 pt. scale.

Frequency Distributions

An orderly arrangement of a group of numbers (or test scores). They show the actual number (or percentage) of observations that fall into a range or category; they provide a summary and picture of group data. Two commonly used methods to portray FD's are: -histograms -tables

Mean

Arithmetic mean or average; Most sensitive to extreme scores.

Mean

Average score in a distribution or sample. M = Ex/N E - "sum of" x - raw test scores N - total number of test scores

Norms

Average scores of some identified group of individuals. These norms provide us with a standard against which we can compare individual test scores. Norms are created by administering a test to a large number of individuals who are carefully selected to be representative of the population that the test is intended to serve. Three popular norms: - percentile ranks - age norms - grade norms

Area Transformations

Change not only the unit of measurement but also the unit of reference. They rely on the normal curve. They magnify the differences between individuals at the middle of the distribution and compress the differences between individuals at the extremes of the distribution. Most common area transformations are: -percentile -stanine

Linear Transformations

Change the unit of measurement but do not change the characteristics of raw data in any way. Most popular linear transformations include: -percentages -z scores -T scores both z and T scores are based on the SD.

Age Norms and Grade Norms

Common types of norms because they allow us to determine at what age level or grade level an individual is performing. That is, they allow us to determine whether an individual's test score is similar to, below, or above the scores of others at the same age or grade level. Frequently used in educational settings.

Equal Interval Scales

Level of measurement in which numbers are assigned with the assumption that each number represents a point that is an equal distance from the points adjacent to it. Likert-type scales (not all professional's believe Likert to be interval but instead ordinal). Lacks a point that indicates absolute absence of the attribute being measured.

Measures of Variability

Like MOCT, MOV describe a set of scores in numerical form. However MOCT tell about the center of a distribution of scores, and MOV represent how spread out a group of scores is and provide more information about individual differences.Three commonly used measures of variability are the: -range -variance -standard deviation

Range

MOV The highest score in the distribution - the lowest score in a distributions = range.

Median

Middle score in a distribution. It is determined by putting all scores in a distribution in order and selecting the middle score. If number is even add two middle scores and divide by 2.

Levels of Measurement

Most measurement experts think in terms of four levels of measurement based on the mathematical operations that can be performed with the data at each level.

Measures of Relationship

Must have at least two sets of score to calculate measures of relationship. The "correlation coefficient" is a statistic that is typically used to describe the relationship between two or more distributions of scores. Using a correlation coefficient, you can related one set of scores to another to see whether the same individuals scored similarly on two different tests (for example, if they scored low on one test, did they also score low on another test?). Such a relationship is called a positive correlation. As opposed to a negative correlation which is when people who score high on one test are likely to score low on another test and vice verse.

Florida Comprehensive Assessment Test Norm-Referenced Test (FCAT)

Part of Florida's overall plan for increasing student achievement in primary and secondary schools by implementing higher standards. Consists of both "criterion-referenced tests" (tests that measure how well a student has learned a specific body of knowledge as defined by some criterion) & "norm-referenced test" (tests that compare a student's performance against how other students in a norm group did on the test). Grade 3-11 Reports scores using percentile ranks and stanines.

Procedures for Interpreting Test Scores

Raw Scores Frequency distributions Normal Curve Descriptive statistics -measures of central tendency -measures of variability -measures of relationship

Linear Transformations - Standard Deviation Units

Refer to how many standard deviations an individual score falls away from the mean.

Pearson Product-Moment Correlation Coefficient

Represented by r, this coefficient measures the linear association between two variables, or sets of test scores, that have been measured on interval or ratio scale. The correlation (r) between two distributions is expressed in terms of a coefficient that can range from -1.00 to +1.00, with -1.00 indicating a perfect negative (or inverse) correlation, 0.00 indicating a complete lack of a relationship, and +1.00 indicating a perfect positive correlation. Positive - increasing diamond karats, increasing price Negative - increasing age, decreasing eyesigh No correlation - height and academic ability

Ordinal (Statistics)

*Frequency *Mode *Median *Percentile Numbers rank or order people or objects based on the attribute being measured. Distances or values between numbers vary. Numbers only have meaning within the group.

Nominal (Statistics)

*Frequency *Mode Numbers represent labels or categories of data. Numbers have no quantitative data.

Equal interval (Statistics)

*Frequency *Mean *Mode *Median *Standard deviation *Correlation *T test *Analysis of Variance (ANOVA) Points on the scale are an equal distance apart. this scale does not contain an absolute zero point (number that indicates complete absence of the attribute).

Ratio (Statistics)

*Frequency *Mean *Mode *Median *Standard deviation *Correlation *T test *Analysis of Variance (ANOVA) *Proportions Points on this scale are an equal distance apart. There is an absolute zero point.

