Chapter 3 Statistical Concepts

Statistical Terms Statistics Descriptive Statistics Inferential Statistics Sample Population

Scales of Measurement Nominal Ordinal Interval Ratio

Nominal Scale The nominal scale of measurement describes qualitative (word) rather than quantitative (number) differences. Any item being measured can only fit into one category. Categories imply no level of ranking or quantitative value.

Ordinal Scale Example: 1st, 2nd, 3rd Ordinal scales are similar to nominal scales in that they classify discrete variables. Unlike nominal scales, ordinal scales rank variables in terms of magnitude. Because ordinal scales are not quantitative in nature, no assumption of equal intervals between values is made.

Interval and/or Ratio Scale Ranks variables by order of magnitude. Unlike ordinal scales, interval scales provide equally spaced levels between variables. Main difference between interval and ratio: Interval has no absolute zero Ratio has absolute zero

Examples of Scales of Measurement

Describing Test Scores Distributions Graphs Measures of Central Tendency Measures of Variability The Normal Curve Measures of Relationship

Distributions Distribution – a set of scores Frequency Distribution – a distribution ranked by the number of responses for each variable. Graphs – Distributions can often be represented on a graph.

Graphical Presentations Histogram

Frequency Polygon

Smoothed Frequency Polygon

Graphical Example of a Normal Curve

Descriptors of Curves Symmetry Symmetrical – one side of the curve mirrors the other Asymmetrical – skew exists in the curve

Skewness – the degree to which the distribution of a curve is asymmetrical. Positive Skew - a distribution with an asymmetrical “tail” extending out to the right. Negative Skew - a distribution with an asymmetrical “tail” extending out to the left.

Kurtosis – a statistic that reflects the peakedness or flatness of a distribution relative to a normal distribution.

Measures of Central Tendency Measures of Central Tendency describe distributions based on the average performance of a test score. Measures of Central Tendency are typically represented through the mean, median, and mode.

Mean Average value for the distribution of scores. Most common measure of central tendency Used with interval and ratio scales Calculated by totaling test scores and dividing the sum by the number of individuals who took the test.

Median The middle score that divides a distribution in half. Used with ordinal, interval, or ratio scales Helpful with highly skewed distributions. Calculated by determining the center score in a distribution. When there is an even number of scores, the two middle scores are averaged.

Mode The most common score or value that appears in a set of scores. Used with nominal, ordinal, interval, and ratio variables. Calculated by counting the score that appears most often in a distribution. It is possible to have a bimodal (2 most common scores) or multimodal (multiple most common scores) distribution.

Distribution Curves and Measures of Central Tendency In a symmetrical (normal) curve, the values for the mean, mode, and median are identical. The mean can be impacted by outlying scores. In asymmetrical distributions, the median may be the best measure of central tendency.

Measures of Variability Variability – the degree to which scores differ from one another. Measures of Variability – the degree to which scores differ from the mean. There are several methods for measuring variability.

Standard Deviation Most common measure of variability. Provides an average distance of test scores from the mean. Larger standard deviations indicate greater variance from the mean and greater variance between scores. Standard Deviation provides a sense of where an individual score stands in relation to the mean.

The Normal Curve The normal curve, often referred to as the normal distribution, represents the theoretical distribution of any set of scores. The normal curve is applied to many constructs in counseling, such as memory and intelligence. Some areas in counseling are not normally distributed, such as depression.

Properties of the Normal Curve It is bell shaped. It is bilaterally symmetrical, which means its two halves are identical. The mean, median and mode are equal to one another. The tails are asymptotic, meaning they approach but never touch the baseline. It is unimodal, which means that it has a single point of maximum frequency or maximum height. 100% of the scores fall between -3 and +3 standard deviations from the mean with approximately 68% of the scores falling between -1 and +1 standard deviations, approximately 95% of the scores falling between -2 and +2 standard deviations, and approximately 99.5% of the scores falling between -3 and +3 standard deviations.

Illustration of a Normal Curve

Graphical Example of a Normal Curve

Standard Scores and T Scores with Normal Curve

Measures of Relationships Between Variables

Measures of Relationship Another important statistical measure in testing involves the measurement of the relationship between two variables. For example, counselors may be interested in the relationship between depression and hours of sleep each evening. Often this is measured with a correlation coefficient

Correlation Coefficient Correlation Coefficients range from -1.00 to +1.00 and indicate the relationship between two variables. The direction of a correlation coefficient is either positive, indicating that when a score in one variable goes up that the score in the other variable will also go up, or negative, indicating that when a score on one variable goes up that a score in the other variable will go down.

Correlation Coefficient, Cont. The strength of a correlation is indicated by the numeric value of the coefficient. A 1.00 correlation, negative or positive, indicates a perfect relationship between two variables. A zero indicates no relationship between two variables. Correlation does not imply causation!