Raw Scores

Scores that have not been averaged, sorted, or processed yet. The basic score calculated when an individual completes a psychological test.

Area Transformations - Stanines

Stanines are expressed in whole numbers from 1 - 9. 1, 2, 3 - typically below mean 4, 5, 6 - typically average or close to mean 7, 8 - above average 9 - exceptional

Variance (s2)

Tells us whether individual scores tend to be similar or substantially different from the mean. In most cases, a large variance tells that individuals scores differ substantially from the mean, and a small variance tells us that individual scores are very similar to the mean. The formula for the variance requires squaring the sum of the deviations (differences) of each score from the mean. The variance calculation changes the unit of measurement, making it difficult to interpret the variance. Therefore, we often take the square root of the variance, which gives us the standard deviation.

Correlation Coefficient (MOR)

The "correlation coefficient" is a statistic that is typically used to describe the relationship between two or more distributions of scores. Using a correlation coefficient, you can relate one set of scores to another to see whether the same individuals scored similarly on two different tests (for example, if they scored low on one test, did they also score low on another test?). Such a relationship is called a positive correlation. As opposed to a negative correlation which is when people who score high on one test are likely to score low on another test and vice verse.

Nominal Scales

The most basic level of measurement. Numbers are assigned to represent groups or categories of information. Generally used for demographic data such as grouping people based on their gender, race, or place of residence. Categorical data - data is grouped according to a common property. Because nominal scales yield only categorical data there are few ways to describe or manipulate the data they yield.

Mode

The most common score in a distribution. Calculated by ordering the scores in a distribution and seeing which score occurs most often. There may be more than one mode or not mode at al.

Standard Deviation

The most commonly used measure of variability in a distribution of test scores.

Normal Probability Distributions

Theoretical distributions that exist in our imaginations as perfect and symmetrical and actually consist of a family of distributions that have the same general bell shape - high in the middle and tapered to the ends. *Also referred to as normal curves.

Measures of Central Tendency

Types of Descriptive Statistics. Three common MOCT are the: -mean (arithmetic average) -the median -the mode

Standard Scores

Universally understood units in testing that allow test users to evaluate a person's performance in reference to other persons who took the same or a similar test. Transformed scores are most often used with standardized tests of aptitude, achievement, and personality and are designed to help us compare compare one individual's scores with group norms. They also help us to compare individuals test scores on two different tests. When we transform raw test scores, we create a more informative scale.

Ordinal Scales

Variables are measured with numbers representing categories that can be placed in a meaningful numerical order but no information regarding the specific size of the interval between variables. (rank on 1-5 scale) Age equivalents, grade equivalents, and percentile scores all represent ordinal scales.

Percentile Ranks

Very common type of norm because it provides us with a way to rank individuals on a scale of 1 to 100%, making it relatively easy to interpret.

Linear Transformations - z Scores

Very similar to standard deviation units except they can represented in whole numbers and decimal points. As with SD units, the mean of a distribution of test scores will always have a z score of 0. A z score of 1 is always 1 standard deviation above the mean etc.

Positively Skewed Distributions

Distribution where most of the scores are at the lower end of the scale. Is when many of the data clusters towards the lower end (left) of the graph. A distribution in which the peak is to the left of the center point and the tail extends towards the right or in the positive direction.

Linear Transformations - Percentages

Divide raw score by total possible score and multiply by 100

Descriptive Statistics

Frequency distributions provide a visual image of a distributions of scores. While, descriptive statistics describe or summarize a distributions of test scores using numbers. Allows you to determine the main points of a group of scores. The descriptive statistics typically relied on in psychological testing include: -measures of central tendency -measures of variability -measures of relationship

Negatively Skewed Distributions

Have one high point and are skewed to the left. There are more high scores than low scores., Contains a preponderance of scores on the high end of the scale. The mean will be lower than the median in a negatively skewed distribution.

Peaked Distributions

Have one high point and result and when many individuals scores near the center of the distribution.

Bimodal Distributions

Have two high points and result when many people score low, many people score high, and few people score in the middle.

Why is it important to understand the different levels of measurement?

The level of measurement informs you about the statistical operations you can perform and what you can and cannot say about test scores.

Area Transformations - Percentiles

The mean of a normal distribution always has a percentile of 50. This means that 50% of individuals scored above the mean and 50% scored below the mean. To calculate an individuals percentile rank, you must first find the number of individuals who scored below the individuals score and the number who scored exactly the same score. Then take number who scored below a specific raw score, add .5 of those who scored exactly the same, and divide it by total number of people to take the test.

Norm-based Interpretation

The process of comparing an individuals test score to a norm group